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Publications mathématiques de BesançonA l g è b r e e t t h é o r i e d e s n o m b r e s

2010

Publ

icat

ions

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ues

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10M. J. Bertin Mesure de Mahler et série L d’une surface K 3 singulière

F. Brunault Régulateurs p-adiques explicites pour le K2 des courbes elliptiques

H. Cohen Some formulas of Ramanujan involving Bessel functions

C. Delaunay et C. Wuthrich Some remarks on self-points on elliptic curves

G. Gras Analysis of the classical cyclotomic approach to Fermat’s last Theorem

F. Hajir Asymptotically good families

D. Solomon Equivariant L-functions at s = 0 and s = 1

L. Thomas On the Galois module structure of extensions of local fields

Revue du Laboratoire de mathématiques de Besançon (CNRS UMR 6623)

P r e s s e s u n i v e r s i t a i r e s d e F r a n c h e - C o m t é

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Éva Bayer, École Polytechnique Fédérale de Lausanne (Suisse) ____________________________ [email protected]

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Cédric Bonnafé, Université de Franche-Comté ___________________________________ [email protected]

Jean-Marc Couveignes, Université Toulouse 2-Le Mirail ____________ [email protected]

Vincent Fleckinger, Université de Franche-Comté _____________________________ [email protected]

Farshid Hajir, University of Massachusetts, Amherst (USA) ____________________________ [email protected]

Jean-François Jaulent, Université Bordeaux 1 _________________________ [email protected]

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Publications mathématiques de BesançonA l g è b r e e t t h é o r i e d e s n o m b r e s - F o n d A t e u r : g e o r g e s g r A s


2010

© Presses universitaires de Franche-Comté, Université de Franche-Comté, 2010

Directeur de la revue : Patrick Hild, directeur du laboratoire



Laboratoire de mathématiques de Besançon (CNRS UMR 6623)




Actes de lA conférence “fonctions l et Arithmétique”besAnçon, 8-12 juin 2009

2010

Sommaire

M. J. Bertin

Mesure de Mahler et série L d’une surface K3 singulière . . . . . . . . . . . . . . . . . . . . . 5-28

F. Brunault

Régulateurs p-adiques explicites pour le K2 des courbes elliptiques . . . . . . . . . . . 29-57

H. Cohen

Some formulas of Ramanujan involving Bessel functions . . . . . . . . . . . . . . . . . . . . . . 59-68

C. Delaunay et C. Wuthrich

Some remarks on self-points on elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69-84

G. Gras

Analysis of the classical cyclotomic approach to Fermat’s last Theorem . . . . . . . 85-119

F. Hajir

Asymptotically good families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121-128

D. Solomon

Equivariant L-functions at s = 0 and s = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129-156

L. Thomas

On the Galois module structure of extensions of local fields . . . . . . . . . . . . . . . . . . 157-194

MESURE DE MAHLER ET SÉRIE L D’UNE SURFACE K3

SINGULIÈRE

par

Marie José Bertin

Résumé. — Nous présentons un exemple de polynôme définissant une surface K3 singulièredont la mesure de Mahler s’exprime comme somme de la série L de la surface et d’un termeproportionnel à la mesure des faces.

Abstract. — We give an example of a polynomial in three variables defining a singular K3-surface, whose logarithmic Mahler measure is expressed as a linear combination of the L-seriesof the surface and a Dirichlet L-series associated to the faces.

1. Introduction

La mesure de Mahler logarithmique m(P ) d’un polynôme de Laurent P ∈ C[X±1

1, ...,X±1

n ] a

été introduite par Mahler en 1962 pour mesurer la taille de certains facteurs d’un polynôme.

Elle s’exprime à l’aide d’une intégrale

m(P ) :=1

(2πi)n

∫

Tn

log |P (x1, ..., xn)|dx1

x1

...dxn

xn

où Tn désigne le n-tore (x1, ..., xn) ∈ Cn/|x1| = ... = |xn| = 1.La mesure de Mahler M(P ) est alors

M(P ) := exp(m(P )).

Pour n = 1 et P ∈ Z[X], unitaire, on déduit de la formule de Jensen

M(P ) =∏

P (α)=0

max(|α|, 1).

Classification mathématique par sujets (2000). — 11R06,14D,14J27,14J28.Mots clefs. — Modular Mahler’s Measure, Eisenstein-Kronecker Series, L-series of K3 hypersurfaces.

Je remercie chaleureusement J. Lewis et N. Yui pour leur invitation au CMS Winter meeting à Toronto(Décembre 2006). Cette Conférence, qui m’a donné l’occasion de rencontrer des experts du domaine, a été lepoint de départ de ce travail. Je suis particulièrement reconnaissante à M. Schütt pour ses conseils et pourtoutes nos discussions. Merci également à D. Boyd, O. Lecacheux et D. Zagier pour leur aide variée et trèsappréciée.

6 Mesure de Mahler et série L d’une surface K3 singulière

On remarque dans ce cas que M(P ) n’est autre que la quantité Ω(P ) définie par Lehmer

dans un article de 1933 [17] où il pose la célèbre question : existe-t-il un polynôme de Z[X],

unitaire, non produit de polynômes cyclotomiques vérifiant

1 < M(P ) < M(P0) = 1, 1762 · · · ,

où

P0(X) = X10 + X9 − X7 − X6 − X5 − X4 − X3 + X + 1

est le polynôme de Lehmer.

La question de Lehmer est toujours ouverte. Seule une réponse partielle a été donnée par

Smyth en 1971 [35], M(P ) > 1, 32 · · · mais sous l’hypothèse très forte P non réciproque. Cette

question a suscité de nombreuses recherches. Je veux seulement citer un résultat remarquable

de Boyd [8]

limn→∞

M(P (x, xn)) = M(P (x, y))

lorsque la limite porte sur une infinité de termes. Ce résultat laissait espérer une réponse à

la question de Lehmer si l’on trouvait des polynômes de 2 variables de petite mesure. Aussi

Boyd [9] évalua-t-il numériquement

m((x + 1)y2 + (x2 + x + 1)y + x(x + 1)) = 0, 2274 · · ·

m(y2 + (x2 + x + 1)y + x2) = 0, 2513 · · · .

(m(P0 = ln(1, 1762 · · · ) ≃ 0, 162288 · · · )Smyth prouvait également [9]

m(x + y − 1) =3√

3

4πL(χ−3, 2) = L′(χ−3,−1),

L(χ−3, s) =∑

n≥1

χ−3(n)

n2

désignant la série de Dirichlet de caractère le symbole de Legendre χ−3(n) =(−3

n

)

et

m(x + y + z + 1) =7

2π2ζ(3).

C’était le début des mesures de Mahler explicites.

Au même moment Bloch (1981) [7] comparait la valeur L(E, 2), de la série L en s = 2 d’une

courbe elliptique E, au second groupe de K-théorie K2(E) relié à des séries d’Eisenstein-

Kronecker. Il cherchait des relations à coefficients rationnels entre L(E, 2) et des séries

d’Eisenstein-Kronecker prises sur des points de torsion.

Quelques années plus tard, en 1996, Deninger [14] conjecturait

m(x +1

x+ y +

1

y+ 1)

?=

15

4π2L(E, 2) = L′(E, 0)

où E désigne la courbe elliptique de conducteur 15 définie par le polynôme ; cependant Cas-

saigne et Maillot (1997) [19] prouvaient la relation

Publications mathématiques de Besançon - 2010

Marie José Bertin 7

m(a + bx + cy) =

1

π (D( |a||b| eiγ) + α log | a | +β log | b | +γ log | c |) si ∆

maxlog | a |, log | b |, log | c | si non ∆.

La condition ∆ signifie que | a |, | b |, | c | sont les côtés d’un triangle d’angles α, β, γ et D

désigne le dilogarithme de Bloch-Wigner.

En fait, le point commun de tous ces résultats est l’expression de la mesure de Mahler d’un

polynôme de deux variables en termes de dilogarithmes, dilogarithmes de Bloch et Wigner

pour les courbes de genre 0, dilogarithmes elliptiques pour les courbes E de genre 1 définies

par un polynôme vérifiant certaines conditions permettant de relier leur mesure de Mahler au

K2(E). Donc ces résultats sont en relation avec les groupes de Bloch de corps de nombres en

genre 0, de courbes elliptiques en genre 1 [2] [5].

Les travaux de Boyd fournissent de nombreux résultats expérimentaux de formules explicites

de mesure de Mahler en deux variables. Peu de formules sont démontrées à l’heure actuelle.

Citons cependant les travaux de Rodriguez-Villegas [26] [27], Lalin [18], Rogers [18], Brunault

[13], Mellit [21].

Pour préciser encore le contexte de notre travail, je parlerai des conjectures de Rodriguez-

Villegas (2004) [11] justifiées par le point de vue de Maillot (2003) [20]

m(1 + x + y + z + t)?= cL(f, 3)

avec f forme parabolique de poids 3 de conducteur 15, la constante c dépendant du conducteur.

La série L(f, 3) est également la série L de la surface K3 définie par

1 + x + y + z + t = 0

1 + 1

x + 1

y + 1

z + 1

t = 0.

De même

m(1 + x + y + z + t + w)?= c1L(g, 4)

avec g forme parabolique de poids 4, de conducteur 6 liée à la série L de la quintique de

Barth-Nieto.

L’idée de Maillot [20] utilise un résultat de Darboux (1875) : la mesure de Mahler d’un

polynôme P non réciproque, intégrale d’une forme différentielle sur une variété est en fait

une intégrale sur une variété plus petite, intersection de la variété définie par P et par son

polynôme réciproque P ∗. La forme de l’expression de la mesure

de Mahler est contenue dans la cohomologie de la plus petite variété.

Dans la première formule de Smyth en deux variables, la “petite variété” est une courbe de

genre 0 ; par suite la mesure de Mahler s’exprime à l’aide d’une série L de Dirichlet. Dans

la deuxième formule de Smyth en trois variables, la “petite variété” est l’intersection de trois

plans d’où l’expression en ζ(3). Dans la première conjecture de Rodriguez-Villegas, la “petite

variété” est la surface K3 modulaire étudiée par Peters, Top, Van der Vlugt définie par un

polynôme réciproque [25]. Sa série L est la série L de la forme modulaire f . Dans la dernière

conjecture on trouve un lien avec la quintique de Barth-Nieto.



Ces quelques exemples et conjectures soulignent l’importance de connaître les mesures de

Mahler explicites de polynômes réciproques. C’est pour cela que j’ai orienté mes recherches

vers la détermination des mesures de Mahler explicites des polynômes

Pk = x +1

x+ y +

1

y+ z +

1

z− k, k ∈ Z.

Ces polynômes définissent des surfaces K3 notées Yk dont le nombre de Picard générique est 19

[24]. Dans des articles précédents [4], nous avons exprimé m(Pk) comme une somme de séries

d’Eisenstein-Kronecker et relié m(P2) et m(P3) aux séries L des variétés correspondantes Y2

et Y3. Nous allons traiter ici le cas k = 10.

Les polynômes P2, P3 et P10 appartiennent au sous-ensemble des polynômes Pk définissant une

surface K3 singulière, c’est-à-dire de nombre de Picard 20. Ce sous-ensemble a été déterminé

expérimentalement par Boyd et confirmé par une preuve de Schütt [12], [29].

Comme on l’explique dans [3], la dérivée de la mesure de Mahler par rapport au paramètre

est une période de la surface K3 ; ainsi seul le réseau transcendant intervient pour la mesure

de Mahler. Lorsque le nombre de Picard de la variété est 20, le réseau transcendant est de

dimension 2 et la série L de la variété correspond à un morceau de dimension 2 du H2 de

dimension 22 de la surface K3, qui est précisément le réseau transcendant.

Après avoir rappelé brièvement les définitions et résultats nécessaires, nous prouverons le

théorème suivant.

Théorème 1.1. — Soit Y10 l’hypersurface K3 associée au polynôme P10

P10 = x2yz + xy2z + xyz2 + t2(xy + xz + yz) − 10xyzt.

Si L(Y10, s) désigne la série L de l’ hypersurface Y10, on a les relations suivantes.

1)

L(Y10, 3) =1

2

′∑

k,m

k2 − 2m2

(k2 + 2m2)3= L(f, 3),

où f est la CM -newform de poids 3 et niveau 8 donnée dans les tables de Schütt [[28], Table

1],

f = q − 2q2 − 2q3 + 4q4 + 4q6 − 8q8 − 5q9 + 14q11 − 8q12 + 16q16 + 2q17 + 10q18 − 34q19 + · · · .

2) Définissons comme dans [4]

d3 :=3√

3

4πL(χ−3, 2) =

2√

3

π3

′∑

m,k

1

(m2 + 3k2)2.

La mesure de Mahler et la série L satisfont l’égalité

m(P10) = 2d3 +1

9

|det TY10|3/2

π3L(Y10, 3),

où TY10désigne le réseau transcendant de la surface Y10.



Remarque 1.2. —

1. Boyd avait remarqué que les deux quotients de période τ2 et τ10 (voir section 2) ap-

partenaient au même corps quadratique imaginaire Q(√−2). Aussi avait-il deviné une

relation de la forme

m(P10)?= 2d3 + 3m(P2).

Cette relation est actuellement prouvée grâce aux résultats de [4] et du présent papier.

2. Les séries L des deux surfaces K3 à savoir Y2 et Y10 sont les mêmes. Aussi, d’après la

conjecture de Tate, on devrait avoir une correspondance algébrique entre Y2 et Y10.

2. Rappels

Pour les principales définitions sur les surfaces K3 utiles dans ce travail, on pourra consulter

[3]. Pour une présentation plus approfondie on pourra lire [37].

2.1. Mesure de Mahler des polynômes Pk. — Rappelons le théorème principal.

Théorème 2.1. — [3] Soit k = t + 1

t et

t = (η(τ)η(6τ)

η(2τ)η(3τ))6, η(τ) = e

πiτ12

∏

n≥1

(1 − e2πinτ ), q = exp 2πiτ .

Alors

m(Pk) =ℑτ

8π3

′∑

m,κ

(−4(2ℜ 1

(mτ + κ)3(mτ + κ)+

1

(mτ + κ)2(mτ + κ)2)

+ 16(2ℜ 1

(2mτ + κ)3(2mτ + κ)+

1

(2mτ + κ)2(2mτ + κ)2)

− 36(2ℜ 1

(3mτ + κ)3(3mτ + κ)+

1

(3mτ + κ)2(3mτ + κ)2)

+ 144(2ℜ 1

(6mτ + κ)3(6mτ + κ)+

1

(6mτ + κ)2(6mτ + κ)2)).

2.2. La série L d’une surface K3. — Soit V une variété projective lisse de dimension d

sur un corps fini à q = pn éléments. On suppose en outre V géométriquement irréductible,

c’est-à -dire irréductible sur toute clôture algébrique de Fq. Soit Nn le nombre de points de

V dans Fq.

La fonction zéta attachée à V est définie par

ZV (T ) := exp

∑

n≥1

NnT n

n

.

D’après les conjectures de Weil (1949) prouvées par Dwork (1960) et Deligne (consulter par

exemple [15]), ZV (T ) est une fonction rationnelle à coefficients dans Q satisfaisant l’équation



fonctionnelle

ZV

(

1

qdT

)

= ±(qd/2T )cZV (T )

pour un certain c ∈ N.

Si V est une surface K3 algébrique définie sur Q, alors pour presque tout p, sa réduction

modulo p est encore une surface K3 notée Vp. De plus ZVp(T ) est de la forme

ZVp(T ) =1

(1 − T )(1 − p2T )P2(T )

avec P2(T ) polynôme de degré 22 satisfaisant P2(0) = 1 et

P2(T ) = Qp(T )Rp(T ),

le polynôme Rp(T ) provenant des cycles algébriques, Rp(Tp ) ∈ Z[T ], et Qp(T ) provenant des

cycles transcendants. Sous certaines conditions, la partie H1,1 de la décomposition de Hodge

du H1 a une dimension 20 dans H2(V, Z). Dans ce cas, la surface K3 est singulière, autrement

dit son nombre de Picard ρ vaut 20 et son réseau transcendant est de rang 2.

Plus précisément, si Fp désigne l’opérateur de Frobénius, on a

P2(T ) = det(

1 − TFp | H2

et(V, Ql))

.

Comme

H2

et(V, Ql) = H2

alg(V, Ql) + H2

tr(V, Ql),

si l’on note

Np = #Vp(Fp)

on a la formule

Np = 1 + p2 + TrH2

alg(V, Ql) + TrH2

tr(V, Ql).

La quantité TrH2

alg(V, Ql) correspond aux 20 cycles algébriques engendrant le groupe de

Néron-Severi de V . Son expression dépend du nombre de cycles algébriques définis sur Fp

ou Fp2 .

La quantité TrH2

tr(V, Ql) correspond aux cycles transcendants.

Par exemple, si les 20 générateurs du groupe de Néron-Severi de la surface K3 singulière sont

définis sur Fp (c’est le cas pour Y2 par exemple [4]) alors

P2(T ) = (1 − pT )20(1 − βT )(1 − β′T )

et

Np = 1 + p2 + 20p + β + β′.

La philosophie de Langlands dit que l’on peut définir Qp(T ) pour p divisant le déterminant

N du réseau transcendant, de telle sorte que

Z(V, s) :=∏

p

1

Qp(p−s)=

∑

n≥1

an

ns

soit la série de Dirichlet d’une forme modulaire parabolique.



Par exemple [34], si l’équation affine de la surface est

t2(x + y)(x + z)(y + z) + xyz = 0,

la forme parabolique de poids 3 et de niveau N = 8 est

f(z) = η(z)2η(2z)η(4z)η(8z)2 ∈ S3(Γ0(8), ǫ8)

où ǫ8 est le caractère associé à Q(√−2).

2.3. Deux résultats de Shioda. — Rappelons l’essentiel concernant les surfaces ellip-

tiques. On pourra consulter l’article de Shioda [31] mais aussi [32], [33].

1) Soit Φ : X → P1 une surface elliptique avec une section. Considérons les sections de cette

fibration elliptique, c’est-à-dire déterminées par les points rationnels de la courbe elliptique

correspondante définie sur le corps des fonctions rationnelles en s. On note r(Φ) le rang du

groupe des sections. Alors le nombre de Picard ρ(X) satisfait l’équation

(1) ρ(X) = r(Φ) + 2 +

h∑

ν=1

(mν − 1)

où h est le nombre de fibres singulières et mν le nombre de composantes irréductibles de la

fibre singulière correspondante.

2) Soit (X,Φ, P1) une surface elliptique avec une section Φ, sans courbe exceptionnelle de

première espèce ( i.e. de self-intersection −1). Notons NS(X) le groupe de Néron-Severi des

classes d’équivalence algébrique des diviseurs de X.

Soit s le point générique de P1 et Φ−1(s) = E la courbe elliptique définie sur K = C(s)

avec le point rationnel o = o(s). Alors, E(K) est un groupe abélien de type fini si j(E) est

transcendant sur C.

Soit r le rang de E(K) et s1, ..., sr les générateurs de E(K) modulo la torsion. En outre, le

groupe E(K)tors est engendré par au plus deux éléments t1 d’ordre e1 et t2 d’ordre e2 tels

que 1 ≤ e2, e2|e1 et | E(K)tors |= e1e2. Le groupe E(K) des points K-rationnels de E est

canoniquement identifié avec le groupe des sections de X sur P1(C).

Pour s ∈ E(K), on note (s) la courbe image dans X de la section correspondant à s.

Définissons

Dα := (sα) − (o) 1 ≤ α ≤ r

D′β := (tβ) − (o) β = 1, 2.

Considérons maintenant les fibres singulières de X sur P1. On pose

Σ := v ∈ P1/Cv = Φ−1(v) soit une fibre singulièreet pour tout v ∈ Σ, Θv,i, 0 ≤ i ≤ mv − 1, les mv composantes irreducibles de Cv.

Soit Θv,0 l’unique composante de Cv passant par o(v).

On peut donc écrire

Cv = Θv,0 +∑

i≥1

µv,iΘv,i, µv,i ≥ 1.



Soit Av la matrice d’ordre mv − 1 dont l’élément d’indice (i, j) est (Θv,i · Θv,j), i, j ≥ 1, où

(D ·D′) est le nombre d’intersection des diviseurs D et D′ sur X. Finalement f désignera une

fibre non singulière, i.e. f = Cu0pour u0 /∈ Σ.

Théorème 2.2. — Le groupe de Néron-Severi NS(X) de la surface elliptique X est engendré

par les diviseurs suivants

f,Θv,i (1 ≤ i ≤ mv − 1, v ∈ Σ)

(o),Dα 1 ≤ α ≤ r, D′β β = 1, 2.

Les seules relations entre ces diviseurs sont au plus deux relations

eβD′β ≈ eβ(D′

β · (o))f +∑

v∈Σ

(Θv,1, ...,Θv,mv−1)eβA−1

v

(D′β · Θv,1)

.

.

.

(D′β · Θv,mv−1)

où ≈ signifie l’équivalence algébrique.

3. Preuve du théorème

Une preuve de ce théorème a déjà été donnée par Bertin [6].

La preuve donnée ici permet de faire l’économie du calcul de la matrice de Gram des géné-

rateurs du groupe de Néron-Severi dont la valeur absolue du déterminant est le discriminant

N10 de la surface K3. Ce discriminant est essentiel pour le calcul de la série L de Y10. Nous

aurons également besoin d’un lemme basé sur le modèle de Néron de courbes elliptiques sur

Q(s) pour déterminer si les composantes des fibres singulières sont définies sur Fp ou Fp2 .

Dans cette nouvelle preuve nous compterons de manière différente les points de Y10(Fp). Ceci

sera explicité dans le lemme 2.

La preuve repose sur trois ingrédients :

– le théorème de modularité de Livné,

– la classification de Schütt des formes modulaires CM de poids 3 à coefficients rationnels,

– le critère de Serre-Livné permettant de comparer les séries L associées à deux représentations

l-adiques rationnelles.

Nous allons donc rappeler ces trois théorèmes.

Théorème 3.1. — (Théorème de modularité de Livné [15])

Soit S une surface K3 définie sur Q, de nombre de Picard 20 de discriminant N . Son réseau

transcendant T (S) est un Gal(Q/Q)-module de dimension 2 définissant une série L, notée

L(T (S), s).

Alors il existe une forme modulaire de poids 3 à multiplication complexe sur Q(√−N) satis-

faisant

L(T (S), s)·= L(f, s)



(·= signifie à un facteur rationnel près).

En outre, si NS(S) est engendré par des diviseurs définis sur Q

L(S, s)·= ζ(s − 1)20L(f, s).

Théorème 3.2. — (Classification de Schütt des surfaces K3 [28])

Considérons les classifications suivantes des surfaces K3 singulières définies sur Q

1. par le discriminant d du réseau transcendant de la surface à facteurs carrés près,

2. par la newform associée à twist près,

3. par le niveau de la newform associée à facteurs carrés près,

4. par le corps CM , Q(√−d), de la newform associée.

Alors toutes ces classifications sont équivalentes. En particulier, Q(√−d) a pour exposant 1

ou 2.

Nous rappellerons plus tard le critère de Serre-Livné.

3.1. Détermination de N10 à facteur carré près. — Notons N10 le discriminant de

la surface Y10 c’est-à-dire le déterminant de son réseau transcendant et Yλ une surface K3

de notre famille. Nous allons utiliser des arguments développés dans Verrill [36] et Peters-

Stienstra [24]. Le comportement de Yλ, quand λ ∈ C varie, est donné par sa structure de

Hodge.

Rappelons que si Yλ est une surface K3, il existe à scalaire près, une unique 2-forme ωλ ∈H2,0(Yλ, C). Si γ1,λ, · · · , γ22,λ est une base de H2(Yλ, Z) alors γ∗

1,λ, · · · , γ∗22,λ est la base

duale correspondante dans H2(Yλ, C), duale pour le produit d’intersection. Autrement dit, si

γ ∈ H2(Yλ, Z), on définit γ∗ ∈ H2(Yλ, Z) tel que pour tout α ∈ H2(Yλ, C))∫

αγ∗ = (γ · α)

où (γ · α) est le nombre d’intersection des deux cycles α et γ. L’intersection des cycles d’ho-

mologie vérifie la dualité de Poincaré pour le produit extérieur en cohomologie i.e. ∀α, γ ∈H2(Yλ, Z), on a

(γ · α) =

∫

Yλ

α∗ ∧ γ∗.

Donc la matrice d’intersection de la base duale est aussi la matrice d’intersection du réseau des

cycles transcendants T (Yλ) = Pic(Yλ)⊥. Pour notre famille, la matrice du réseau transcendant

T dans la base γ1, γ2, γ3 est donnée par Peters-Stienstra [24]

T =

0 0 1

0 12 0

1 0 0

.

Comme l’intégrale de ωλ sur tout cycle algébrique est nulle, si l’on identifie H2(Yλ, Z) ⊗ C et

H2(Yλ, C) alors ωλ ∈ T (Yλ) ⊗ C.



Donc ωλ ∈ H2,0 s’écrit

ωλ(τ) = a(τ)γ∗1 + b(τ)γ∗

2 + c(τ)γ∗3

avec

12b(τ)2 + 2a(τ)c(τ) = 0,

car ωτ ∧ ωτ ∈ H(4,0) = (0).

On en déduit que l’élément

ω(τ) = G(τ)γ1 + τG(τ)γ2 − 6τ2G(τ)γ3

appartient à T (Yλ(τ)). Ici G(τ), τG(τ), τ2G(τ) satisfaitl’équation différentielle de Picard-

Fuchs des périodes de la famille.

Par suite, dire que Xλ est de nombre de Picard 20 est équivalent à l’existence d’un vecteur

pγ1 + qγ2 + rγ3 ∈ Tλ devenant algébrique, donc vérifiant∫

pγ1+qγ2+rγ3ω(τ) = 0, soit

−6pτ2 + 12qτ + r = 0.

Donc τ vérifie une équation quadratique sur Q dont le coefficient du terme de plus haut degré

est un diviseur de 6 si l’on normalise le vecteur avec p = 1.

Pour la surface Y10 on a 2τ2 + 1 = 0. On en déduit que le vecteur γ1 − 3γ3 ∈ Tλ devient

algébrique dans Y10.

Choisissons alors une nouvelle base orthogonale de T10,

γ′1, γ′

2, γ′

3 = γ1 − 3γ3, γ2, γ1 + 3γ3.

Dans cette base la matrice de Gram de T10 vaut

−6 0 0

0 12 0

0 0 6

.

Ceci nous donne un sous-réseau de T10 de matrice de Gram

(

12 0

0 6

)

de déterminant 72.

On en déduit qu’à un facteur carré près, N10 = 72. Or d’après le théorème de Livné la surface

Y10 est modulaire. D’après le théorème et le tableau de Schütt, le niveau de la CM -newform

est 8. Par suite N10 = 72 ou 8.

Remarque 3.3. — Si l’on veut prouver que N10 = 72, on peut par exemple prendre la

méthode donnée dans Bertin [6] ou bien utiliser un autre argument comme dans 3.3.7.



3.2. Fibres singulières. — La surface K3, Y10, est un revêtement double de la surface

elliptique rationnelle de Beauville [1] définie par

(x1 + y1)(x1 + z1)(y1 + z1) + ux1y1z1 = 0.

Rappelons la nature des fibres singulières de cette dernière :

à u = ∞ de type I6

à u = 0 de type I3

à u = 1 de type I2

à u = −8 de type I1.

En coupant Y10 par l’hyperplan t = s(x + y + z), on obtient, après simplification, la fibration

elliptique

s2(x + y)(x + z)(y + z) + (s2 − 10s + 1)xyz = 0.

Posant alors u = (s2−10s+1)/s2, on déduit de ce qui précède, la nature des fibres singulières

de cette fibration elliptique de la surface Y10 :

à s = 0 de type I12

à s = ∞ de type I2

à s = 1/10 de type I2

à s = α (α2 − 10α + 1 = 0) de type I3

à s = β (β2 − 10β + 1 = 0) de type I3

à s = 1 de type I1

à s = 1/9 de type I1.

Puisque ρ = 20 ([24]), il résulte de la formule de Shioda que r = 1 ; pour décrire le groupe de

Néron -Severi, nous devons chercher une section infinie sur la surface.

La surface Y10 possède 7 points doubles :

P01 = (1 : 0 : 0 : 0) P02 = (0 : 1 : 0 : 0) P03 = (0 : 0 : 1 : 0) P04 = (0 : 0 : 0 : 1)

P12 = (1 : −1 : 0 : 0) P13 = (1 : 0 : −1 : 0) P23 = (0 : 1 : −1 : 0).

Les points doubles P12, P13, P23 sont dans toutes les fibres singulières. Les points doubles P01,

P02, P03 sont seulement dans la fibre singulière au-dessus de 0.

Les 4 droites P01P03P13, P02P03P23, P01P02P12, P12P13P23 d’équations respectives

y = 0 t = 0 x = 0 t = 0 z = 0 t = 0 x + y + z = 0 t = 0

passant chacune par trois points doubles et les droites P03P04, P01P04, P02P04 d’équations

respectives

x = 0 y = 0 y = 0 z = 0 x = 0 z = 0

passant chacune par deux points doubles, sont sur la surface.

On va compléter les générateurs de NS(Y10) avec la section infinie (Ps).



Pour cela, nous utilisons le modèle de Weierstrass W0 au voisinage de 0,

(W0) Y 2

0+ (s2 − 10s + 1)X0Y0 = X0(X0 − s4)(X0 + s2 − 10s3)

déduit du modèle W∞ au voisinage de l’infini,

(W∞) Y 2

∞ + (σ2 − 10σ + 1)X∞Y∞ = X∞(X∞ − 1)(X∞ + σ2 − 10σ).

Le changement de variable

x =−Y∞ − (σ2 − 10σ + 1)X∞

X∞ + σ2 − 10σ

y =Y∞

X∞ + σ2 − 10σ

fait passer du modèle

(x + y)(x + 1)(y + 1) + (σ2 − 10σ + 1)xy = 0

au modèle de Weierstrass W∞.

Pour obtenir W0 à partir de W∞, il suffit de faire les changements

σ 7→ 1

s, X∞ 7→ X0

s4, Y∞ 7→ Y0

s6.

La section infinie a été trouvée par Lecacheux [16]. Nous donnons ses coordonnées dans le

modèle W∞

X∞ =−1

432(σ − 5)2(σ − 2)2(σ − 8)2

Y∞ =1

15552

√−3(σ3 + σ2(−15 + 4

√−3) + σ(42 − 40

√−3) + 40 + 4

√−3)

(σ2 + σ(−10 +√−3) + 13 − 5

√−3)(σ − 5 +

√−3)(σ − 5)(σ − 2)(σ − 8).

3.3. Corps de définition des fibres singulières. — Pour savoir si les composantes des

fibres singulières sont définies sur Q, nous devons connaître le début du modèle de Néron au

voisinage de chaque s définissant une fibre singulière [22] ou, ce qui est équivalent le début

de l’algorithme de Tate [30]. Cet algorithme repose sur le théorème suivant.

Théorème 3.4. — (Néron [22]) Soit Ws un modèle de Weierstrass d’une courbe elliptique

définie sur C[s]. Notons v la valuation s-adique. On suppose que (Ws)s=0 possède un point

double à tangentes distinctes et que v(j(Ws)) = −m < 0, ce qui équivaut à Ws de type Im

dans la classification de Kodaira.



Alors pour tout entier l > m2, il existe un modèle de Weierstrass Es déduit de Ws par une

transformation de la forme

X = x + qz(2)

Y = y + ux + rz(3)

Z = z(4)

avec q, r, u ∈ C[s]. Le modèle de Weierstrass Es est donné par

Y 2Z + λXY Z + µY Z2 = X3 + αX2Z + βXZ2 + γZ3,

les coefficients satisfaisant

v(λ2 + 4α) = 0, v(µ) ≥ l, v(β) ≥ l, v(γ) = m, v(j(Es)) = −m.

Ce théorème est valable en toutes caractéristiques et si la caractéristique est différente de 2,

est équivalent au début de l’algorithme de Tate [30].

Lemme 3.5. — On suppose Ws défini sur Q[s].

1. Les composantes rationnelles de la fibre singulière Im sont définies sur le corps de

nombres Q(√

λ2

0+ 4α0).

2. Si Ps = (X(s) : Y (s) : 1) est un point sur Es vérifiant v(X(s)) = a, v(Y (s)) = b, avec

b < a < 0, alors Ps appartient à la zéro section.

La preuve de ce lemme résulte immédiatement des travaux de Néron [22].

3.3.1. Le modèle de Néron pour s = 0. — A l’aide des transformations

x = X0(s2 − 10s + 1) + Y0 X = X0 − 2s6Z0

y = −Y0 Y = Y0 + sX0 + s6Z0

z = s2(Z0(10s3 − s2) − X0) Z = Z0,

on obtient le modèle de Néron Es au-dessus de s = 0

Y 2Z + XY Z(s2 − 12s + 1) + Y Z2(2s8 − 24s7) =

X3 + X2Z(s − 10s2 − 9s3 − s4 + 6s6) + XZ2(2s7 − 39s8 − 36s9 − 4s10 + 12s12)

+Z3(−s12 − 38s14 − 36s15 − 4s16 + 8s18).

On déduit alors du lemme précédent que toutes les composantes rationnelles de la fibre sin-

gulière I12 sont définies sur Q(√

1 + 4 × 0) = Q.

3.3.2. Le modèle de Néron pour s = ∞. — De même, le modèle de Néron Eσ au-dessus de

l’infini est

Y 2Z + XY Z(9 − 30σ + 3σ2)+Y Z2180σ2(σ − 10)

= X3 − (27 − 180σ)X2Z + XZ2σ2(810σ + 2619)

+ Z3(24300σ2 − 274860σ3 + 48600σ4).

On déduit alors du lemme précédent que toutes les composantes rationnelles de la fibre sin-

gulière I2 sont définies sur Q(√

81 − 4 × 27) = Q(√−3).



3.3.3. La fibre singulière au-dessus de α (ou β). — Elle est de type I3. Grâce aux transfor-

mations précédentes, on voit que Θα,0 est la droite x + y = 0 t = αz, Θα,1 est la droite

y + z = 0 t = αx, Θα,2 est la droite x + z = 0 t = αy.

Ces composantes sont définies sur Q(√

6).

3.3.4. La fibre singulière au-dessus de s = 1/10. — Elle est de type I2.

On remarque que le changement de variable σ 7→ 10 − σ laisse le modèle W∞ inchangé. En

outre, ce changement de variable laisse fixe la fibre singulière I12, échange les deux fibres I1,

les deux fibres I2 et les deux fibres I3. Donc le modèle de Néron pour s = 1

10est le même

que celui pour s = ∞. On en déduit que les composantes rationnelles de la fibre singulière I2

au-dessus de s = 1

10sont définies sur Q(

√−3).

3.3.5. La section infinie. — Les coordonnées de la section infinie (Ps) sont données par les

formules dans le modèle Es au-dessus de s = 0 :

X = − 1

432(1 − 30s + 357s2 − 2140s3 + 6756s4 − 10560s5 + 6400s6 + 864s8)s

Y = − 1

15552((130752 − 524800

√−3)s9 + (1013760

√−3 − 1572480)s8

+(−611640√−3 + 2517768)s7 + (58776

√−3 − 1687896)s6

+(590274 + 80550√−3)s5 + (−116172 − 37872

√−3)s4 + (7665

√−3 + 12924)s3

+(−819√−3 − 756)s2 + (45

√−3 + 18)s −

√−3)

Z = s3

On en déduit

x + y = − 1

432

(s2 − 10s + 1)(2s − 1)2(5s − 1)2(8s − 1)2

s2.

Par suite, d’après le lemme 1, la section infinie coupe la zéro section. En outre, d’après ce qui

précède, elle ne coupe aucun des Θ0,i, 1 ≤ i ≤ 11, aucun des Θα,i, 1 ≤ i ≤ 2, aucun des Θβ,i,

1 ≤ i ≤ 2, ni Θ∞,1, ni Θ1/10,1.

3.3.6. Les sections de torsion. — Dans le modèle W0, on obtient avec Pari [23] les 6 points

de torsion

s6 = (s2(10s − 1) : 0 : 1)

2s6 = (s4 : 0 : 1)

3s6 = (0 : 0 : 1)

4s6 = (s4 : −s4(s2 − 10s + 1) : 1)

5s6 = (−s2 + 10s3 : −s2(10s − 1)(s2 − 10s + 1) : 1)

(0).

3.3.7. Le discriminant de la variété. — Notons Θ le réseau trivial de NS(Y10),

Θ = 〈(0), f,Θv,i, v ∈ 0,∞, 1/10, α, β, 1 ≤ i ≤ mv − 1〉.D’après Shioda [31],

NS(Y10)/Θ ≃ E(C(s))



Par suite

|detNS(Y10)| =|det Θ| × |det MW (E(Cs))|

|E(C(s))tors|2.

Puisque la section infinie (Ps) coupe la zéro section et ne coupe aucune des composantes des

fibres singulières ne rencontrant pas la zéro section, les contributions aux fibres singulières

sont nulles. D’après [30], on a

h(Ps) = 2χ + 2(Ps).(0) = 2 × 2 + 2 = 6.

Si Ps engendrait le groupe de Mordell-Weil, le discriminant de la variété vaudrait

|det NS(Y10)| =12 × 32 × 22 × 6

62= 72.

Nous allons montrer qu’il en est effectivement ainsi donc qu’il n’existe pas de point Q vérifiant

3Q = Ps + ks6, 0 ≤ k ≤ 5.

Puisque la section s6 est de 6-torsion, elle ne coupe pas la zéro section, elle coupe Θ0,2 ou

Θ0,10, coupe Θ∞,1, Θ1/10,1, Θα,1 ou Θα,2, Θβ,1 ou Θβ,2. En effet, on doit avoir

h(s6) = 4 − 2 × 10

12− 1 − 4

3= 0.

Si le groupe de Mordell-Weil était engendré par Q tel que 3Q = Ps, alors Q couperait Θ0,4 ou

Θ0,8, Θ∞,0, Θ1/10,0, Θα,0 ou Θα,1 ou Θα,2, Θβ,0 ou Θβ,1 ou Θβ,2. On aurait alors h(Q) = 2 ou

h(Q) = 10/3, ce qui est impossible car dans ce cas le discriminant de la variété serait différent

de 72 et de 8.

De même, si le groupe de Mordell-Weil était engendré par Q tel que 3Q = Ps + ks6, 1 ≤k ≤ 5, en regardant l’intersection avec la fibre singulière au-dessus de 0, on aurait k = 3. Ceci

donnerait cont0(Q) = 2×10

12, cont∞(Q) = cont1/10(Q) = 1

2, contα(Q) = contβ(Q) = 2

3, soit

h(Q) = 2. Comme précédemment, ce cas est impossible.

Par suite

|det NS(Y10)| = 72.

3.4. La série L de Y10. — Elle est donnée par les Ap = β + β′ tels que Q2(T ) = (1 −βT )(1 − β′T ). Le calcul des Ap repose sur le lemme suivant.

Lemme 3.6. — Soit X une surface K3 singulière et elliptique définie sur Q. On suppose la

section infinie engendrant le groupe de Mordell-Weil définie sur Q(√

d). Alors

Ap = −∑

x∈P1(Fp), Ex lisse

ap(x) −∑

x∈P1(Fp), Ex singulière

ǫp(x) −(

d

p

)

p

où ap(x) est donné par la formule

ap(x) = p − 1 − #Ex(Fp)

et la contribution ǫp(x) est définie par



ǫp(x) =

0, si Ex a réduction additive

1, si Ex a réduction multiplicative déployée

−1, si Ex a réduction multiplicative non déployée

.

Démonstration. — Suivant que les composantes des fibres singulières sont définies ou non sur

Fp, le polynôme P2(T ) prend la forme

P2(T ) = (1 − pT )k1(1 + pT )k2(1 −(

d

p

)

pT )

avec k1 + k2 = 19 et(

dp

)

désignant le symbole de Legendre. D’après 2.2, on a donc la formule

Np = 1 + p2 + p(k1 + (−1)k2) +

(

d

p

)

p + Ap.(5)

En évaluant alors Np à l’aide de la fibration Φ, on obtient :

Np =∑

x ∈ P1(Fp)

Ex lisse

#Ex(Fp) +∑

x ∈ P1(Fp)

Ex singulière

#Ex(Fp)

+∑

x ∈ P1(Fp)

Ex singulière

#Ex(Fp)∑

j 6=0

#Θj,x

=∑

x ∈ P1(Fp)

Ex lisse

(p + 1 − ap(x))

+∑

x ∈ P1(Fp)

Ex singulière

(p + 1 − ǫp(x)) + p(k1 − 2 + (−1)k2).

Ce qui donne

Np = 1 + p2 −∑

x ∈ P1(Fp)

Ex lisse

ap(x) −∑

x ∈ P1(Fp)

Ex singulière

ǫp(x) + p(k1 + (−)k2)

En comparant ces deux expressions de Np, on obtient le résultat annoncé.

Remarque 3.7. — En fait, selon que(

λ2+4αp

)

= ±1 la réduction est déployée ou non.



3.5. Modularité de la fonction L. — La surface K3, Y10, définie sur Q, admet un système

ρ = (ρl) de représentations l-adiques de dimension 2 de GQ,

ρl : GQ → AutH2

trc(Y10, Ql) = V10

où V10 est un Ql-espace vectoriel.

Le système ρ = (ρl) a une fonction L

L(s, ρ) =∏

p 6=2,3,5

1

1 − App−s + ±p2p−2s.

On veut identifier cette fonction L avec la fonction L associée à une forme de Hecke à co-

efficients rationnels et multiplication complexe. Pour cela nous allons utiliser le critère de

Serre-Livné.

Lemme 3.8. — Soit ρl, ρ′l : GQ → AutVl deux représentations l-adiques avec TrFp,ρl

=

TrFp,ρ′l

pour un ensemble de nombres premiers p de densité 1 (i.e. pour tous les nombres

premiers sauf un nombre fini). Si ρl et ρ′l définissent deux systèmes strictement compatibles,

les fonctions L associées à ces deux systèmes sont les mêmes.

La grande idée de Serre reprise par Livné [15] est de remplacer cet ensemble de nombres

premiers de densité 1 par un ensemble fini.

Définition 3.9. — Un ensemble fini T de nombres premiers est appelé ensemble de test

effectif pour une représentation de Galois ρl : GQ → AutVl si l’on peut appliquer le lemme

précédent en remplaçant l’ensemble de densité 1 par T .

Définition 3.10. — Soit P l’ensemble des nombres premiers, S un sous-ensemble fini de Pà r éléments, S′ = S∪−1. Définissons pour chaque t ∈ P, t 6= 2 et chaque s ∈ S′ la fonction

fs(t) :=1

2(1 −

(s

t

)

)

et si T ⊂ P, T ∩ S = ∅,f : T → (Z/2Z)r+1

tel que

f(t) = (fs(t))s∈S′ .

Théorème 3.11. — (Critère de Livné)

Soit ρ et ρ′ deux GQ-représentations 2-adiques non ramifiées hors d’un ensemble fini S de

nombres premiers, satisfaisant

TrFp,ρ ≡ TrFp,ρ′ ≡ 0 (mod 2)

et

detFp,ρ ≡ detFp,ρ′ (mod 2)

pour tout p /∈ S ∪ 2.Tout ensemble fini T de nombres premiers disjoint de S tel que f(T ) = (Z/2Z)r+1 \0 est

un ensemble de test effectif pour ρ par rapport à ρ′.



Pour appliquer le critère de Livné nous devons montrer que la représentation de Galois ρ

associée à Y10 vérifie TrFp,ρ ≡ 0 (2), c’est-à dire que Ap est

pair, ce qui est équivalent d’après (5) à Np pair puisque 1 + p2 ≡ 0 (2) et k1 + k2 + 1 = 20.

Définissons

Y := (x : y : z : t) ∈ P3/xyz(x + y + z) + t2(xy + xz + yz) − 10xyzt = 0.La surface Y10 est obtenue à partir de Y par éclatement des 7 points doubles

P01 P02 P03 P04 P12 P13 P23.

Notons P(Y ′) la variété projective correspondant à

Y ′ := (x : y : z : 1) ∈ P3/(x : y : z : 1) ∈ Y, xyz 6= 0et Np(Y

′) le nombre de ses points.

Or si (x : y : z : 1) ∈ Y ′ avec au moins une coordonnée x, y ou z différente de ±1, par exemple

z, le point (x : y : 1/z : 1) ∈ Y ′. En outre le cas x = ±1, y = ±1 et z = ±1 est impossible.

Par suite Np(Y′) est pair. Il suffit donc de compter les points de Y10 provenant de Y avec au

moins une coordonnée nulle. On a ainsi

– p − 2 points avec une seule coordonnée nulle (ces points sont de la forme (x : y : z : 0) et

vérifient x + y + z = 0),

– 6(p−1) points avec exactement deux coordonnées nulles sur les droites doubles t = 0 x = 0,

t = 0 y = 0, t = 0 z = 0,

– 3(p − 1) points avec exactement 2 coordonnées nulles sur les droites x = 0 y = 0, x =

0 z = 0, y = 0 z = 0,

– 4 points avec trois coordonnées nulles P01, P02, P03 P04,

– les 3 points d’intersection des droites doubles P01, P02, P03,

– 10p points provenant de l’éclatement de tous les points doubles, les points P01, P02, P03

étant à éclater deux fois.

Au total on a 20p − 4 points soit un nombre pair de points. Finalement, on a bien

Ap ≡ Np ≡ 0 (2).

La représentation de Galois ρ associée à Y10 est ramifiée au plus pour 2 et 3 puisque Y10(p) est

singulière seulement pour ces nombres premiers. Par suite r = 2. En outre, on peut prendre

T = 5, 7, 11, 13, 17, 19, 23.Pour calculer Ap, p ∈ T , avec la formule du lemme 2, nous devons connaître ǫx pour x =

0, 1,∞. Nous avons vu que ǫ0 = 1 et ǫ∞ =(

−3

p

)

. Pour obtenir ǫ1, nous utilisons un modèle

minimal au voisinage de 1

Y 2 + XY (8 + 8S − S2) + Y (−24S + 3S2) =

X3 − X2(19 − 8S − S2) − X(40S − 5S2) + 192S − 24S2.

Comme λ2

0+ 4α0 = −12, on en déduit ǫ1 =

(

−3

p

)

.

On calcule alors les Ap, p ∈ T avec les ordres Pari [23]



e(s) = ellinit([s2 − 10 ∗ s + 1, s2 ∗ (1 − 10 ∗ s − s2), 0, 10 ∗ s7 − s6, 0])

Ap = −sum(s = 2, p − 1, ellak(e(s), p) − 1 − 2 ∗ kronecker(−3, p) − kronecker(−3, p) ∗ p

On trouve alors le tableau suivant

p 5 7 11 13 17 19 23

Ap 0 0 14 0 2 -34 0

Or la série L(f, s) pour f newform à multiplication complexe de poids 3 à coefficients de

Fourier rationnels de niveau 8 [28] est également de trace paire et a les mêmes coefficients ap

pour p ∈ T . Par suite la série L de la surface Y10 vérifie

L(Y10, 3) = L(f, 3).

En outre,

1

2

′∑

k,m

k2 − 2m2

(k2 + 2m2)3

est la série L de la forme modulaire F := [θ1, θ2], pour θa =∑

n∈Z qan2

forme modulaire

de poids 1/2 pour le groupe de congruence Γ0(4). Le crochet de Rankin-Cohen des formes

modulaires g et h de poids respectifs k et l pour un groupe de congruence Γ, est une forme

modulaire de poids k + l + 2 pour Γ définie par

[g, h] := kgh′ − lg′h.

Donc F = [θ1, θ2] est une forme modulaire de poids 3 pour Γ0(4) de trace paire. Pour montrer

l’égalité de L(f, s) = L(F, s) il suffit de montrer qu’elles coincident sur T . Ceci achève la

preuve du 1).

Remarque 3.12. — Comme me l’a suggéré M. Schütt, dans notre cas on peut se passer de

la méthode de Livné. Pour cela on renvoie au théorème 5 de Schütt ainsi qu’à sa preuve [28].

La série L de Y10 ne peut différer de la forme modulaire f que par un caractère de Dirichlet

quadratique. Puisqu’il n’y a pas de ramification pour p 6= 2, 3, le caractère quadratique ne

peut être que χ±1, χ±2, χ±3 ou χ±6, ce qui fait 8 possibilités. Les valeurs χ±2(5) = −1,

χ±3(5) = −1, χ−1(7) = −1, χ6(7) = −1 et χ−6(13) = −1 excluent les caractères quadratiques

non triviaux et montrent qu’il suffit de comparer Ap et ap pour p ≤ 13 pour déduire que

L(Y10, 3) = L(f, 3).

3.6. Preuve du théorème 1.1 2). — Notons comme dans [3]

Djτ = (mjτ + κ)(mjτ + κ).



Alors

m(Pk) =ℑτ

8π3

′∑

m,κ

[−4(m(τ + τ) + 2κ)2

D3τ

+4

D2τ

+ 16(2m(τ + τ) + 2κ)2

D3

2τ

− 16

D2

2τ

− 36(3m(τ + τ) + 2κ)2

D3

3τ

+36

D2

3τ

+ 144(6m(τ + τ) + 2κ)2

D3

6τ

− 144

D2

6τ

]

Pour k = 10, on a τ = −i√2

et

Dτ =1

2(m2 + 2κ2)

D2τ = 2m2 + κ2

D3τ =1

2(9m2 + 2κ2)

D6τ = 18m2 + κ2.

m(P10) =

√2

16π3[16 × 4

′∑

m,κ

m2 − 2κ2

(m2 + 2κ2)3

−36 × 84k2

(9m2 + 2k2)3+ 36 × 4

1

(9m2 + 2k2)2

−36 × 44k2

(18m2 + k2)3+ 36 × 4

1

(18m2 + k2)2].

D’où l’expression

m(P10) =

√2

16π3[16 × 4

′∑

m,κ

m2 − 2κ2

(m2 + 2κ2)3

+36 × 89m2 − 2κ2

(9m2 + 2κ2)3− 36 × 4

1

(9m2 + 2κ2)2

+36 × 8κ2 − 18m2

(κ2 + 18m2)3+ 36 × 4

1

(18m2 + κ2)2].

Lemme 3.13. — (Zagier [38]) On peut montrer les formules suivantes :

1.

A(s) :=

′∑

(− 1

(9m2 + 2k2)s+

1

(k2 + 18m2)s) = 2

∞∑

n=1

(−3

n

)

rn

ns

où

rn :=1

2#(k,m)/k2 + 2m2 = n;



plus précisément

A(2) =π2

2√

6L(χ−3, 2).

2.′

∑ k2 − 18m2

(k2 + 18m2)s+

′∑ 9m2 − 2k2

(9m2 + 2k2)s= (1 +

2

3s+

27

32s)

′∑ m2 − 2k2

(m2 + 2k2)s.

Démonstration. — 1) Soit n = k2 + 2m2.

– Si 3 ∤ k et 3 | m, alors k2 + 2m2 = k2 + 18m′2 ≡ 1 mod 3.

– Si 3 | k et 3 ∤ m, alors k2 + 2m2 = 9k′2 + 2m2 ≡ −1 mod 3.

– Dans les deux autres cas, 3 | n.

Ainsi

A(s) = 2

∞∑

n=1

(−3

n

)

rn

ns= 2L(χ−3, s)L(χ24, s)

puisque∞

∑

n=1

rn

ns= ζ(s)L(χ−8, s).

Si s = 2, puisque 2L(χ24, 2) = π2

2√

6, on déduit

A(2) =π2

2√

6L(χ−3, 2).

2) Notons

B(s) :=′

∑ k2 − 18m2

(k2 + 18m2)s+

′∑ 9m2 − 2k2

(9m2 + 2k2)s

et définissons

S :=1

2

′∑

k,m

k2 − 2m2

(k2 + 2m2)s.

Maintenant

S =

∞∑

n=1

an

ns= L(f, s)

avec f forme parabolique de poids 3 et niveau 8. En outre, la fonction L(f, s), a un dévelop-

pement en produit d’Euler

L(s, f) =1

1 − a33−s + 33−1−2s

∏

p 6=3

Lp(f, s).

Puisque a3 = −2, il s’ensuit

L(s, f) =1

1 + 2

3s + 9

32s

∏

p 6=3

Lp(f, s).



Mais

B(s) =

′∑

3|k,3∤m

m2 − 2k2

(m2 + 2k2)s+

′∑

3|m,3∤k

m2 − 2k2

(m2 + 2k2)s+ 2

′∑

3|k,3|m

m2 − 2k2

(m2 + 2k2)s,

′∑

3|k,3∤m

m2 − 2k2

(m2 + 2k2)s+

′∑

3∤k,3|m

m2 − 2k2

(m2 + 2k2)s= 2(1 +

2

3s+

9

32s)S

et

′∑

3|k,3|m

m2 − 2k2

(m2 + 2k2)s= 91−s

′∑ m2 − 2k2

(m2 + 2k2)s.

Aussi,

B(s) = 2S(1 +2

3s+

9

32s) + 4 × 91−sS

= 2S(1 +2

3s+

27

9s).

On déduit du lemme précédent

m(P10) = 2d3 +3 × 8

√2

π3

′∑ m2 − 2κ2

(m2 + 2κ2)3.

Finalement

m(P10) = 2d3 +1

9

|det TY10|3/2

π3L(Y10, 3),

ce qui achève la preuve du théorème 1 2).

Remarque 3.14. — On a également prouvé la relation conjecturée par Boyd [12]

m(P10) = 2d3 + 3m(P2).

Références

[1] A. Beauville, Les familles stables de courbes elliptiques sur P1 admettant quatre fibres singulièresC.R.Acad.Sci. Paris Sér. I Math. t.294 (1982), 657-660.

[2] M.J. Bertin, Mesure de Mahler d’une famille de polynômes, J. reine angew. Math. 569 (2004),175-188.

[3] M.J. Bertin, Mesure de Mahler d’hypersurfaces K3, J. of Number Theory, 128 (2008), No 11,2890-2913.

[4] M.J. Bertin, Mahler’s measure and L-series of K3 hypersurfaces, in Mirror Symmetry V, (Eds. N.Yui, S.-T. Yau, J. D. Lewis), AMS/IP Studies in Advanced Mathematics, vol. 38 (2007).

[5] M.J. Bertin, Mahler’s measure : from number theory to geometry, in Number Theory and po-lynomials, 20-32, London Math. Soc. Lecture Note Ser., 352, Cambridge Univ. Press, Cambridge,(2008).

[6] M.J. Bertin, The Mahler measure and the L-series of a singular K3-surface, arXiv :0803.0413v1[math. NT] 4 Mar 2008.



[7] S. Bloch & D. Grayson, K2 and L-functions of elliptic curves computer calculations, Contemp.Math. 55, Part I, (1986), 79-88.

[8] D.W. Boyd, Kronecker’s Theorem and Lehmer’s Problem for Polynomials in several Variables, J.Number Theory 13 (1981), 116-121.

[9] D.W. Boyd, Speculations concerning the Range of Mahler’s Measure, Canad. Math. Bull. 24 (1981),453-469.

[10] D.W. Boyd, Mahler’s measure and special values of L-functions, Experiment. Math. 7 (1998),37-82.

[11] D.W. Boyd, Explicit formulas for Mahler measure, Bulletin CRM, Autumn 2005, 14-15 (CRM-Fields Prize lecture).

[12] D.W.Boyd, Personal communication.[13] F. Brunault, Etude de la valeur en s = 2 de la fonction L d’une courbe elliptique, Thèse deDoctorat, Université Paris 7, (2005).

[14] C. Deninger, Deligne periods of mixed motives, K-theory and the entropy of certain Zn-actions,J. Amer. Math. Soc.10 :2 (1997), 259-281.

[15] K. Hulek, R. Kloosterman & M. Schütt, Modularity of Calabi-Yau varieties, preprint (2006), inGlobal Aspects of Complex Geometry, Springer Verlag 2006 (F. Catanese, H. Esnault, A. Huckleberry,K. Hulek and T. Peternell, eds.), 271-309.

[16] O. Lecacheux, Personal communication.[17] D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), 461-479.

[18] M. Lalin & M. Rogers, Functional equations for Mahler measures of genus-one curves, AlgebraNumber Theory 1 (2007), No 1, 87-117.

[19] V. Maillot, Géométrie d’Arakelov des grassmanniennes, des variétés toriques et de certaines hy-persurfaces, Thèse Université Paris 7 (1997).

[20] V. Maillot, Unpublished lecture at the BIRS workshop, The Many Aspects of Mahler’s Measure,April 26-May 1, 2003.

[21] A. Mellit, Elliptic dilogarithms and parallel lines, Mathematische Arbeitstagung 2009, Preprintof the Max-Planck-Institut für Mathematik (Juin 2009).

[22] A. Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Publ. Math.,Inst. Hautes Étud. Sci., 21, 128 (1964).

[23] PARI/GP, http ://pari.math.u-bordeaux.fr/ belabas/pari/

[24] C. Peters & J. Stienstra, A pencil of K3 surfaces related to Apery’s recurrence for ζ(3) and Fermisurfaces for potential zero, Arithmetic of Complex Manifolds (Erlangen, 1988) (W.-P. Barth and H.Lange, eds.), Lecture Notes in Math., vol. 1399, Springer, Berlin 1989, 110-127.

[25] C. Peters, J. Top & M. van der Vlugt, The Hasse zeta function of a K3 surface related to thenumber of words of weight 5 in the Mela’s codes, J. reine angew. Math. 432 (1992), 151-176.

[26] F. Rodriguez-Villegas, Modular Mahler Measures,http ://www.ma.utexas.edu/users/villegas/research.html, Preprint (1996).

[27] F. Rodriguez-Villegas, Modular Mahler measures I, Topics in Number Theory (S.D. Ahlgren,G.E. Andrews & K. Ono, ed.), Kluwer, Dordrecht (1999), 17-48.

[28] M. Schütt, CM newforms with rational coefficients, Ramanujan J. 19 (2009), No 2, 187-205.[29] M. Schütt, Personal communication.[30] M. Schütt & T. Shioda, Elliptic surfaces, arXiv :0907.0298v1 [math.AG] 2 Jul 2009.[31] T. Shioda, On the Mordell-Weil Lattices, Comm. Math. Univ. St. Pauli, 39 (1990), 211-240.[32] T. Shioda, On elliptic modular surfaces, J. Math. Soc. Japan, vol. 24, No 1, (1972), 20-59.



[33] T. Shioda & H. Inose, On singular K3 surfaces in Complex analysis and algebraic geometry(Baily, Shioda T. ed.), Cambridge (1977), 119-135.

[34] J. Stienstra & F. Beukers, On the Picard-Fuchs Equation and the Formal Brauer Group of CertainElliptic K3-Surfaces, Math. Ann. 271, (1985), 269-304.

[35] Ch. J. Smyth, On the product of conjugates outside the unit circle of an algebraic integer, Bull.London Math. Soc. 3 (1971), 169-175.

[36] H. Verrill, Root Lattices and Pencils of Varieties, Ph. D. Cambridge University, (July 1994).[37] N. Yui, Arithmetic of Calabi-Yau varieties, Matematisches Institut Seminars (Y. Tschinkel, ed.),Universität Göttingen, (2004), 9-29.

[38] D. Zagier, Personal communication.

16 février 2010

Marie José Bertin • E-mail : [email protected], Université Pierre et Marie Curie (Paris 6),Institut de Mathématiques, 175 rue du Chevaleret, 75013 PARIS, France


RÉGULATEURS p-ADIQUES EXPLICITES POUR LE K2 DES

COURBES ELLIPTIQUES

par

François Brunault

Résumé. — Dans cet article, nous utilisons le système d’Euler de Kato et la théorie dePerrin-Riou pour établir une formule reliant la valeur en 0 de la fonction L p-adique d’unecourbe elliptique définie sur Q, et un régulateur p-adique sur la courbe modulaire X(N). Enparticulier, nous obtenons une relation explicite entre fonction L p-adique et régulateur p-adiquepour la courbe elliptique X0(20).

Abstract (Explicit p-adic regulators for K2 of elliptic curves). — In this article, weuse Kato’s Euler system and Perrin-Riou’s theory to get an explicit formula linking the valueat 0 of the p-adic L-function of an elliptic curve defined over Q, and a p-adic regulator on themodular curve X(N). In particular, we give an explicit relation between the p-adic L-functionand the p-adic regulator, in the case of the elliptic curve X0(20).

Les fonctions ζ et fonctions L de nature arithmétique, et leur comportement aux points

entiers sont l’objet de conjectures profondes et mystérieuses. Par exemple, dans le cas d’une

courbe elliptique définie sur Q, la conjecture de Birch et Swinnerton-Dyer prédit que l’ordre

d’annulation en 1 de la fonction L est égal au rang du groupe des points rationnels. Pour les

entiers ≥ 2, la situation est différente car la fonction L ne s’y annule pas. Les valeurs spéciales

sont alors décrites conjecturalement par Beilinson, en termes de régulateurs.

De nombreux travaux (citons ici Mazur, Tate et Teitelbaum [MTT86], Bloch et Kato [BK90],

Fontaine et Perrin-Riou [FPR94], Besser [Bes00a, Bes00b]. . . ) permettent d’envisager un

Classification mathématique par sujets (2000). — 11F67,11G40,19F27.Mots clefs. — cohomologie étale, conjectures de Beilinson, courbe elliptique, courbe modulaire, élémentszêta, fonction L, fonction L p-adique, forme modulaire, K-théorie, régulateur, régulateur p-adique, systèmed’Euler de Kato.

Ce texte est la version longue d’un exposé donné à l’occasion de la conférence Fonctions L et arithmétique,qui s’est déroulée du 8 au 12 juin 2009 à Besançon. C’est avec grand plaisir que je remercie ChristopheDelaunay, Christian Maire et Xavier-François Roblot de m’y avoir invité, ainsi que pour l’ambiance chaleureuseet la bonne organisation de la conférence. Je remercie également Laurent Berger, Massimo Bertolini, HugoChapdelaine, Pierre Colmez, Henri Darmon, Rob de Jeu, Olivier Fouquet, Lionel Fourquaux, Matthew T.Gealy, Xavier-François Roblot, Floric Tavares Ribeiro et Stefano Vigni pour des discussions très stimulantessur les fonctions L p-adiques.

30 Régulateurs p-adiques explicites pour le K2 des courbes elliptiques

analogue de ces conjectures dans le monde p-adique, où il s’agit de relier les valeurs spéciales

des fonctions L p-adiques et les régulateurs p-adiques.

Indiquons plus précisément les principaux résultats connus en direction de ces conjectures,

dans le cas particulier de la valeur en 2 de la fonction L d’une courbe elliptique. Vers 1978,

Bloch [Blo00] construit, pour toute courbe elliptique E définie sur C, un régulateur complexe

rE,∞ ∶ K2(E) → C. Pour certaines courbes elliptiques E définies sur Q à multiplication

complexe, il montre que L(E,2) est, à un facteur rationnel explicite non nul près, le régulateur

d’un élément explicite de K2(E) ⊗Q. Dans les années 1980, Beilinson [Bei84] propose une

vaste généralisation de l’approche de Bloch et formule une conjecture décrivant les valeurs

spéciales des fonctions L de toutes les variétés algébriques en tous les entiers. De plus, pour

toute courbe elliptique (modulaire) définie sur Q, il montre que L(E,2) est proportionnel au

régulateur d’un élément a priori inexplicite de K2(E) ⊗Q. En 1988, Coleman et de Shalit

[CdS88] définissent, pour toute courbe elliptique E sur Qp ayant bonne réduction en p, un

régulateur p-adique rE,p ∶ K2(E) → Qp. Dans le cas des courbes à multiplication complexe,

ils obtiennent aussi un analogue p-adique de la formule de Bloch. Il est intéressant de noter

que leur formule fait intervenir le même élément dans K2, et essentiellement le même facteur

rationnel.

La motivation de ce travail est de donner des exemples explicites de lien entre fonction L

p-adique et régulateur p-adique dans le cas des courbes elliptiques sans multiplication com-

plexe. La stratégie adoptée ici pour atteindre cet objectif ne surprendra pas les experts. Nous

utilisons de manière essentielle les résultats fondamentaux et très profonds de Kato [Kat04],

en particulier sa construction d’un système d’Euler pour toute forme modulaire f , ainsi que

son lien avec la fonction L p-adique de f . D’autres ingrédients interviennent également : l’ap-

plication exponentielle de Perrin-Riou [PR94], et la loi de réciprocité explicite, démontrée

notamment par Colmez [Col98]. Nous espérons que l’approche explicite suivie dans ce texte

pourra contribuer à familiariser certains lecteurs avec le système d’Euler de Kato.

De manière plus précise, nous montrons (théorème 4.1) un lien entre la valeur en 0 de la

fonction L p-adique d’une courbe elliptique E sans multiplication complexe, et un régulateur

p-adique explicite sur la courbe modulaire paramétrant E. On peut ici choisir l’une des courbes

X(N), X1(N) ou X0(N) (voir la proposition 9.1). La formule obtenue n’est pas optimale :

la constante rationnelle devant la valeur spéciale peut s’annuler. Enfin, signalons que Gealy

[Gea03, Gea05] a obtenu une formule plus générale, reliant la valeur spéciale de la fonction

L p-adique d’une forme modulaire en tout entier ≤ 0, à un régulateur p-adique défini sur la

variété de Kuga-Sato.

Voici le plan de l’article. Dans les trois premières sections, nous faisons des rappels sur la

fonction L p-adique, la représentation p-adique et le régulateur p-adique associés à une courbe

elliptique E définie sur Q. Le résultat principal du texte (théorème 4.1) est énoncé dans la

section 4. Nous faisons ensuite des rappels sur la cohomologie d’Iwasawa dans la section 5,

en vue de définir le système d’Euler de Kato, ou plutôt sa partie locale, dans la section 6.

Dans la section 7, nous utilisons la théorie développée par Perrin-Riou pour énoncer la loi de

réciprocité explicite. La section 8 est consacrée à la démonstration du théorème 4.1. Enfin,


François Brunault 31

en guise d’application, nous donnons dans la section 9 des exemples explicites, notamment le

cas de la courbe elliptique X0(20).

1. La fonction L p-adique

Soit E une courbe elliptique définie sur Q, de conducteur N . On note f(z) = ∑n≥1 ane2iπnz ∈S2(Γ0(N)) la forme parabolique primitive, de poids 2 et de niveau N , associée à E.

La fonction L de Hasse-Weil de E, donnée par L(E,s) = L(f, s) = ∑n≥1 ann−s pour R(s) > 32,

se prolonge en une fonction holomorphe sur C. Pour tout caractère de Dirichlet χ ∶ (Z/mZ)∗ →C∗ (m ≥ 1), la série L de E tordue par χ, donnée par L(E,χ, s) = ∑n≥1 anχ(n)n−s, se prolonge

en une fonction holomorphe sur C.

Soit ωE ∈ Ω1(E/Q) la forme différentielle associée à une équation de Weierstraß minimale de

E sur Z. Notons ΛE le réseau des périodes de ωE. On définit les périodes réelle et imaginaire

pure de E par ΛE ∩ R = Ω+E ⋅ Z et ΛE ∩ iR = Ω−E ⋅ Z, avec Ω+E ∈ R>0 et Ω−E ∈ iR>0. Il

résulte du théorème de Manin-Drinfeld et de la théorie des symboles modulaires [Man72]

que L(E,χ,1)/Ωχ(−1)E est algébrique pour tout χ. Notons τ(χ) = ∑a∈(Z/mZ)∗ χ(a)e2iaπ/m la

somme de Gauss de χ.

Soit p un nombre premier. Le facteur d’Euler en p de L(E,s) vaut (1 − app−s + ε(p)p1−2s)−1,avec ε(p) = 1 si p ∤ N , et ε(p) = 0 si p ∣N .

On note ordp la valuation sur Qp qui prolonge la valuation p-adique usuelle sur Qp. Pour

pouvoir définir une fonction L p-adique associée à E, les conditions équivalentes suivantes

doivent être satisfaites :

1. Il existe α ∈Qp tel que ordp(α) < 1 et 1 −αT ∣ 1 − apT + ε(p)pT 2.

2. La courbe elliptique E est semi-stable en p.

3. p2 ∤ N .

Supposons ces conditions vérifiées et donnons-nous α vérifiant (1). La fonction L p-adique de

E, qui dépend de ce choix de α, est construite par interpolation p-adique de valeurs tordues

de la fonction L complexe de E, au moyen du théorème suivant.

Rappelons qu’une distribution sur Z∗p à valeurs dans Qp est une forme linéaire continue sur

l’espace des fonctions localement analytiques de Z∗p dans Qp. Fixons des plongements QQp

et Q C.

Théorème 1.1 ([Man73, MSD74]). — Il existe une unique distribution µE,α d’ordre vp(α)sur Z∗p, à valeurs dans Qp(α), telle que

(1) ∫Z∗p

µE,α = (1 −α−1)1+ε(p)L(E,1)Ω+E

,

et telle que pour tout n ≥ 1 et tout χ ∶ (Z/pnZ)∗ →Q∗

primitif, on ait



(2) ∫Z∗p

χ(x)µE,α = τ(χ)αn

L(E,χ,1)Ωχ(−1)E

.

Posons q = p si p est impair, et q = 4 si p = 2. Tout x ∈ Z∗p s’écrit de manière unique

x = ω(x)⟨x⟩ où ω(x) est une racine ϕ(q)-ième de l’unité dans Zp, et ⟨x⟩ ∈ 1 + qZp. Pour

y ∈ pZp et s ∈ Zp, on définit (1 + y)s = ∑+∞n=0 (sn) yn. Pour s ∈ Zp, l’application x ↦ ⟨x⟩sest un caractère continu de Z∗p (on peut également montrer que ⟨x⟩s = expp(s logp x), avec

logp ∶ Z∗p → qZp et expp ∶ qZp

≅Ð→ 1 + qZp).

Définition 1.2. — La fonction L p-adique de E est donnée par

(3) Lp,α(E,s) = ∫Z∗p

⟨x⟩s−1 ⋅ µE,α (s ∈ Zp).Pour tout caractère continu χ ∶ Z∗p →C∗p, on pose également

(4) Lp,α(E,χ, s) = ∫Z∗p

χ(x)⟨x⟩s−1µE,α.On dispose d’une équation fonctionnelle reliant les valeurs Lp,α(E,s) et Lp,α(E,2−s). Posons

WNf = w(f)f , où WN est l’involution d’Atkin-Lehner sur S2(Γ0(N)), et w(f) est l’opposé

du signe de l’équation fonctionnelle de L(E,s). On a alors [MTT86, §17, Cor. 2]

Lp,α(E,s) = −w(f)⟨N⟩1−sLp,α(E,2 − s) si p ∤N,(5)

Lp,α(E,s) = apw(f)⟨Np⟩1−sLp,α(E,2 − s) si p ∣N.(6)

Pour tout caractère continu χ ∶ Z∗p →C∗p, on a également

Lp,α(E,χ, s) = −w(f)χ−1(−N)⟨N⟩1−sLp,α(E,χ−1,2 − s) si p ∤ N,(7)

Lp,α(E,χ, s) = apw(f)χ−1(−Np)⟨Np⟩1−sLp,α(E,χ−1,2 − s) si p ∣ N.(8)

2. Représentations p-adiques

Si X est une courbe lisse définie sur Q ou sur Qp, la cohomologie étale p-adique VX ∶=H1(X

Qp,Qp) est un Qp-espace vectoriel muni d’une action continue de Gal(Qp/Qp), ce qui

en fait une représentation p-adique.

Le module de Tate p-adique TpE = lim←ÐE[pn] est un Zp-module libre de rang 2, muni d’une

action continue de Gal(Qp/Qp). De plus VpE ∶= TpE ⊗Qp est isomorphe à VE(1).



La courbe modulaire ouverte Y1(N) et sa compactifiée X1(N) sont définies sur Q, lisses

et géométriquement irréductibles. Par fonctorialité, les espaces VY1(N) et VX1(N) sont munis

d’une action des opérateurs de Hecke Tn (n ≥ 1).

Définition 2.1. — On pose Vf ∶= VY1(N)/⟨Tn − an, n ≥ 1⟩.Grâce aux théorèmes de comparaison entre cohomologie étale p-adique et cohomologie de de

Rham, on sait que Vf est une représentation p-adique de dimension 2. De plus, si j ∶ Y1(N) X1(N) désigne l’inclusion et φ ∶ X1(N)→ E est une paramétrisation modulaire, l’application

(φ j)∗ induit un isomorphisme VE≅Ð→ Vf . Cependant, cet isomorphisme dépend du choix de

φ.

Grâce au cup-produit, on a une dualité parfaite VE × VE → Qp(−1), d’où un isomorphisme

V ∗E (1) ≅ VE(2).Soit BdR le corps des périodes p-adiques. Fontaine a montré que la représentation VE est de

de Rham, c’est-à-dire que DdR(VE) ∶= (BdR ⊗Qp V )Gal(Qp/Qp) est un Qp-espace vectoriel de

dimension 2. On sait que DdR(VE) s’identifie à H1dR(EQp) et que la filtration naturelle de

DdR(VE) correspond à la filtration de Hodge :

(9) FiliDdR(VE) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

H1dR(EQp) si i ≤ 0,

Ω1(EQp) si i = 1,

0 si i ≥ 2.

Rappelons que Dcris(VE) ∶= (Bcris ⊗ VE)Gal(Qp/Qp) s’identifie à un sous-espace vectoriel de

DdR(VE), et est muni d’un Frobenius ϕ. D’après Saito [Sai97, Sai00], on a

(10) dimQp Dcris(VE) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

2 si p ∤ N,1 si p ∣ N et p2 ∤N,0 si p2 ∣ N,

et le polynôme caractéristique de ϕ sur Dcris(VE) est donné par

(11) det(1 − ϕT ∣Dcris(VE)) = 1 − apT + ε(p)pT 2.

Si E a bonne réduction en p, l’endomorphisme ϕ de Dcris(VE) est diagonalisable après ex-

tension des scalaires à Qp(α), ses valeurs propres étant α et β ∶= p/α. Puisque ap ∈ Z, on a

nécessairement α ≠ β.

Si E a réduction multiplicative en p, on a ϕ = α = ap = ±1 surDcris(VE), qui est de dimension 1.

Pour n ≥ 0 et k ∈ Z, notons

(12) expn,k ∶ DdR(VE(k)/Qp(ζpn)) →H1(Qp(ζpn), VE(k))



l’exponentielle de Bloch-Kato associée à VE(k) vue comme représentation de Gal(Qp/Qp(ζpn))[BK90, 3.10]. L’application naturelle

Qp(ζpn)⊗DdR(VE)→DdR(VE(k)/Qp(ζpn))a⊗ v ↦ at−kv

est un isomorphisme, ce qui permet d’identifier ces deux espaces. Via cette identification, on

a FiliDdR(VE(k)) = Fili+kDdR(VE) pour i, k ∈ Z.

Donnons maintenant la dimension des groupes de cohomologie galoisienne associés aux tor-

dues de VE. Rappelons que pour une représentation p-adique V , on note H1e (Qp, V ) (resp.

H1f(Qp, V ), H1

g (Qp, V )) le noyau de H1(Qp, V ) → H1(Qp,B ⊗ V ) avec B = Bϕ=1cris (resp.

B =Bcris, B =BdR). Notons h1∗(V ) = dimQp H

1∗(Qp, V ) pour ∗ ∈ e, f, g,∅.

Proposition 2.2. — Si E n’a pas réduction multiplicative déployée en p, alors h1∗(VE(k))

est donnée par la table suivante :

(13)

k h1e(VE(k)) h1

f(VE(k)) h1g(VE(k)) h1(VE(k))

≤ 0 0 0 0 2

1 1 1 1 2

≥ 2 2 2 2 2

Si E a réduction multiplicative déployée en p, alors h1∗(VE(k)) est donnée par la table suivante :

(14)

k h1e(VE(k)) h1

f(VE(k)) h1g(VE(k)) h1(VE(k))

< 0 0 0 0 2

0 0 1 1 3

1 1 1 1 2

2 2 2 3 3

> 2 2 2 2 2

Démonstration. — Il suffit d’utiliser (11) et les résultats de Bloch-Kato [BK90, Prop. 3.8 et

3.8.4]. L’image de exp0,k est H1e (Qp, VE(k)), et son noyau a pour dimension

dimDcris(VE(k))ϕ=1 + dimFil0DdR(VE(k)) − dimVE(k)Gal(Qp/Qp)(15)

= dimDcris(VE)ϕ=pk

+ dimFilkDdR(VE) − dimVE(k)Gal(Qp/Qp).

De plus, on a

h1f(VE(k)) = h1

e(VE(k)) + dimDcris(VE(k))/(1 − ϕ)Dcris(VE(k))(16)

= h1e(VE(k)) + dimDcris(VE(k))ϕ=1.

Remarquons qu’on a une inclusion naturelle



VE(k)Gal(Qp/Qp) Dcris(VE(k))ϕ=1(17)

v ↦ 1⊗ v.

Comme V ∗E (1) ≅ VE(2), on a une dualité parfaite en cohomologie galoisienne

(18) H1(Qp, VE(k)) ×H1(Qp, VE(2 − k)) →H2(Qp,Qp(1)) ≅Qp,

pour laquelle H1e (resp. H1

f ) est l’orthogonal de H1g (resp. H1

f ), d’où

h1(VE(k)) = h1f(VE(k)) + h1

f(VE(2 − k))(19)

h1(VE(k)) = h1g(VE(k)) + h1

e(VE(2 − k)).(20)

Il suffit donc de déterminer h1e(VE(k)) et h1

f(VE(k)) pour tout k ∈ Z.

Si E a bonne réduction en p, les valeurs propres α et β de ϕ vérifient ∣α∣∞ = ∣β∣∞ =√p et ne

peuvent donc être des puissances de p. Par suite, pour tout k ∈ Z, on a Dcris(VE)ϕ=pk = 0, et

aussi VE(k)Gal(Qp/Qp) = 0 d’après (17). On obtient alors

(21) h1e(VE(k)) = h1

f(VE(k)) = 2 − dimFilkDdR(VE) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

0 si k ≤ 0,

1 si k = 1,

2 si k ≥ 2.

Les formules (19) et (20) permettent de compléter la table (13).

Si E a réduction additive en p, on aDcris(VE) = 0 ; si E a réduction multiplicative non déployée

en p, on a ϕ = −1 sur Dcris(VE) : dans ces deux cas, la table (13) reste valable.

Supposons enfin que E a réduction multiplicative déployée en p. D’après la théorie de Tate,

VE admet un vecteur non nul invariant sous Galois :

(22) dimVE(k)GQp = dimDcris(VE(k))ϕ=1 =⎧⎪⎪⎨⎪⎪⎩1 si k = 0,

0 si k ≠ 0.

La formule (15) montre alors que h1e(VE(k)) est donné par la même formule que précédem-

ment, d’où la première colonne de la table (14). On obtient la deuxième colonne par (16), et

la table complète par dualité, grâce aux formules (19) et (20).

Remarque 2.3. — Lorsque exp0,2 ∶ DdR(VE(2)) →H1(Qp, VE(2)) est bijective, c’est-à-dire

lorsque E n’a pas réduction multiplicative déployée en p, on peut définir l’isomorphisme

réciproque

(23) log ∶H1(Qp, VE(2)) ≅Ð→DdR(VE(2)) ≅DdR(VE).



La forme différentielle ωf = 2iπf(z)dz définit un élément canonique, encore noté ωf , dans

Fil1DdR(Vf) [Kat04, 9.2.2]. On a un isomorphisme canonique V ∗f (1) ≅ Vf(2) et donc une

dualité parfaite

(24) DdR(Vf(2)) ×DdR(Vf) [⋅,⋅]ÐÐ→Qp.

3. Le régulateur étale p-adique

Les groupes K2 considérés ici sont les groupes de K-théorie algébrique définis par Quillen.

Soit X une courbe lisse définie sur Qp. On dispose d’une application classe de Chern en

cohomologie étale continue [Jan88]

(25) chX ∶K2(X)→ H2(X,Zp(2)).La suite spectrale Ea,b2 = Ha(Qp,H

b(XQp,Zp(2))), qui converge vers Ha+b(X,Zp(2)), est

concentrée en degrés 0 ≤ a, b ≤ 2. Puisque

(26) H2(XQp,Zp(2)) ≅

⎧⎪⎪⎨⎪⎪⎩0 si X est affine,

Zp(1) si X est projective,

on a dans tous les cas E0,22 = 0. De plus, par dualité locale en cohomologie galoisienne,

E2,02 =H2(Qp,Zp(2)) est isomorphe à

(27) HomZp(H0(Qp, (Qp/Zp)(−1)),Qp/Zp) = 0.

La suite spectrale dégénère donc en E2 et l’on a un isomorphisme

(28) H2(X,Zp(2)) ≅H1(Qp,H1(X

Qp,Zp(2))).

En composant chX avec (28), puis en tensorisant par Q, on obtient l’application régulateur

(29) reg(p)X∶K2(X)⊗Q→H1(Qp, VX(2))

où l’on a posé VX(2) ∶=H1(XQp,Qp(2)).

Remarque 3.1. — Dans le cas où X a bonne réduction, Besser a défini un régulateur p-

adique plus intrinsèque, le régulateur syntomique, et a fait le lien avec le régulateur présenté

ici [Bes00a, 9.11]. Il a également montré le lien entre le régulateur syntomique et le régulateur

de Coleman et de Shalit défini par intégration p-adique [Bes00b, Thm 3].



Appliquons ce qui précède à X = EQp . Grâce à l’application naturelle K2(E)→K2(EQp), on

obtient

(30) reg(p)E ∶K2(E)⊗Q→ H1(Qp, VE(2)).

Nous venons de définir le régulateur p-adique sur K2(E). Cependant, il sera commode d’uti-

liser un régulateur p-adique défini sur le K2 d’une courbe modulaire, de la manière suivante.

Soit Y (N) la courbe modulaire définie sur Q paramétrant les courbes elliptiques E munies

d’un isomorphisme E[N] ≅ (Z/NZ)2. Le morphisme fini Y (N)→ Y1(N) induit un morphisme

de trace VY (N) → VY1(N), et on dispose de la projection canonique VY1(N) → Vf . En composant

reg(p)

Y (N)et le morphisme induit par VY (N) → Vf , on obtient

(31) reg(p)f∶K2(Y (N))⊗Q→H1(Qp, Vf(2)).

Maintenant, on dispose dans K2(Y (N))⊗Q des éléments de Beilinson-Kato, obtenus comme

cup-produits d’unités de Siegel. Plus précisément, pour tout (a, b) ∈ (Z/NZ)2, on a une unité

de Siegel ga,b ∈ O∗(Y (N)) ⊗Q [Kat04, §1]. Pour tout (a b

c d) ∈ M2(Z/NZ), on peut donc

considérer

(32) ρ(a b

c d) ∶= ga,b, gc,d ∈K2(Y (N))⊗Q.

Définition 3.2. — On pose zN = ρ(I2) = g1,0, g0,1 ∈K2(Y (N))⊗Q.

Le but de ce texte est d’obtenir une formule explicite pour reg(p)f(zN). Remarquons que si E

n’a pas réduction multiplicative déployée en p, il est licite de considérer

(33) log reg(p)f∶K2(Y (N))⊗Q→DdR(Vf(2)).

4. Énoncé du théorème principal

On suppose dans cette section p2 ∤N . En utilisant le théorème de semi-stabilité de Vf [Fal02,

Tsu99], on obtient l’admissibilité du ϕ-module Dcris(Vf) et donc [Kat04, Thm 16.6(1)] qu’il

existe un unique ηα ∈Dcris(Vf)⊗Qp Qp(α) tel que

(34) ϕ(ηα) = αηα et [ωf , ηα] = 1,

où le crochet de dualité a été défini en (24), et où l’on considère ωf comme un élément de

DdR(Vf) ≅DdR(Vf(2)). On peut penser à ηα comme à un substitut, dans le contexte p-adique,

d’une 1-forme antiholomorphe.

Notons ⟨f, f⟩ ∶= i2 ∫X1(N)(C)

ωf ∧ ωf le carré scalaire de Petersson de f .



Théorème 4.1. — Supposons p impair et E sans multiplication complexe. Si E a bonne

réduction en p, alors

[log reg(p)f(zN), ηα] =(∏

ℓ∣N

1 − aℓ)L(E,1)Ω−Ei⟨f, f⟩ ⋅

⋅ (1 − pα−1)−1(1 − p−1α−1)−1Lp,α(E,ω−1,0),(35)

où le produit porte sur les diviseurs premiers de N . Si E a réduction multiplicative non déployée

en p, alors

[log reg(p)f(zN), ηα] =(∏

ℓ∣Nℓ≠p

1 − aℓ)L(E,1)Ω−Ei⟨f, f⟩ ⋅

⋅1 − α

1 − p−1α−1Lp,α(E,ω−1,0).

(36)

Remarque 4.2. — Soit φ ∶ X1(N) → E une paramétrisation modulaire de E. Posons φ∗ωE =cωf avec c ∈Q∗. On a

i⟨f, f⟩ = − 1

2c2∫X1(N)(C)

φ∗ωE ∧ φ∗ωE = −degφ

2c2∫E(C)

ωE ∧ ωE.

De plus ∫E(C)ωE ∧ωE = −c∞Ω+EΩ−E, où c∞ est le nombre de composantes connexes de E(R).Comme L(E,1)/Ω+E ∈Q [Man72], on en déduit que le facteur L(E,1)Ω−E/(i⟨f, f⟩) dans (35)

et (36) est rationnel et vaut

(37)L(E,1)Ω−Ei⟨f, f⟩ = 2c2

c∞ degφ

L(E,1)Ω+E

.

Remarquons que L(E,1)/Ω+E peut aussi s’exprimer en termes de Lp,α(E,1) lorsque la réduc-

tion est bonne ou multiplicative non déployée.

Remarque 4.3. — Kato a donné une formule pour le régulateur complexe de zN faisant

intervenir la dérivée en 0 de L(N)(E,s), la fonction L complexe de E privée de tous ses

mauvais facteurs d’Euler [Kat04, 2.6]. La formule obtenue ici pour le régulateur p-adique est

similaire : on peut en effet remarquer que

(38)d

ds(L(N)(E,s))

s=0= (∏

ℓ∣N

1 − aℓ)L′(E,0).La présence de facteurs d’Euler en p supplémentaires dans (35) et (36) est un phénomène bien

connu (comparer avec la formule (2) de [CdS88]).

Remarque 4.4. — Le théorème 4.1 n’est pas optimal en ce sens que le membre de droite

de (35) et (36) peut s’annuler : par exemple, s’il existe ℓ premier divisant N tel que aℓ = 1.



Comme nous le verrons dans les sections suivantes, la présence des facteurs 1 − aℓ reflète

l’idée que le système d’Euler de Kato, qui est parfaitement normalisé, l’est en cohomologie

d’Iwasawa, mais pas au niveau des groupes K2.

Question 4.5. — Supposons L(E,1) = 0. La trace (voir la section 9, définition 9.3) de zNdans K2(E)⊗Q est-elle nulle ? La même question se pose s’il existe ℓ tel que aℓ = 1.

Remarque 4.6. — On peut également calculer le régulateur p-adique de ρ(M) pour tout

M ∈ GL2(Z/NZ) (voir la définition (32) de ρ(M)). Cependant, il ne semble pas possible

de calculer reg(p)f(ρ(M)) avec les méthodes de cet article lorsque la matrice M n’est pas

inversible.

Remarque 4.7. — Je ne sais pas si le théorème 4.1 reste valable dans le cas où E est à

multiplication complexe. Cela vient du fait que dans ce cas, la construction (antérieure) du

système d’Euler par Rubin est de nature différente : elle utilise les unités elliptiques à la place

des éléments de Beilinson-Kato.

5. Cohomologie d’Iwasawa

On fixe un système compatible (ζpn)n≥0 de racines primitives pn-ièmes de l’unité dans Qp.

Pour n ≥ 0, posons Gn = Gal(Qp(ζpn)/Qp) ≅ (Z/pnZ)∗ et G∞ = lim←ÐGn ≅ Z∗p. Pour tout

x ∈ Z∗p, notons σx ∈ G∞ l’élément correspondant.

L’algèbre d’Iwasawa est définie par Λ = Zp[[G∞]] ∶= lim←ÐZp[Gn]. On a une application cano-

nique de G∞ dans Λ.

Si T est un Zp-module de type fini muni d’une action continue de Gal(Qp/Qp), le groupe

de cohomologie d’Iwasawa de T est défini par H1(Qp, T ) = lim←ÐH1(Qp(ζpn), T ), où la limite

projective est relative aux applications de corestriction. Il existe une unique structure de Λ-

module sur H1(Qp, T ) compatible, via les projections naturelles Λ → Zp[Gn], à la structure

de Gn-module de H1(Qp(ζpn), T ) pour chaque n.

Si V est un Qp-espace vectoriel de dimension finie muni d’une action continue de Gal(Qp/Qp),le groupe H1(Qp, V ) ∶=H1(Qp, T )⊗Qp, où T est un Zp-réseau de V stable par Galois, ne dé-

pend pas du choix de T , et est un Λ⊗Qp-module. On note πn ∶H1(Qp, V )→H1(Qp(ζpn), V )

l’application canonique.

Pour chaque n, on a une forme Zp-bilinéaire

(39) H1(Qp(ζpn), T ) ×H1(Qp(ζpn), T ∗(1)) ⟨⋅,⋅⟩nÐÐ→H2(Qp(ζpn),Zp(1)) ≅ Zp.

En posant, pour x = (xn)n≥0 ∈H1(Qp, T ) et y = (yn)n≥0 ∈H1(Qp, T∗(1))

(40) ⟨x, y⟩ ∶= ( ∑σ∈Gn

⟨σ−1xn, yn⟩n[σ])n≥0∈ Λ,

on obtient [PR94, 3.6.1] une application Zp-bilinéaire



(41) H1(Qp, T ) ×H1(Qp, T∗(1)) ⟨⋅,⋅⟩ÐÐ→ Λ

qui est Λ-linéaire à gauche, et Λ-antilinéaire à droite, c’est-à-dire

⟨x,λ ⋅ y⟩ = ι(λ)⟨x, y⟩ (λ ∈ Λ)où ι est l’involution de Λ déduite de l’involution σ ↦ σ−1 de G∞. En tensorisant (41) par Qp,

on obtient

(42) H1(Qp, V ) ×H1(Qp, V∗(1)) ⟨⋅,⋅⟩ÐÐ→ Λ⊗Qp.

Si χ ∶ G∞ → C∗p est un caractère continu, on peut évaluer les éléments de Λ ⊗Qp en χ, et

obtenir un morphisme de Qp-algèbres Λ⊗Qp →Cp. Pour x ∈H1(Qp, V ), y ∈H1(Qp, V∗(1))

et χ ∶ G∞ →C∗p caractère se factorisant par Gn (n ≥ 0), on a

(43) ⟨x, y⟩(χ) = ∑σ∈Gn

⟨σ−1πn(x), πn(y)⟩nχ(σ).L’isomorphisme H1(Qp, T ) ≅Ð→ lim←ÐH

1(Qp(ζpn), T /pnT ) permet de définir des isomorphismes

H1(Qp, T ) ≅Ð→H1(Qp, T (k))(44)

z = (zn)n≥0 ↦ z(k) ∶= (zn ⊗ ζ⊗kpn )n≥0.ainsi que H1(Qp, V ) ≅H1(Qp, V (k)). Notons λ↦ λ(k) l’automorphisme de Λ⊗Qp envoyant

[σx] sur xk[σx] pour tout x ∈ Z∗p. On a la formule

(45) (λ ⋅ z)(k) = λ(−k) ⋅ z(k) (λ ∈ Λ⊗Qp, z ∈H1(Qp, V )).Rappelons que q = p si p est impair, et q = 4 si p = 2. Le sous-groupe G1

∞≅ 1 + qZp de

G∞ ≅ Z∗p est isomorphe à Zp, engendré topologiquement par σ1+q. On définit de manière

analogue Zp[[G1∞]], qui est une sous-algèbre de Λ. En posant ∆ = (G∞)tors ≅ µφ(q), on a un

isomorphisme canonique G∞ ≅ G1∞×∆ et Zp[∆] s’identifie à une sous-algèbre de Λ. Posons

(46) H = ∑n≥0

anxn ∈Qp[[x]];∃h ≥ 0, ∣an∣p = On→+∞(nh).

On a un isomorphisme

Zp[[x]] ≅Ð→ Zp[[G1∞]](47)

x↦ [σ1+q] − [1].qui s’étend en un morphisme injectif H → Qp[[G1

∞]]. Notons H1

∞son image, et posons

H∞ = H1∞⊗Zp Zp[∆]. On a des inclusions naturelles Λ⊗Qp ⊂ H∞ ⊂Qp[[G∞]].



La projection naturelle H∞ →Qp[Gn] munit H1(Qp(ζpn), V ) d’une structure de H∞-module.

On étend πn en une application H∞-linéaire

(48) πn ∶H1(Qp, V )⊗Λ⊗Qp H∞ →H1(Qp(ζpn), V ).

Par extension des scalaires de (42), on a aussi une application

(49) (H1(Qp, V )⊗H∞) × (H1(Qp, V∗(1)) ⊗H∞) ⟨⋅,⋅⟩ÐÐ→H∞

qui est H∞-linéaire à gauche et H∞-antilinéaire à droite (les produits tensoriels sont pris

au-dessus de Λ⊗Qp).

Enfin, si χ ∶ G∞ → C∗p est un caractère d’ordre fini, on peut encore évaluer les éléments de

H∞ en χ. On vérifie alors que la formule (43) reste valable pour x ∈ H1(Qp, V ) ⊗ H∞ et

y ∈H1(Qp, V∗(1)) ⊗H∞. En particulier, on obtient

(50) ⟨x, y⟩(1) = ⟨π0(x), π0(y)⟩0pour x ∈H1(Qp, V )⊗H∞ et y ∈H1(Qp, V

∗(1)) ⊗H∞.

6. Le système d’Euler de Kato

Nous allons détailler la construction par Kato [Kat04, 12.5,13.9] d’un système d’Euler pour

la représentation Vf . En fait, nous ne définirons que sa partie locale, qui est un élément

z(p)Kato∈H1(Qp, Vf).

Considérons la tour infinie des courbes modulaires Y (Npn), avec n ≥ 0. On dispose de mor-

phismes finis Y (Npn+1)→ Y (Npn), qui induisent des morphismes de trace K2(Y (Npn+1)) →K2(Y (Npn)) et VY (Npn+1)(2) → VY (Npn)(2). On a un diagramme commutatif

(51) K2(Y (Npn+1))⊗Q

reg(p)

Y (Npn+1)//

trace

H1(Qp, VY (Npn+1)(2))trace

K2(Y (Npn))⊗Qreg(p)

Y (Npn)// H1(Qp, VY (Npn)(2)).

D’autre part, on a un morphisme fini Y (Npn)→ Y1(N)⊗Q(ζpn) et par le lemme de Shapiro,

on a un isomorphisme

(52) H1(Qp, VY1(N)⊗Q(ζpn )(2)) ≅H1(Qp(ζpn), VY1(N)(2)).En utilisant la trace VY (Npn) → VY1(N)⊗Q(ζpn ) et l’isomorphisme (52), on obtient un morphisme

(53) H1(Qp, VY (Npn)(2)) → H1(Qp(ζpn), VY1(N)(2)).



En composant (51) et (53), on obtient un diagramme

(54) K2(Y (Npn+1)) //

trace

H1(Qp(ζpn+1), VY1(N)(2))cores

K2(Y (Npn)) // H1(Qp(ζpn), VY1(N)(2)).Le point fondamental est la commutativité du diagramme (54). Pour obtenir un élément

de H1(Qp, VY1(N)(2)), il suffit donc de fabriquer une famille d’éléments de K2(Y (Npn))compatible pour la trace.

Proposition 6.1 ([Kat04], 2.3). — La famille (zNpn)n≥1 d’éléments de K2(Y (Npn))⊗Q

est compatible pour la trace.

Remarque 6.2. — Dans les calculs, il faut faire attention à ce que zN n’est pas égal à la

trace de zNp lorsque p ∤ N . C’est là, d’ailleurs, la propriété clé d’un système d’Euler (voir la

formule (91)).

Pour chaque n, le groupe GL2(Z/NpnZ) agit par automomorphismes sur Y (Npn). Ces

actions sont compatibles via les morphismes canoniques GL2(Z/Npn+1Z) → GL2(Z/NpnZ)et Y (Npn+1) → Y (Npn). Pour tout α ∈ SL2(Z), la famille (α∗zNpn)n≥1 d’éléments de

K2(Y (Npn))⊗Q est donc encore compatible pour la trace.

Définition 6.3. — On note z(p)N,α(2) ∈H1(Qp, VY1(N)(2)) l’image de la famille (α∗zNpn)n≥1

par le diagramme (54). De plus, on note z(p)f,α(2) ∈ H1(Qp, Vf(2)) l’image de z

(p)N,α(2) par le

morphisme induit par VY1(N)(2)→ Vf(2).Compte tenu des isomorphismes (44), on dispose en fait d’éléments z

(p)f,α(k) ∈ H1(Qp, Vf(k))

pour tout k ∈ Z. Il reste à les normaliser. Pour cela, Kato utilise la structure de Λ-module de

H1(Qp, Vf).La classe de la forme différentielle ωf = 2iπf(z)dz définit un élément de H1(Y1(N)(C),C).Cet espace vectoriel est muni d’une appplication C-linéaire c∗, induite par la conjugaison

complexe c sur Y1(N)(C). Pour x ∈H1(Y1(N)(C),C), posons x± = 12(x ± c∗x).

Définition 6.4. — On pose Vf,Q =H1(Y1(N)(C),Q)/⟨Tn − an;n ≥ 1⟩.On a dimQ Vf,Q = 2 et par les théorèmes de comparaison, on peut voir Vf,Q comme

une Q-structure du Qp-espace vectoriel Vf . De plus, on a un isomorphisme Vf,Q ⊗Q C ≅H1(Y1(N)(C),C)/⟨Tn − an;n ≥ 1⟩.Définition 6.5. — On note γE ∈ Vf,Q l’unique élément vérifiant

(55) [ωf ] = γ+E ⊗Ω+E + γ−

E ⊗Ω−E

où [ωf ] désigne l’image canonique de ωf dans Vf,Q ⊗C.



Soit H1(X1(N)(C),ptes,Z) le groupe d’homologie relative de X1(N)(C) à support dans les

pointes. On a une dualité parfaite

(56) H1(X1(N)(C),ptes,Z) ×H1c (Y (N)(C),Z) → Z

où H1c désigne le groupe de cohomologie à support compact. Après extension des scalaires à

C, l’application (56) est donnée par l’intégration. On dispose également de la dualité parfaite

de Poincaré

(57) H1(Y1(N)(C),Z) ×H1c (Y1(N)(C),Z) → Z.

Après extension des scalaires à C, l’application (57) est donnée par (ω,ν) ↦ ∫Y1(N)(C)ω ∧ ν.

On déduit de (56) et (57) un isomorphisme

(58) Ψ ∶H1(X1(N)(C),ptes,Z) ≅Ð→H1(Y1(N)(C),Z).Définition 6.6. — Pour tout α ∈ SL2(Z), notons ξ(α) la classe du chemin α0, α∞ dans

H1(X1(N)(C),ptes,Z) et posons

(59) δ(α) ∶= Ψ(ξ(α−1)) ∈H1(Y1(N)(C),Z).De plus, notons δf(α) l’image de δ(α) dans Vf,Q.

Dans les notations de [Kat04, 5.5,6.3], on a δ(α) = δ1,N(2,1, α) et δf(α) = δ(f,1, α). La

théorie des symboles modulaires [Man72] montre que le groupe H1(Y1(N)(C),Z) est engen-

dré par les δ(α) lorsque α parcourt SL2(Z). En particulier, les δf(α) engendrent le Q-espace

vectoriel Vf,Q.

Définition 6.7. — On pose µ = ∏ℓ∣N,ℓ≠p 1 − aℓℓ−2σ−1ℓ ∈ Λ, où le produit est étendu aux

diviseurs premiers ≠ p de N .

Kato montre [Kat04, 12.5,13.9,13.10] qu’il existe une unique application Qp-linéaire

Vf →H1(Qp, Vf )(60)

γ ↦ z(p)γ

vérifiant

(61) µ ⋅ z(p)

δf (α)= z(p)

f,α(α ∈ SL2(Z)).

Définition 6.8. — On pose z(p)Kato = z(p)γE

∈H1(Qp, Vf).



La propriété fondamentale du système d’Euler de Kato est son lien avec la fonction L p-adique

de E. Plus précisément, Kato montre que la fonction L p-adique de E est l’image de z(p)Kato(2)

par un homomorphisme de Λ⊗Qp-modules

(62) Lη ∶H1(Qp, Vf(2)) →H∞.Ce point de vue donne en fait une nouvelle construction de la fonction L p-adique de E (il est

intéressant de noter qu’aucune hypothèse n’est faite sur le type de réduction de E en p). Le

morphisme Lη, que l’on peut voir comme une extension de l’application logarithme définie en

(23), est défini à l’aide de l’exponentielle de Perrin-Riou. La section suivante est consacrée à

la définition et aux propriétés de Lη.

7. Exponentielle de Perrin-Riou

Nous travaillerons dans cette section avec la représentation VE , mais il est bien sûr possible

de remplacer VE par Vf .

Notons Γ∞ = Gal(Qp/Qp(ζp∞)). D’après [PR92, 2.1.4], on peut identifier V Γ∞E à un sous-

Λ⊗Qp-module de H1(Qp, VE). Perrin-Riou a défini une application exponentielle

(63) ΩVE∶ Dcris(VE)→ (H1(Qp, VE)/V Γ∞

E)⊗Λ⊗Qp H∞

qui interpole, en un sens que nous préciserons plus tard, les applications expn,k pour tout

n ≥ 0 et k ≥ 1. Nous ne détaillerons pas ici la définition de ΩVE, pour laquelle nous renvoyons

à [PR99] et [PR01].

Pour η ∈ Dcris(VE), on peut alors poser

Lη ∶H1(Qp, VE(2)) →H∞(64)

x↦ ⟨x,ΩVE(η)⟩,

où ⟨⋅, ⋅⟩ est le crochet de dualité (49) pour la représentation V = VE(2). Cette définition est

licite car V Γ∞E est de dimension finie sur Qp, donc de torsion dans H1(Qp, VE). Par linéarité

à gauche de ⟨⋅, ⋅⟩, l’application Lη est un morphisme de Λ⊗Qp-modules.

Le théorème suivant est un cas particulier de la loi de réciprocité explicite, conjecturée par

Perrin-Riou et démontrée par Colmez [Col98].

Théorème 7.1. — Supposons que p est impair et que E n’a pas réduction multiplicative

déployée en p. Alors pour tout x ∈H1(Qp, VE(2)) et η ∈ Dcris(VE), on a

(65) Lη(x)(1) = [log π0(x), (1 − p−1ϕ−1)(1 −ϕ)−1η]avec [⋅, ⋅] ∶DdR(VE(2)) ×DdR(VE)→Qp.



Remarque 7.2. — Le fait que la réduction de E n’est pas multiplicative déployée assure que

log π0(x) est bien défini et que 1 − ϕ est inversible sur Dcris(VE), ce qui montre que (65) a

bien un sens. Dans le cas multiplicatif déployé, il devrait être possible également de calculer

Lη(x), puisque la loi de réciprocité de Colmez s’applique à toute représentation de de Rham.

Plaçons-nous sous les hypothèses du théorème 7.1. Aucune puissance de p n’est alors valeur

propre de ϕ sur Dcris(VE) et cela entraîne V Γ∞E = 0 [PR94, 3.4.3].

Soit η ∈ Dcris(VE). Posons g = (1 + x) ⊗ η ∈ H ⊗Qp Dcris(VE). Nous allons définir les objets

intervenant dans la construction de ΩVE(η) = ΩVE

(g).Le Frobenius ϕ ∶H → H est le morphisme injectif d’anneaux défini par

ϕ(f) = f((1 + x)p − 1) (f ∈H).Il existe une unique application Qp-linéaire ψ ∶H →H telle que

ϕ(ψ(f)) = 1

p∑ζ∈µp

f(ζ(1 + x) − 1) (f ∈H).On vérifie que ψ(1 + x) = 0, ce qui fait que g ∈Hψ=0 ⊗Dcris(VE). De plus ψ ϕ = idH. Notons

Φ l’opérateur ϕ⊗ϕ sur H ⊗Dcris(VE).Lemme 7.3. — Il existe G0 ∈H⊗Dcris(VE) tel que (1 −Φ)G0 = g.Démonstration. — En posant η = (1−ϕ)η′ avec η′ ∈ Dcris(VE), on a g = (1−Φ)(1⊗η′)+x⊗η.La série

(66) ∑n≥0

Φn(x⊗ η) = ∑n≥0

ϕn(x)⊗ϕn(η) = ∑n≥0

((1 + x)pn

− 1)⊗ ϕn(η)converge dans H⊗Dcris(VE) puisque (1+x)pn

−1n→+∞ÐÐÐ→ 0 dansH et que ϕn(η) est bornée dans

Dcris(VE). En notant F la somme de cette série, on a (1−Φ)F = x⊗ η et donc G0 = 1⊗ η′ +F

convient.

L’anneau H est muni d’une dérivation D définie par

(67) Df = (1 + x)f ′(x) (f ∈H).On a Dϕ = pϕD et ψD = pDψ. De plus D ∶H → H est surjective et kerD =Qp. On en déduit

que D est un isomorphisme sur Hψ=0. On étend D à H ⊗Dcris(VE) en posant D =D ⊗ 1.

Lemme 7.4. — Il existe une famille (Gk)k∈Z de H⊗Dcris(VE) telle que pour tout k ∈ Z, on

ait DGk = Gk+1 et (1 − pkΦ)(Gk) = g.Démonstration. — On remarque que Dg = g et que D(1 − pkΦ) = (1 − pk+1Φ)D pour tout

k ∈ Z. Par conséquent, pour tout k ≥ 1, l’élément Gk =DkG0 vérifie bien (1 − pkΦ)Gk = g.Pour les entiers k ≤ 0, raisonnons par récurrence. Supposons (1 − pkΦ)Gk = g avec k ≤ 0.

Comme D est surjective sur H, il existe Gk−1 ∈ H ⊗Dcris(VE) telle que DGk−1 = Gk. Alors



(1−pk−1Φ)Gk−1−g est annulé par D, donc est de la forme 1⊗θ avec θ ∈ Dcris(VE). En écrivant

θ = (1 − pk−1ϕ)θ′, ce qui est possible car les valeurs propres de ϕ ne sont pas des puissances

de p, on vérifie que Gk−1 = Gk−1 − 1⊗ θ′ convient.

Remarquons que η = g(0) = (1 − ϕ)G0(0). On étend ψ à H ⊗Dcris(VE) en posant ψ = ψ ⊗ 1.

Pour k ∈ Z, on a alors ψ(g) = ψ(1 − pkΦ)Gk = ψ(Gk) − (pk ⊗ ϕ)Gk et donc

(68) ψ(Gk) = (pk ⊗ ϕ)Gk (k ∈ Z).En suivant [PR01, 5.2.2], définissons Ξn,k ∈Qp(ζpn)⊗Dcris(VE) par

(69) Ξn,k = (pn(k−1) ⊗ϕ−n)G−k(ζpn − 1) (n ≥ 1, k ∈ Z).Par construction de ΩVE

(η) = ΩVE(g), on a

(70) πn(ΩVE(η)(k)) = (−1)k−1(k − 1)! expn,k Ξn,k (n,k ≥ 1).

C’est en ce sens que ΩVEinterpole les applications expn,k avec k ≥ 1. Pour obtenir des informa-

tions sur πn(ΩVE(η)(k)) lorsque k ≤ 0, on a recours à la loi de réciprocité explicite démontrée

par Colmez [Col98]. Pour n ≥ 0 et k ∈ Z, introduisons l’exponentielle duale

(71) exp∗n,k ∶ H1(Qp(ζpn), VE(k)) →DdR(VE(k)/Qp(ζpn)),

obtenue par dualité à partir de expn,2−k. La loi de réciprocité énoncée dans [PR01, 5.3.2]

donne alors

(72) exp∗n,k πn(ΩVE(η)(k)) = 1

(−k)!Ξn,k (n ≥ 1, k ≤ 0).Pour n = 1 et k = 0, on obtient en particulier

(73) exp∗1,0 π1(ΩVE(η)) = Ξ1,0 = (p−1 ⊗ ϕ−1)G0(ζp − 1).

Lemme 7.5. — Pour tout n ≥ 0, on a un diagramme commutatif

(74) H1(Qp(ζpn+1), VE) exp∗//

cores

Qp(ζpn+1)⊗DdR(VE)tr⊗1

H1(Qp(ζpn), VE) exp∗// Qp(ζpn)⊗DdR(VE)

Notons tr la trace de Qp(ζp) ⊗DdR(VE) à DdR(VE). En appliquant le lemme précédent, il

vient



exp∗ π0(ΩVE(η)) = exp∗ coresπ1(ΩVE

(η))= tr(p−1 ⊗ϕ−1)G0(ζp − 1)= p−1ϕ−1(trG0(ζp − 1)).(75)

En évaluant en x = 0 l’identité

(ϕ⊗ 1)ψG0 = 1

p∑ζ∈µp

G0(ζ(1 + x) − 1),il vient

(ψG0)(0) = 1

p∑ζ∈µp

G0(ζ − 1).En évaluant aussi (68) en 0, on obtient alors

ϕ(G0(0)) = (ψG0)(0) = 1

p(trG0(ζp − 1) +G0(0))

En reportant dans (75) et en tenant compte du fait que G0(0) = (1 −ϕ)−1η, il vient

exp∗ π0(ΩVE(η)) = p−1ϕ−1(pϕ − 1)(1 − ϕ)−1η

= (1 − p−1ϕ−1)(1 − ϕ)−1η.(76)

Nous pouvons finalement donner la démonstration de la formule (65). Soit x ∈H1(Qp, VE(2)).En utilisant la définition (64) de Lη, ainsi que la formule d’évaluation (50), il vient

Lη(x)(1) = ⟨x,ΩVE(η)⟩(1)

= ⟨π0(x), π0(ΩVE(η))⟩0

= [log π0(x), exp∗ π0(ΩVE(η))]

par définition de exp∗. On conclut grâce à (76).

8. Démonstration du théorème principal

Commençons par énoncer le théorème de Kato, qui va nous permettre de démontrer le théo-

rème 4.1.

Théorème 8.1 ([Kat04], 16.6). — Supposons E sans multiplication complexe, et soit z(p)Kato

le système d’Euler de Kato défini dans la section 6. Soit ηα ∈ Dcris(Vf)⊗Qp Qp(α) comme en

(34). Alors pour tout caractère continu χ ∶ G∞ ≅ Z∗p →C∗p, on a

(77) Lp,α(E,χω−1,0) = Lηα(z(p)Kato(2))(χ).



Notons que comme χ est arbitraire, le théorème de Kato donne en fait une expression de

toutes les valeurs spéciales de Lp,α(E,s), en termes du système d’Euler z(p)Kato

.

Choisissons α1, α2 ∈ SL2(Z) tels que δf(α1)+ ≠ 0 et δf(α2)− ≠ 0, et écrivons γE ∈ Vf,Q sous la

forme

(78) γE = b1δf(α1)+ + b2δf(α2)− (bi ∈Q∗).Rappelons que z

(p)Kato= z(p)γE

(définition 6.8) et donc

(79) z(p)Kato(2) = b1z(p)δf (α1)+

(2) + b2z(p)δf (α2)−(2).

Le lemme suivant étudie le comportement de l’application γ ↦ z(p)γ vis-à-vis de la conjugaison

complexe.

Lemme 8.2. — Pour γ ∈ Vf,Q, on a z(p)c∗γ = −σ−1z(p)γ .

Démonstration. — Il suffit de le montrer pour γ = δf(α) avec α ∈ SL2(Z). En posant ǫ =(−1 0

0 1), on a c∗ξ(α) = ξ(α′) avec α′ = ǫαǫ. Puisque la conjugaison complexe sur X1(N)(C)

renverse l’orientation, on a Ψ(c∗γ) = −c∗Ψ(γ) pour γ ∈ H1(X1(N)(C),ptes,Z). Par suite

c∗δf(α) = −δf(α′).Il reste à montrer que z

(p)δf (α′)

= σ−1z(p)δf (α). Comme µ n’est pas diviseur de zéro dans Λ, et grâce

à la propriété (61), il suffit de montrer z(p)f,α′= σ−1z(p)f,α. Comme les unités de Siegel vérifient

g−a,−b = ga,b pour tout a, b, il vient ǫ∗zNpn = zNpn pour tout n, et donc

(80) reg(p)

Y (Npn)((α′)∗zNpn) = ǫ∗ reg

(p)

Y (Npn)(α∗zNpn).

Or le morphisme π ∶ Y (Npn) → Y1(N) ⊗Q(ζpn) vérifie π ǫ = σ−1 π, où l’on a noté σ−1 =−1 ∈ (Z/pnZ)∗ ≅ Gal(Q(ζpn)/Q). On en déduit z

(p)N,α′= σ−1z(p)N,α et le résultat.

Pour z ∈ H1(Qp, Vf(k)) (k ∈ Z), posons z± = 12(z ± σ−1z). D’après le lemme 8.2, on a z

(p)γ± =

(z(p)γ )∓ pour γ ∈ Vf,Q. Remarquons que d’après (45), on a z(2)± = z±(2) pour z ∈H1(Qp, Vf).Il suit

(81) z(p)Kato(2) = b1(z(p)δf (α1)

(2))− + b2(z(p)δf (α2)(2))+.

Appliquons maintenant π0 ∶H1(Qp, Vf(2)) → H1(Qp, Vf(2)). Comme π0(σ−1z) = π0(z) pour

z ∈H1(Qp, Vf(2)), on obtient

(82) π0(z(p)Kato(2)) = b2π0(z(p)δf (α2)

(2)).Écrivons δf(I2) sous la forme



(83) δf(I2) = b′1δf(α1)+ + b′2δf(α2)− (b′i ∈Q).En remplaçant γE par δf(I2) dans le raisonnement ci-dessus, on obtient une formule analogue

à (82), à savoir

(84) π0(z(p)δf (I2)(2)) = b′2π0(z(p)δf (α2)(2)).

En combinant (82) et (84), il vient donc

(85) π0(z(p)δf (I2)(2)) = b′

2

b2π0(z(p)Kato

(2)).Lemme 8.3. — Pour α ∈ SL2(Z), on a

(86) π0(z(p)f,α(2)) = (∏ℓ∣Nℓ≠p

1 − aℓ)π0(z(p)δf (α)(2)).

Démonstration. — En tordant deux fois vers la droite la propriété (61) grâce à (45), on a

(87) z(p)f,α(2) = (µ ⋅ z(p)

δf (α))(2) = µ(−2) ⋅ z(p)

δf (α)(2)

avec µ(−2) = ∏ℓ∣N,ℓ≠p 1 − aℓσ−1ℓ . Il reste à appliquer π0 et à remarquer que π0(µ(−2)) =

∏ℓ∣Nℓ≠p

1 − aℓ.

En appliquant le lemme 8.3 à α = I2 et compte tenu de (85), il vient

(88) π0(z(p)f,I2(2)) = (∏ℓ∣Nℓ≠p

1 − aℓ)b′2b2π0(z(p)Kato(2)).

Dans (88), le membre de gauche est lié au régulateur p-adique, tandis que le membre de droite

va donner la fonction L p-adique, grâce au théorème de Kato.

Considérons le membre de gauche de (88). Par définition de z(p)f,I2

, on a

(89) π0(z(p)f,I2(2)) = reg(p)f(trY (Np)→Y (N)(zNp)).

Lemme 8.4. — On a

(90) reg(p)f(trY (Np)→Y (N)(zNp)) =

⎧⎪⎪⎨⎪⎪⎩(1 − ap + p) reg(p)f (zN) si p ∤N,reg(p)f(zN) si p ∣N.

De plus 1 − ap + p = (1 −α)(1 − pα−1) si p ∤ N .



Démonstration. — Si p divise N , alors trY (Np)→Y (N)(zNp) = zN d’après [Kat04, 2.3], d’où le

lemme dans ce cas.

Si p ∤ N alors d’après [Kat04, 2.4], on a

(91) trY (Np)→Y (N)(zNp) = (1 − T ′(p)(1/p 0

0 1)∗

+ p(1/p 0

0 1/p)∗

) zN ,

où T ′(p) est l’opérateur de Hecke sur Y (N) défini en [Kat04, 2.9]. L’opérateur T ′(p)(1/p 0

0 1)∗

induit l’adjoint de Tp sur VY1(N) [Kat04, 5.4], donc ap sur Vf . Puisque le caractère de f

est trivial, l’automorphisme (1/p 0

0 1/p) induit l’identité sur Vf . Par compatibilité du régu-

lateur étale p-adique aux opérateurs de Hecke et aux automorphismes, on trouve la formule

annoncée.

La dernière assertion résulte du fait que 1−ap+p = det(1−ϕ) = (1−α)(1−β) avec αβ = p.Calculons maintenant la constante rationnelle b′2/b2 dans (88). Remarquons que δf(I2)− =(b′2/b2)γ−E et donc

(92)b′2b2= ⟨δf (I2)−, ωf ⟩⟨γ−

E, ωf ⟩ ,

où l’on considère ωf comme un élément de H1c (Y1(N)(C),C), et où l’application ⟨⋅, ωf ⟩ ∶

Vf,Q → C est induite par (57).

Lemme 8.5. — On a

(93) ⟨δf(I2)−, ωf ⟩ = −L(f,1)Démonstration. — Remarquons que ξ(I2) = 0,∞ est invariant par la conjugaison complexe,

donc δf(I2) = δf(I2)−. On a alors

(94) ⟨δf(I2), ωf ⟩ = ∫Y1(N)(C)

δ(I2) ∧ ωf = ∫ξ(I2)

ωf

par définition de la dualité de Poincaré. Un calcul classique [Man72] donne ∫ ∞0 2iπf(z)dz =−L(E,1), d’où le résultat.

Lemme 8.6. — On a

(95) ⟨γ−E , ωf ⟩ = −iΩ−E

⟨f, f⟩.



Démonstration. — Remarquons que c∗ωf = ωf . D’après la définition 6.8 de γE , on a

(96) γ−E = 1

Ω−Eω−f = 1

2Ω−E(ωf − ωf)

et par suite

(97) ⟨γ−E , ωf ⟩ = 1

2Ω−E∫Y1(N)(C)

−ωf ∧ ωf = −iΩ−E⟨f, f⟩.

On déduit de (92) et des lemmes 8.5 et 8.6 que

(98)b′2b2= L(E,1)Ω−E

i⟨f, f⟩ .

Enfin, considérons le terme π0(z(p)Kato(2)) dans (88).

Proposition 8.7. — On a

(99) [log π0(z(p)Kato(2)), ηα] = 1 −α

1 − p−1α−1⋅Lp,α(E,ω−1,0).

Démonstration. — Par le théorème 7.1 appliqué à x = zKato(2) et η = ηα, on a

(100) Lηα(z(p)Kato(2))(1) = [log π0(z(p)Kato

(2)), 1 − p−1α−11 − α

⋅ ηα].Le théorème 8.1 de Kato pour χ = 1 entraîne Lηα(z(p)Kato

(2))(1) = Lp,α(E,ω−1,0), d’où le

résultat.

En mettant ensemble (88), (89), (98), le lemme 8.4 et la proposition 8.7, on obtient le théorème

4.1.

9. Exemples explicites

Le théorème 4.1 montre un lien entre Lp,α(E,ω−1,0) et le régulateur p-adique de zN ∈K2(Y (N))⊗Q. Puisque le régulateur étale p-adique est compatible aux morphismes de trace

associés aux morphismes finis et localement libres, il est naturel de chercher une formule

exprimant Lp,α(E,ω−1,0) comme le régulateur d’un élément explicite de K2(E)⊗Q.

En général, la paramétrisation modulaire φ ∶ X1(N)→ E est difficile à calculer explicitement.

De plus, et c’est là une difficulté plus sérieuse, la trace est une opération hautement non

triviale au niveau des groupes K2. Il existe un algorithme pour la calculer [RT83], mais il

ne semble pas raisonnable de pouvoir espérer l’utiliser lorsque le degré de φ est grand. Nous

devons donc nous contenter de cas particuliers.



Auparavant, nous montrons qu’il est quand même toujours possible de calculer la trace de zNdans K2(Y0(N)) ⊗Q.

Proposition 9.1. — Soit π ∶ Y (N) → Y0(N) le morphisme canonique, et u = π∗g1,0. On a

la formule

(101) trY (N)→Y0(N)(zN) = 2

ϕ(N)2 u,WNu,où ϕ(N) est l’indicatrice d’Euler de N , et WN est l’involution d’Atkin-Lehner sur X0(N).Démonstration. — Soit v = π∗g0,1. Notons π = π0 π1 avec π1 ∶ Y (N) → Y1(N) et π0 ∶

Y1(N) → Y0(N). Rappelons que Y1(N) (resp. Y0(N)) est le quotient de Y (N) par le sous-

groupe suivant de GL2(Z/NZ) :

(102) Γ1 = (∗ ∗0 1

) (resp. Γ0 = (∗ ∗0 ∗)).

D’autre part, les unités de Siegel vérifient ga,b∣γ = g(a,b)γ pour tout a, b ∈ Z/NZ et

γ ∈ GL2(Z/NZ). On en déduit que g0,1 est invariante par Γ1, d’où g0,1 = π∗1w avec

w ∈O∗(Y1(N))⊗Q. Alors

(103) (π1)∗zN = (π1)∗g1,0,wD’autre part

(104) π∗1(π1)∗g1,0 = ∏a∈(Z/NZ)∗

b∈Z/NZ

ga,b

est invariante par Γ0, d’où π∗1(π1)∗g1,0 = π∗g avec g ∈ O∗(Y0(N)) ⊗Q. Par injectivité de π∗1 ,

on a (π1)∗g1,0 = π∗0g, d’où

(105) π∗zN = g, (π0)∗w.On a en fait

(106) u = (π0)∗(π1)∗g1,0 = (π0)∗π∗0g = g ⊗ (deg π0)et aussi

(107) v = (π0)∗(π1)∗g0,1 = (π0)∗(π1)∗π∗1w = (π0)∗w ⊗ (deg π1),d’où l’on déduit



(108) π∗zN = 1

degπu, v.

Insistons sur le fait que la formule (108), qui peut sembler naturelle à première vue, est

particulière au symbole zN = g1,0, g0,1.On a degπ = card(Γ0/±I2) = Nϕ(N)2/2. Pour montrer (101), il suffit donc d’établir que

v = (WNu)N , ou encore

(109) v( −1Nτ) = u(τ)N (τ ∈ h),

où h désigne le demi-plan de Poincaré, et l’égalité étant entendue dans C∗ ⊗Q. En notant

ν ∶ h→ Y (N)(C) l’application canonique, on a d’une part

u(τ) = (π∗u)(ν(τ)) = ( ∏γ∈Γ0/±1

g1,0∣γ)(ν(τ))(110)

= ∏a∈(Z/NZ)∗

b∈Z/NZ

ga,b(ν(τ))ϕ(N)/2.

D’autre part, on a ν(−1/τ) = σ ⋅ ν(τ) avec σ = (0 −1

1 0), et donc

v( −1Nτ) = (π∗v)(ν( −1

Nτ)) = ( ∏

γ∈Γ0/±1

g0,1∣γσ)(ν(Nτ))(111)

= ∏a∈(Z/NZ)∗

ga,0(ν(Nτ))Nϕ(N)/2.En utilisant l’expression des unités de Siegel en termes de q-produits [Kat04, 1.9], on montre

que

(112) ga,0(ν(Nτ)) = ∏b∈Z/NZ

ga,b(ν(τ)) (τ ∈ h)(comparer avec [Kat04, 2.12]), ce qui achève la preuve.

Remarque 9.2. — Supposons N premier. Comme le signe de l’équation fonctionnelle de

L(E,s) est aN , on a toujours (1 − aN)L(E,1) = 0 et le membre de droite de (35) est nul.

La proposition 9.1 confirme ce fait : puisque X0(N) n’a que deux pointes 0 et ∞, qui sont

échangées par WN , l’unité modulaire WNu est de la forme λ/u avec λ ∈ Q∗, et la trace de

zN est proportionnelle à λ,u (on peut montrer que λ = Nϕ(N)/2, mais ce n’est pas essentiel

ici).



Par une construction de Bloch [Nek94, 7.4], on peut ajouter au symbole u,WNu ∈K2(Y0(N))⊗Q de la proposition 9.1 des symboles de la forme λ,w, où λ est une constante

et w est une unité modulaire, de manière à obtenir un élément u,WNu′ ∈K2(X0(N))⊗Q.

Définition 9.3. — Si φ0 ∶ X0(N)→ E est une paramétrisation modulaire, posons

(113) zE,φ0∶= (φ0)∗( 2

φ(N)2 u,WNu′) ∈K2(E)⊗Q.

On déduit du théorème 4.1 et de la proposition 9.1 la formule suivante pour le régulateur

p-adique de zE,φ0. On suppose p2 ∤ N . Soit ηE,α l’unique élément de Dcris(VE) ⊗Qp(α) tel

que

(114) ϕ(ηE,α) = αηE,α et [ωE, ηE,α] = 1

où l’on considère ωE comme élément de DdR(VE) ≅DdR(VE(2)). Posons φ∗0ωE = 2iπcf(z)dzavec c ∈Q∗, et notons c∞ le nombre de composantes connexes de E(R).Théorème 9.4. — Supposons p impair et E sans multiplication complexe. Si E a bonne

réduction en p, alors

[log reg(p)E(zE,φ0

), ηE,α] = 2c

c∞(∏ℓ∣N

1 − aℓ)L(E,1)Ω+E

⋅

⋅ (1 − pα−1)−1(1 − p−1α−1)−1Lp,α(E,ω−1,0),(115)

Si E a réduction multiplicative non déployée en p, alors

[log reg(p)E (zE,φ0

), ηE,α] = 2c

c∞(∏ℓ∣Nℓ≠p

1 − aℓ)L(E,1)Ω+E

⋅

⋅1 − α

1 − p−1α−1Lp,α(E,ω−1,0).

(116)

Démonstration. — Prenons φ = φ0 π0 ∶X1(N)→ E avec π0 ∶X1(N)→ X0(N) le morphisme

canonique. Notons φ∗ ∶ Vf≅Ð→ VE l’isomorphisme induit par φ, et transportons les formules

(35) et (36) dans VE . Par compatibilité du régulateur aux morphismes de trace, on a

(117) φ∗ reg(p)f(zN) = reg

(p)E (zE,φ0

).D’après φ∗ωE = cωf et comme φ∗ φ

∗ est la multiplication par degφ dans VE, donc dans

DdR(VE), il vient φ∗ωf = (degφ/c)ωE et donc ηE,α = (degφ/c)φ∗ηα. En tenant compte de

(37), on obtient les formules annoncées.



Donnons quelques exemples où zE,φ0est non nul et peut être explicité. D’après la formule pour

le régulateur complexe de zE,φ0montrée par Kato, on a zE,φ0

≠ 0 dès que les deux conditions

suivantes sont remplies :

1. Pour tout diviseur premier ℓ de N , on a aℓ ≠ 1.

2. On a L(E,1) ≠ 0.

Pour ne pas avoir à calculer de trace dans K2, considérons le cas où φ0 est un isomorphisme,

c’est-à-dire le cas où E =X0(N). Ces courbes elliptiques ont été étudiées en détail par Ligozat

[Lig75]. La courbe X0(N) est de genre 1 pour douze valeurs de N :

(118) N ∈ 11,14,15,17, 19,20, 21, 24,27, 32,36, 49.Pour ces courbes, la condition (2) est toujours vérifiée et la condition (1) est vérifiée pour six

valeurs de N :

(119) (∀ℓ ∣ N,aℓ ≠ 1)⇔ N ∈ 20,24,27,32, 36,49.Parmi ces six courbes,X0(20) et X0(24) sont sans multiplication complexe. Elles sont données

par les équations minimales sur Z suivantes :

X0(20) ∶ y2 = x3+ x2

+ 4x + 4,(120)

X0(24) ∶ y2 = x3− x2

− 4x + 4.(121)

Considérons la courbe E =X0(20). Un calcul explicite assez long donne

u = ( x − 4

x(x + 1))⊗4

3(122)

W20u = ( (x − y + 2)(x − 4)2x2(x2 + 5y + 7x + 6))⊗

4

3(123)

De plus, en posant zE = zE,φ0, les techniques de Goncharov et Levin [GL98] permettent de

montrer

(124) zE = 1

3x, y + 2x + 2′ ∈K2(E)Z ⊗Q.

On a c = 1 [Lig75, 4.2.7.1], c∞ = 1, a2 = 0, a5 = −1 et L(E,1)/Ω+E = 1/6. En appliquant le

théorème 9.4, on obtient donc

[log reg(p)E x, y + 2x + 2, ηE,α] = 2Lp,α(E,ω−1,0)(1 − pα−1)(1 − p−1α−1) (p ≠ 2,5),(125)

[log reg(p)E x, y + 2x + 2, ηE,α] = 5

3Lp,α(E,ω−1,0) (p = 5).(126)



Pour terminer, signalons le cas N = 64 : la courbe X0(64) n’est autre que la quartique de

Fermat et la trace de z64 définit un élément de K2(X0(64))⊗Q. D’après la formule complexe

de Kato, cet élément est non nul. Il serait intéressant de calculer son régulateur p-adique.

Références

[Bei84] A. A. Beilinson. Higher regulators and values of L-functions. In Current problems in mathe-matics, Vol. 24, Itogi Nauki i Tekhniki, pages 181–238. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. iTekhn. Inform., Moscow, 1984.

[Bes00a] A. Besser. Syntomic regulators and p-adic integration. I. Rigid syntomic regulators. In Pro-ceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jeru-salem, 1998), volume 120 (part B), pages 291–334, 2000.

[Bes00b] A. Besser. Syntomic regulators and p-adic integration. II. K2 of curves. In Proceedings ofthe Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998),volume 120 (part B), pages 335–359, 2000.

[BK90] S. J. Bloch and K. Kato. L-functions and Tamagawa numbers of motives. In The GrothendieckFestschrift, Vol. I, volume 86 of Progr. Math., pages 333–400. Birkhäuser Boston, Boston, MA, 1990.

[Blo00] S. J. Bloch. Higher regulators, algebraic K-theory, and zeta functions of elliptic curves, vo-lume 11 of CRM Monograph Series. American Mathematical Society, Providence, RI, 2000.

[CdS88] R. Coleman and E. de Shalit. p-adic regulators on curves and special values of p-adic L-functions. Invent. Math., 93(2) :239–266, 1988.

[Col98] P. Colmez. Théorie d’Iwasawa des représentations de de Rham d’un corps local. Ann. of Math.(2), 148(2) :485–571, 1998.

[Fal02] G. Faltings. Almost étale extensions. Astérisque, 279 :185–270, 2002. Cohomologies p-adiqueset applications arithmétiques (II).

[FPR94] J.-M. Fontaine and B. Perrin-Riou. Autour des conjectures de Bloch et Kato : cohomologiegaloisienne et valeurs de fonctions L. In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos.Pure Math., pages 599–706. Amer. Math. Soc., Providence, RI, 1994.

[Gea03] M. T. Gealy. Special values of p-adic L-functions associated to modular forms. Preprint, 2003.[Gea05] M. T. Gealy. On the Tamagawa Number Conjecture for Motives Attached to Modular Forms.PhD thesis, California Institute of Technology, Pasadena, California, December 2005. Available athttp://resolver.caltech.edu/CaltechETD:etd-12162005-124435.

[GL98] A. B. Goncharov and A. M. Levin. Zagier’s conjecture on L(E,2). Invent. Math., 132(2) :393–432, 1998.

[Jan88] U. Jannsen. Continuous étale cohomology. Math. Ann., 280(2) :207–245, 1988.[Kat04] K. Kato. p-adic Hodge theory and values of zeta functions of modular forms. Astérisque,295 :ix, 117–290, 2004. Cohomologies p-adiques et applications arithmétiques (III).

[Lig75] G. Ligozat. Courbes modulaires de genre 1. Société Mathématique de France, Paris, 1975.Bull. Soc. Math. France, Mém. 43, Supplément au Bull. Soc. Math. France Tome 103, no. 3.

[Man72] Y. I. Manin. Parabolic points and zeta functions of modular curves. Math. USSR Izvestija,6(1) :19–64, 1972.

[Man73] Y. I. Manin. Periods of cusp forms, and p-adic Hecke series. Mat. Sb. (N.S.), 92(134) :378–401, 503, 1973.

[MSD74] B. Mazur and P. Swinnerton-Dyer. Arithmetic of Weil curves. Invent. Math., 25 :1–61, 1974.[MTT86] B. Mazur, J. Tate, and J. Teitelbaum. On p-adic analogues of the conjectures of Birch andSwinnerton-Dyer. Invent. Math., 84(1) :1–48, 1986.


http://resolver.caltech.edu/CaltechETD:etd-12162005-124435


[Nek94] J. Nekovář. Beilinson’s conjectures. In Motives (Seattle, WA, 1991), volume 55 of Proc. Sym-pos. Pure Math., pages 537–570. Amer. Math. Soc., Providence, RI, 1994.

[PR92] B. Perrin-Riou. Théorie d’Iwasawa et hauteurs p-adiques. Invent. Math., 109(1) :137–185,1992.

[PR94] B. Perrin-Riou. Théorie d’Iwasawa des représentations p-adiques sur un corps local. Invent.Math., 115(1) :81–161, 1994. With an appendix by Jean-Marc Fontaine.

[PR99] B. Perrin-Riou. Théorie d’Iwasawa et loi explicite de réciprocité. Doc. Math., 4 :219–273(electronic), 1999.

[PR01] B. Perrin-Riou. Théorie d’Iwasawa des représentations p-adiques semi-stables. Mém. Soc.Math. Fr. (N.S.), 84 :vi+111, 2001.

[RT83] S. Rosset and J. Tate. A reciprocity law for K2-traces. Comment. Math. Helv., 58(1) :38–47,1983.

[Sai97] T. Saito. Modular forms and p-adic Hodge theory. Invent. Math., 129(3) :607–620, 1997.[Sai00] T. Saito. Weight-monodromy conjecture for l-adic representations associated to modularforms. A supplement to : “Modular forms and p-adic Hodge theory” [Invent. Math. 129 (1997), no.3, 607–620 ; MR1465337 (98g :11060)]. In The arithmetic and geometry of algebraic cycles (Banff,AB, 1998), volume 548 of NATO Sci. Ser. C Math. Phys. Sci., pages 427–431. Kluwer Acad. Publ.,Dordrecht, 2000.

[Tsu99] T. Tsuji. p-adic étale cohomology and crystalline cohomology in the semi-stable reductioncase. Invent. Math., 137(2) :233–411, 1999.

18 mars 2010

F. Brunault, Université de Lyon, ÉNS Lyon - UMPA, 46 allée d’Italie, F-69007 Lyon, FranceE-mail : [email protected] Url : http://www.umpa.ens-lyon.fr/~brunault/


SOME FORMULAS OF RAMANUJAN INVOLVING BESSEL

FUNCTIONS

by

Henri Cohen

Abstract. — We generalize a number of summation formulas involving the K-Bessel function,due to Ramanujan, Koshliakov, and others.

Résumé. — Nous généralisons des formules sommatoires faisant intervenir les fonctions K deBessel, et qui sont dues à Ramanujan, Koshliakov et d’autres.

1. Introduction and Tools

In his notebooks, Ramanujan has given a number of interesting summation formulas involvingthe K-Bessel function. These were generalized by Koshliakov and Berndt–Lee–Sohn in [1].The purpose of the present paper is to give a general framework for these formulas, and togeneralize them. We only give a bare sketch of the proofs.

First, we recall that the K-Bessel function Kν(x) with ν ∈ C and x ∈ R>0 can be defined bythe formula

Kν(x) =1

2

∫ ∞

0

tν−1e−(x/2)(t+1/t) dt .

Since K−ν = Kν , we will always assume implicitly that Re(ν) ≥ 0. These functions possessmany important properties, but the most important for us will be their behavior as x → ∞,and to a lesser extent as x → 0: as x → ∞ we have

Kν(x) ∼( π

2x

)1/2

e−x

(note that the right hand side is independent of ν), and as x → 0 we have Kν(x) ∼2−ν−1Γ(ν)x−ν when Re(ν) > 0, and K0(x) ∼ − log(x).

Second, we recall that the Riemann zeta function ζ(s) which is usually defined by the series∑

n≥1n−s for Re(s) > 1, can be extended to the whole complex plane into a meromorphic

function having a single pole, simple, at s = 1, and satisfies the functional equation Λ(1−s) =Λ(s), with Λ(s) = π−s/2Γ(s/2)ζ(s).

Third, we recall that σs(n) denotes the sum of the sth powers of the positive divisors of n:σs(n) =

∑

d|n ds.

60 Some formulas of Ramanujan involving Bessel functions

Fourth, we recall that the real dilogarithm function Li2(x) is defined by Li2(x) = −∫ x0

log(|1−t|)/t dt, so that Li2(x) =

∑

n≥1xn/n2 when |x| ≤ 1. This function has many functional

equations, and in particular Li2(x) + Li2(1/x) = C(x)π2/6 − log2(x)/2, where C(x) = 2 forx > 0 and C(x) = −1 for x < 0, which immediately implies the behavior of Li2(x) whenx → ∞.

Fifth, we define the standard nonholomorphic Eisenstein Series G(τ, s) for Re(s) > 1 andIm(τ) > 0 by

G(τ, s) =1

2

∑ ′(m,n)∈Z2

ys

|mτ + n|2s,

where∑ ′

means that the term (m,n) = (0, 0) must be omitted, and τ = x+iy. This functionis invariant (in τ) under the usual action of SL2(Z), in other words it is a nonholomorphicmodular function of weight 0. It is easy to compute its Fourier expansion, which is given by

G(τ, s) = ζ(2s)ys +π1/2Γ(s − 1/2)

Γ(s)ζ(2s − 1)y1−s

+ 4y1/2πs

Γ(s)

∑

n≥1

σ2s−1(n)

ns−1/2Ks−1/2(2πny) cos(2πnx) .

In particular this expansion shows that G(τ, s) has a meromorphic continuation to the wholecomplex s-plane, with a single pole, at s = 1, which is simple with residue π/2 (note that thisis independent of τ), and if we set

G(τ, s) = π−sΓ(s)G(τ, s)

we have the functional equation G(τ, 1 − s) = G(τ, s).

2. The First Theorem and Specializations

We implicitly consider that variables such as x, b, and so forth are real and strictly positive,in particular different from 0.

Theorem 2.1. — We have

Λ(s)(x(1−s)/2 − x(s−1)/2) + 4x1/2∑

n≥1

σs(n)

ns/2Ks/2(2πnx)

=Λ(−s)(x−(1+s)/2 − x(1+s)/2) + 4x−1/2∑

n≥1

σs(n)

ns/2Ks/2

(

2πn

x

)

.

This immediately follows from the Fourier expansion of G(τ, s).

Corollary 2.2. —

4∑

n≥1

σs(n)

ns/2Ks/2

(

2πn

x

)

= 4x∑

n≥1

σs(n)

ns/2Ks/2(2πnx)

+ Λ(s)(x1−s/2 − xs/2) + Λ(s + 1)(x1+s/2 − x−s/2) .


Henri Cohen 61

We now specialize to the first few integer values of s.

Corollary 2.3. — Let as usual d(n) = σ0(n) be the number of divisors of n. We have

γ − log(4πx) + 4∑

n≥1

d(n)K0

(

2πn

x

)

= x

(

γ − log

(

4π

x

)

+ 4∑

n≥1

d(n)K0(2πnx)

)

.

Corollary 2.4. — We have

π

12x+∑

n≥1

σ(n)

ne−2πn/x =

π

12x +

∑

n≥1

σ(n)

ne−2πnx − log(x)

2.

Both corollaries follow quite easily from the theorem with s = 0 and s = 1 respectively.This second corollary is in fact the transformation formula of the logarithm of Dedekind’s etafunction. More precisely:

Corollary 2.5. — Set

L(x) = −πx

12+∑

n≥1

log(

1 − e−2πnx)

,

which is equal to log(η(ix)). We have

L

(

1

x

)

= L(x) +log(x)

2.

Corollary 2.6. —

π

6(1 − x) + 4x

∑

n≥1

σ2(n)

nK1(2πnx) =

ζ(3)

2π

(

1

x− x2

)

+ 4∑

n≥1

σ2(n)

nK1

(

2πn

x

)

.

This is the theorem for s = 2.

For s ≥ 3 odd, we obtain from the theorem formulas which are closely linked to the theory ofEichler–Shimura of (k−2)-fold integrals of modular forms of weight k, giving pseudo-modularforms of weight 2 − k. For instance, for s = 3 we obtain directly:

Corollary 2.7. —

ζ(3)

2π

(

1

x− x

)

+ 2∑

n≥1

σ3(n)

n2

(

1 +1

2πnx

)

e−2πnx

=π2

90

(

1

x2− x2

)

+ 2∑

n≥1

σ3(n)

n2

(

1 +x

2πn

)

e−2πn/x .

However, as mentioned above, this is not the point. A consequence is the following:

Corollary 2.8. — Set

H(x) = − π3

180x3 − π3

72x +

ζ(3)

2+∑

n≥1

σ3(n)

n3e−2πnx .



Then H ′′′(x) = −(π3/30)E4(ix), where E4(τ) is the usual holomorphic Eisenstein series of

weight 4, and H satisfies the functional equation

H

(

1

x

)

= − 1

x2H(x) ,

in other words is a “pseudo-modular form” of weight −2.

This is quite easily proved by integration, except that there is an a priori unknown “constantof integration”, which in the above corollary is the coefficient of x. To compute this constantexplicitly, we need to compute the asymptotic expansion of both sides, and after some workwe obtain the above formula. As mentioned above, this type of formula exists for all oddintegral values of s, with the series

∑

n≥1(σs(n)/ns)e−2πnx in the right-hand side.

3. The Second Theorem and Specializations

3.1. Integral Representations. — Note the following integral representations of the sumsthat we will consider:

Proposition 3.1. — For x > 0 and all s ∈ C we have∫ ∞

0

dt

ts+1(e2πxt − 1)(e2π/t − 1)= 2xs/2

∑

n≥1

ns/2σ−s(n)Ks(4π√

nx) .

Corollary 3.2. — Under the same conditions, for all j ≥ 0 we have∫ ∞

0

Pj

(

e2πxt)

ts+1(e2πxt − 1)j+1(e2π/t − 1)dt = (−1)j2xs/2

∑

n≥1

ns/2+jσ−s−j(n)Ks(4π√

nx) ,

where Pj(X) is the sequence of polynomials defined by induction by P0(X) = 1 and Pj(X) =X(X − 1)P ′

j−1(X) − jXPj(X) for j ≥ 1.

Corollary 3.3. — We have∫ ∞

0

log(

1 − e−2πxt)

ts+1(e2π/t − 1)= −2xs/2

∑

n≥1

ns−3/2σ1−s(n)Ks(4π√

nx) .

These results are proved by replacing Ks by its integral definition, and then integrating severaltimes.The main results of this section give formulas for the right hand side of the proposition.

3.2. The Case s /∈ Z. —

Theorem 3.4. — Let s /∈ Z such that Re(s) ≥ 0. For any integer k such that k ≥ ⌊(Re(s) +1)/2⌋ we have the identity

8πxs/2∑

n≥1

ns/2σ−s(n)Ks(4π√

nx) = A(s, x)ζ(s) + B(s, x)ζ(s + 1)

+2

sin(πs/2)

(

∑

1≤i≤k

ζ(2i)ζ(2i − s)x2i−1 + x2k+1∑

n≥1

σ−s(n)ns−2k − xs−2k

n2 − x2

)

,


Henri Cohen 63

where

A(s, x) =xs−1

sin(πs/2)− (2π)1−sΓ(s) and B(s, x) = (2π)−s−1Γ(s + 1)

2

x− πxs

cos(πs/2).

Remarks.

(1) In the above theorem, if x = n ∈ Z≥1 the expression (ns−2k − xs−2k)/(n2 − x2) is ofcourse to be interpreted as its limit as x tends to n, in other words as (s/2−k)ns−2k−2.

(2) The condition Re(s) ≥ 0 is not restrictive since the left hand side of the identity isinvariant under the change of s into −s.

(3) Although the theorem is valid when k ≥ ⌊(Re(s)+1)/2⌋, the convergence of the series onthe right hand side is as fast as possible without acceleration only when k ≥ ⌈Re(s)/2⌉,in other words k ≥ ⌊(Re(s) + 2)/2⌋ if Re(s) /∈ 2Z.

The proof of the theorem involves integrating the formula of the theorem of the precedingsection, and doing a careful extension process to show that it is valid for all s as given.The specializations of the above theorem to s = 1/2 and s = 3/2 are as follows:

Corollary 3.5. —

2π∑

n≥1

σ−1/2(n)e−4π√

nx = 2x∑

n≥1

σ−1/2(n)

(n1/2 + x1/2)(n + x)

+

(

1

4πx− πx1/2

)

ζ(3/2) −(

π − 1

x1/2

)

ζ(1/2) .

Corollary 3.6. —

∑

n≥1

σ−3/2(n)(4π√

nx + 1)e−4π√

nx = −4x3∑

n≥1

σ−3/2(n)

(nx1/2 + xn1/2)(n + x)

+

(

3

8π2x+ 2πx3/2

)

ζ(5/2) +

(

2x1/2 − 1

2

)

ζ(3/2) +2π2

3xζ(1/2) .

3.3. The Case s ∈ 2Z. — As usual, we assume that s ≥ 0. The result corresponding tos = 2m with m ∈ Z≥1 is as follows:

Theorem 3.7. — Let m ∈ Z≥1. We have the identity

8(−1)m∑

n≥1

σ2m(n)

nmK2m(4π

√nx) =

4

π2

xm+1∑

n≥1

σ2m(n)

n2m

log(n/x)

n2 − x2−

∑

1≤i≤m−1

ζ(2i)ζ ′(2i − 2m)x2i−m−1

+2

π2xm−1(ζ(2m) log(2πx) + ζ ′(2m)) +

B2m

2mxm

+

(

(−1)m4(2m)!

(2π)2m+2xm+1− xm

)

ζ(2m + 1) .



Note that by the functional equation, if desired we can replace ζ ′(2i − 2m) by its expressionin terms of ζ(2m + 1 − 2i).

For instance, by simple replacement, for s = 2, in other words for m = 1, we obtain

Corollary 3.8. — We have the identity

−8∑

n≥1

σ2(n)

nK2(4π

√nx) =

4

π2x2∑

n≥1

σ2(n)

n2

log(n/x)

n2 − x2

+1

3

(

log(2πx) +ζ ′(2)ζ(2)

)

+1

12x−(

1

2π4x2+ x

)

ζ(3) .

The result corresponding to s = m = 0 is as follows:

Theorem 3.9. — We have

4∑

n≥1

d(n)K0(4π√

nx) =2x

π2

∑

n≥1

d(n)log(n/x)

n2 − x2−(

γ +log(x)

2+

log(2πx1/2)

π2x

)

.

Here again we need to do a careful limiting process.

Corollary 3.10. —

−4π3x−1/2∑

n≥1

n1/2d(n)K1(4π√

nx) =∑

n≥1

d(n) log(n/x)n2 + x2

(n2 − x2)2

−∑

n≥1

d(n)1

n2 − x2+

1

4

(

2 log(2πx1/2) − 1

x2− π2

x

)

.

Simply compute derivatives.

3.4. The Case s ∈ 1 + 2Z. — Once again we assume that s ≥ 0. The result correspondingto s = 2m + 1 with m ∈ Z≥1 is as follows:


8π(−1)mx1/2∑

n≥1

σ2m+1(n)

nm+1/2K2m+1(4π

√nx) = −2xm+2

∑

n≥1

σ2m+1(n)

n2m+2(n + x)

+ 2∑

1≤i≤m

ζ(2i)ζ(2i − 2m − 1)x2i−m−1

+

(

xm + (−1)m+1(2m)!

(2π)2mxm

)

ζ(2m + 1) +B2m+2

2(m + 1)xm+1

+ 2xm+1((log(x) + γ)ζ(2m + 2) + ζ ′(2m + 2))

The result corresponding to m = 0, in other words to s = 1, is as follows:


Henri Cohen 65

Theorem 3.12. — We have the identity

8πx1/2∑

n≥1

σ(n)

n1/2K1(4π

√nx) = −2x2

∑

n≥1

σ(n)

n2(n + x)

+ log(2πx) + γ +1

12x+ 2x((log(x) + γ)ζ(2) + ζ ′(2)) ,

where as usual σ(n) = σ1(n).

Corollary 3.13. — We have the identities

8π2∑

n≥1

σ(n)K0(4π√

nx) =∑

n≥1

σ(n)

(

1

n2− 1

(n + x)2

)

− π2

6− 1

2x+

1

24x2− ((log(x) + γ)ζ(2) + ζ ′(2)) ,

8π3x1/2∑

n≥1

n1/2σ(n)K1(4π√

nx) = −x∑

n≥1

σ(n)

(n + x)3+

π2

12− 1

4x+

1

24x2,

and

16π4∑

n≥1

nσ(n)K0(4π√

nx) =∑

n≥1

σ(n)n − 2x

(n + x)4− 1

4x2+

1

12x3.

3.5. Asymptotics of Sums∑

n≥1σ−m(n)f(x/n). — Set Tm(f) =

∑

n≥1σ−m(n)f(x/n).

A careful study of these sums for a large and useful class of functions f is necessary, so as toobtain their asymptotic expansions as x → ∞. This is quite nontrivial, and requires severalpages of computations. We only give the results for the functions that we need:

Example 1. f(t) = log(1 + t) − t and m ≥ 2. In that case:

Tm(f)(x) = −ζ(m + 1)x log(x) + (−ζ ′(m + 1) + (1 − γ)ζ(m + 1))x

− (ζ(m)/2) log(x) − ζ ′(m)/2 − (log(2π)/2)ζ(m) + o(1) .

Example 2. f(t) = log(t) log(|1 − t2|) and m ≥ 2. In that case:

Tm(f)(x) = (π2/2)ζ(m + 1)x − ζ(m) log2(x) − 2(ζ ′(m) + log(2π)ζ(m)) log(x)

− ζ ′′(m) − 2 log(2π)ζ ′(m) + 2ζ ′′(0)ζ(m) + o(1) .

Example 3. f(t) = Li2(t2) and m ≥ 2. In that case:

Tm(f)(x) = ζ(m) log2(x) + 2(ζ ′(m) + log(2π)ζ(m)) log(x) + ζ ′′(m)

+ 2 log(2π)ζ ′(m) − (2ζ ′′(0) + π2/6)ζ(m) + o(1) .

4. Integration of the Formulas for s ∈ Z

We can integrate the results that we have obtained for m ∈ Z. Integration term by termis trivially justified, but the whole difficulty lies in the determination of the constant ofintegration. This is done by comparing the asymptotic expansions of both sides, that of theright-hand side being obtained thanks to the study mentioned above, and in particular of thespecific examples.



4.1. The Case s ∈ 2Z, s 6= 0. —


4

πx−m+1/2(−1)m−1

∑

n≥1

σ2m(n)

nm+1/2K2m−1(4π

√nx) =

1

π2

∑

n≥1

σ2m(n)

n2m

(

2 log(x

n

)

log

(∣

∣

∣

∣

1 − x2

n2

∣

∣

∣

∣

)

+ Li2

(

x2

n2

)

+ 4∑

1≤i≤m−1

ζ(2i)ζ ′(2i − 2m)x2i−2m

2m − 2i

)

+2

π2((ζ(2m) log(2π) + ζ ′(2m)) log(x) +

ζ(2m)

2log2(x))

− B2m

2m(2m − 1)x2m−1+

(

(−1)m−14(2m − 1)!

(2π)2m+2x2m− x

)

ζ(2m + 1) + C2m ,

where C2m is a constant given by

C2m = ζ ′′(2m) + 2 log(2π)ζ ′(2m) + (π2/6 − 2ζ ′′(0))ζ(2m) .

As mentioned above, the only difficulty is in the computation of C2m, which is done by makingx → ∞ and comparing asymptotic expansions.

4.2. The Case s = 0. —

Theorem 4.2. —

−2

πx1/2

∑

n≥1

d(n)

n1/2K1(4π

√nx)

=1

2π2

∑

n≥1

d(n)

(

2 log(x

n

)

log

(∣

∣

∣

∣

1 − x2

n2

∣

∣

∣

∣

)

+ Li2

(

x2

n2

))

−((

γ − 1

2

)

x +x log(x)

2+

log(2π)

π2log(x) +

log2(x)

4π2

)

+ C0

where C0 is an explicit constant which I have not had the time to compute.

4.3. The Case s = 2m + 1, m ≥ 1. —


Henri Cohen 67

Theorem 4.3. —

4(−1)m+1x−m∑

n≥1

σ2m+1(n)

nm+1K2m(4π

√nx)

= 2∑

n≥1

σ2m+1(n)

n2m+1

(

log(

1 +x

n

)

− x

n

)

+ 2∑

1≤i≤m

ζ(2i)ζ(2i − 2m − 1)

2i − 2m − 1x2i−2m−1

+

(

log(x) + (−1)m(2m − 1)!

(2π)2mx2m

)

ζ(2m + 1) − B2m+2

(2m + 1)(2m + 2)x2m+1

+ 2(x log(x)ζ(2m + 2) + x((γ − 1)ζ(2m + 2) + ζ ′(2m + 2))) + C2m+1

= −2∑

n≥1

log(Γ(1 + x/n))

n2m+1

+ 2∑

1≤i≤m

ζ(2i)ζ(2i − 2m − 1)

2i − 2m − 1x2i−2m−1

+

(

log(x) + (−1)m(2m − 1)!

(2π)2mx2m

)

ζ(2m + 1) − B2m+2

(2m + 1)(2m + 2)x2m+1

+ 2(x log(x)ζ(2m + 2) + x(−ζ(2m + 2) + ζ ′(2m + 2))) + C2m+1

where C2m+1 is a constant given by

C2m+1 = ζ ′(2m + 1) + log(2π)ζ(2m + 1) .

4.4. The Case s = 1. —

Theorem 4.4. — We have the identity

−2∑

n≥1

σ(n)

nK0(4π

√nx) =

∑

n≥1

σ(n)

n

(

log(

1 +x

n

)

− x

n

)

+1

4log2(x) +

log(2π) + γ

2log(x) + C1 −

1

24x

+π2

6x log(x) + x((γ − 1)ζ(2) + ζ ′(2))

where C1 is a constant given by

C1 =5π2

48+

log2(2π)

4+

γ log(2π)

2− γ2

4− γ1

=π2

16− log2(2π)

4+

γ log(2π)

2+

γ2

4− ζ ′′(0) ,

where γ1 = limN→∞(

∑

1≤n≤N log(n)/n − log2(N)/2)

.



For the convenience of the reader, here are some numerical values:

γ1 = −0.0728158454836767248605863758749013191377363383343 · · ·ζ ′′(0) = −2.0063564559085848512101000267299604381989949101609 · · ·

C1 = 2.39247890056761851538254526825624242310295177862365 · · ·

References

[1] B. Berndt, Y. Lee, and J. Sohn, Koshliakov’s formula and Guinand’s formula in Ramanujan’s lost

notebook , in K. Alladi (ed.), Surveys in number theory, Springer (2008), p. 21–42.

[2] H. Cohen, Number Theory vol I: Tools and Diophantine Equations, Graduate Texts in Math. 239,Springer-Verlag (2007).

[3] H. Cohen, Number Theory vol II: Analytic and Modern Tools , Graduate Texts in Math. 240,Springer-Verlag (2007).

January 17, 2010

Henri Cohen, Université Bordeaux I, Institut de Mathématiques, U.M.R. 5251 du C.N.R.S, 351 Cours de laLibération, 33405 TALENCE Cedex, FRANCE • E-mail : [email protected]


SOME REMARKS ON SELF-POINTS ON ELLIPTIC CURVES

by

Christophe Delaunay & Christian Wuthrich

Abstract. — In a previous article ([DW09]), we studied self-points on elliptic curves of primeconductors, p. The non-triviality of these points was proved in general using a local argumentand the modular parametrization over the field Qp. In this paper, we focus on the special case ofNeumann-Setzer curves and we give an alternative proof of the non-triviality of self-points usingthe complex side of the modular parametrization. To obtain this, we prove several estimates,which can be used further to get results about Neumann-Setzer curves and their modularity.

Résumé. — Dans un article précédent ([DW09]), nous avons étudié les self-points des courbeselliptiques de conducteurs premiers, p. La non trivialité de ces points a été établie en généralen utilisant un argument local et la paramétrisation modulaire sur le corps Qp. Dans ce papier,nous nous concentrons sur le cas particulier des courbes de Neumann-Setzer et nous donnonsune démonstration différente de la non-trivialité des self-points grâce à l’aspect complexe de laparamétrisation modulaire. Pour cela, nous obtenons plusieurs estimations que nous utilisonsensuite pour prouver d’autres résultats sur les courbes de Neumann-Setzer et sur leurs aspectsmodulaires.

1. Introduction

Let E be an elliptic curve defined over Q of conductor N . We denote by X0(N) the modularcurve of level N , it is well known, from the modularity properties of E, that there exists amodular parametrization:

ϕ : X0(N) ≻ E

sending the cusp ∞ ∈ X0(N) to the neutral element O of E. A non-cuspidal point y ∈ X0(N)can be understood as an isomorphism class of pairs (F,C) where F is an elliptic curve andC is a cyclic subgroup of order N of F . It is a classical and natural problem to studymiscellaneous properties of the points x = ϕ(y) ∈ E whenever y have some specific and“interesting” description in X0(N).

The first author is supported by the ANR project no. 07-BLAN-0248 "ALGOL". He is also a member of theproject "TraSecure" financed by the "Région Rhônes-Alpes" in France.

70 Some remarks on self-points on elliptic curves

For instance, if y is a cusp in X0(N), the theory of Manin-Drinfeld gives that ϕ(y) is a torsion-point in E and that its order can be controlled. Modular symbols also allow us to computethe point ϕ(y) in this case ([Cre97, chapter 2]).An other example is given by Heegner points. An Heegner point has the form y = (F,C) ∈X0(N), where F and F/C have complex multiplication by the same order O of an imaginaryquadratic field. The Gross-Zagier theorem ([GZ86]) gives necessary and sufficient conditionsthat x = ϕ(y) is a non-torsion point in E(H) where H is some number field associated toO. The Gross-Zagier theorem has many theoretical and explicit applications. In particular,in combination with the complex interpretation of Heegner points, it leads to a very efficientalgorithm for computing a generator of the Mordell-Weil group E(Q) whenever E has analyticrank 1 (for example, [Coh07, chapter 8]).The images of some other natural points have also been considered (see [Har79], [Kur73],etc.) with the perspective to study the rank of the E(L) in some infinite Iwasawa extensionsL. A special case of these points are the so-called “self-points” in the title. They were definedand have also been investigated in [DW09] and [Wut09].

Definition 1.1. — A self-point, PC ∈ E, is a point PC = ϕ(yC) where yC is of the formyC = (E,C) ∈ X0(N).

Note that there are #P1(Z/NZ) cyclic subgroups, C ⊂ E, of order N . The question of the

rank generated by the self-points (and also by the “higher" self-points) has been studied ingenerality in [Wut09]. One of the key ingredient is to determine when the point PC is anon-torsion point. This can indeed be done in most of the cases by considering the modularparametrization over the local field Qp where p is some well-chosen prime dividing N .Whenever the conductor N = p is prime, the local argument is always valid and it can beshown ([DW09]) that the self-points PC are non-torsion points in E(Q(C)), where Q(C) isthe field of definition of C. This implies that the points (PC)C , where C is running throughthe p + 1 cyclic subgroups of order p, generate a group of rank p in E(K) where K is thecompositum of the fields Q(C). Note that the Galois group of K/Q is G ≃ PGL2(Z/pZ).

The aim of this paper is to focus on the special cases when E are Neumann-Setzer curves andto show that by considering the modular parametrization ϕ over the complex field C (ratherthan Qp) may also provide some results on these points.In section 2, we will briefly sum up the results in [DW09] about self-points on elliptic curvesof prime conductor.Neumann-Setzer curves are special curves of prime conductor p and will be described insection 3.Then, we will restrict our attention to these curves. In section 4, we will give a precisedescription of the modular parametrization over C. This will allow us to study the map ϕ.In particular, we will obtain an alternative proof that the self-points are non-torsion.This requires some technical and more or less precise estimates. We will also use them inorder to give additional remarks that are not exactly related to the study of self-points butthat we believe to be interesting nonetheless. In section 5, we will study the growth of themodular degree of the Neumann-Setzer curves and give an explicit way for computing theanalytic order of the Tate-Shafarevich groups of the Neumann-Setzer curves.


Christophe Delaunay and Christian Wuthrich 71

2. Self-points on elliptic curves of prime conductor

Let E be an elliptic curve defined over Q with prime conductor p. We give here some basicproperties related to the self-points of E, see [DW09] for proofs and more details.

The number field Q(E[p]) obtained by adjoining the coordinates of the p-torsion points ofE to Q is Galois and the Galois group Gal(Q(E[p])/Q) is identified with GL2(Fp) via theclassical Galois representation:

ρp : Gal(Q(E[p])/Q) ≻ GL2(Fp),

which is an isomorphism.Let C ⊂ E be a cyclic subgroup of order p, then the field Q(C) is the subfield of Q(E[p])fixed by a Borel subgroup of GL2(Fp) and so it is a primitive non-Galois field of degreep + 1. From the fact that Q(C) does not contain any non-trivial subfield, we can deducethat E(Q(C))tors = E(Q)tors.As C is running through all the p + 1 cyclic subgroups of order p, the fields Q(C) are allconjugate. Their Galois closure is the field K ⊂ Q(E[p]) and Gal(K/Q) is identified withPGL2(Fp) via ρp. Since the map ϕ is defined over Q, the self-points inherit of the algebraicproperties of Q(C):

Proposition 2.1. — We have:

– The point PC lies in E(Q(C)).– The set PCC form a single orbit under the action of Gal(K/Q) in E(K).

It follows immediately from this proposition and from E(Q(C))tors = E(Q)tors that if a self-point PC were a torsion point it would be rational all the other self-points should also berational and equal. This is trivially impossible if deg(ϕ) < p + 1 (see remark 5.1.1 about thisfact). Furthermore, we have that trK/Q PC =

∑

CPC is a torsion point, so if PC were rational

then PC would be a torsion point.

In [DW09], we proved that this case can not occur since we obtained:

Theorem 2.2. — With the previous notations, we have:

– The self-points PC are of infinite order.– The p + 1 self-points PCC generate a rank p group in E(K) and

∑

CPC is the rational

torsion point ϕ(0) ∈ E(Q).

The first point is proved by considering the p-adic interpretation of ϕ. The second point comesfrom a linear argument (using the irreducibility of the Steinberg representation of PGL2(Fp))and from the first point. This second point is in fact a corollary of the first one. In section4, we will give a new proof of the first point using the complex interpretation of the modularparametrization in the case when E is a Neumann-Setzer curve.

3. Neumann-Setzer curves

Let u ∈ Z be an odd integer such that u ≡ 3 (mod 4). Suppose that p = u2+64 is prime. Sucha prime will be called a Neumann-Setzer prime. The Neumann-Setzer curves of conductor p



are the following two isogenous curves:

Ep : y2 + xy = x3 − u + 1

4x2 + 4x − u

Fp : y2 + xy = x3 − u + 1

4x2 − x

We will write E and F if the Neumann-Setzer prime is understood. The curves E and F areisogenous by an isogeny of degree 2.

The discriminant of E is ∆ = −p2 and its j-invariant is −(u2 − 192)3/p2. The 2-divisionpolynomial of E is given by:

P (x) = (4x − u)(x2 + 4)

The group E(Q) contains a rational 2-torsion point which is given by (u/4,−u/8). The twoother points of order 2 are defined over Q(

√−1) and are the points ±(2

√−1,

√−1).

The discriminant of F is ∆ = p and its j-invariant is (u2 +48)3/p2. The 2-division polynomialof F is given by:

P (x) = 4x(

x2 +u

4x − 1

)

.

The points of order 2 of F are:

(0, 0),

(

u +√

p

8,−u +

√p

16

)

and

(

u −√p

8,−u −√

p

16

)

.

So that in this case, Q(E[2]) = Q(√

p).

The curves of prime conductor and, in particular, Neumann-Setzer curves have been studiedby many authors: [Miy73], [Neu71], [Neu73], [Set75], [SW04],... . From these sourcesand from [AU96], we have the following theorem (see [DW09], for details):

Theorem 3.1. — Let p = u2 + 64 be a Neumann-Setzer prime with u ≡ 3 (mod 4). Let Eand F be the Neumann-Setzer curves as above.

– We have E(Q) ≃ F (Q) ≃ Z/2Z.– We have X(E/Q)[2] ≃ X(F/Q)[2] ≃ 0.– The local Tamagawa number of E and F at the prime p is cp = 2.– The curve E is the strong Weil curve in its isogeny class and the Manin constant of E

is equal to 1.– The modular degree of E is even if and only if u ≡ −1 (mod 8).

Furthermore, if G is an elliptic curve of prime conductor such that G(Q)tors ≃ Z/2Z then Gis a Neumann-Setzer curve.

If E is an elliptic curve of prime conductor which is not a Neumann-Setzer curve then E hasconductor 11, 17, 19, 37 or E(Q)tors is trivial. In the last case, E is the only curve in itsisogeny class (and note that its rank can be positive).

It is not known if there exist infinitely many elliptic curves of prime conductor. Nevertheless,classical conjectures can be applied for the number of Neumann-Setzer primes.



Conjecture 3.2. — Let πNS(x) be the number of Neumann-Setzer prime p 6 x and let:

C =1

2

∏

p prime

(

1 − χ(p)

p − 1

)

where χ(·) =(−4

·)

is the primitive Dirichlet character modulo 4. As x ≻ ∞, we have:

πNS(x) ∼ C

∫

√x

2

dt

log t.

The infinite product defining C is converging but is not absolutely convergent since:

1 − χ(p)

p − 1∼ 1 − χ(p)

p

The number C is the Hardy-Littlewood constant of the polynomial x2 + 64 (this is the sameHardy-Littlewood constant as for the polynomial x2 + 1). One can find in [Coh] how tocompute such constants numerically. In particular, we have:

C ≈ 0.686406731409123004556096348363509434089166546754.

Of course, the conjecture above implies the less precise conjectural estimate πNS(x) ∼2C

√x/ log x.

As expected, the comparison of the conjectural estimate with the exact values of the functionπNS(x) for small x is quite convincing.

x 106 1012 1018

πNS(x) 119 53996 34898579

C

∫

√x

2

dt

log t≈ 121.19 ≈ 53969.76 ≈ 34903256.44

4. Complex side of Neumann-Setzer curves

We give a description of the analytic point of view of the modular parametrization. Weassume here that p = u2 + 64 is a Neumann-Setzer prime with u ≡ 3 (mod 4). We considerthe Neumann-Setzer curve E = Ep as in the previous section:

E : y2 + xy = x3 − u + 1

4x2 + 4x − u.



4.1. Complex points of E. — It is well known that there exists an analytic isomorphismbetween the complex points E(C) of E and C/Λ where

Λ = Zω2 ⊕ Zω1

is the period lattice associated to E. This isomorphism is expressed with the Weierstrassfunction and its derivative; we denote it by ℘ so that:

℘ : C/Λ∼≻ E(C).

Following [Coh93, chapter 7], the numbers ω1 and ω2 can be given by:

(1) ω1 =2π

p1/4 agm

(

1,

√

1

2

(

1 + u√p

)

) ∈ R

(2) ω2 =ω1

2+ i · π

p1/4 agm

(

1,

√

1

2

(

1 − u√p

)

)

Here agm(·, ·) denotes the classical arithmetic-geometric mean.

4.2. Complex L-function of E. — We denote by (an)n>1 the coefficients of the L-functionof E:

L(E, s) =∑

n>1

ann−s , for ℜ(s) > 3/2.

It is easy to see that the curve E has split multiplicative reduction at p hence ap = 1. Fromthe modularity of E, the function L(E, s) has an analytic continuation to the whole complexplane and satisfies a functional equation. The sign of this functional equation is ap = +1 sowe have:

Λ(E, s) :=

(√p

2π

)s

Γ(s)L(E, s) = Λ(E, 2 − s).

Furthermore, the function f(τ) =∑

n>1anqn with q = e2iπτ is a newform of weight 2 on

Γ0(p). From the theory of Atkin-Lehner, we have:

f (Wpτ) = −pτ2f(τ)

where Wp =

(

0 −1p 0

)

is the Fricke involution.

4.3. Complex modular parametrization of E. — Let H be the upper half-plane, H =τ ∈ C,ℑ(τ) > 0. The complex points of the space X0(p) can be interpreted as the quotientof H ∪ Q ∪ i∞ by the congruences subgroup Γ0(p). Then the modular parametrization ofE factorizes as ϕ = ℘ φ:

X0(p)ϕ ≻ E(C)

C/Λ

℘

≻

φ

≻



where φ is given by the following converging series for τ ∈ X0(p) \ cusps:φ : X0(p) ≻ C/Λ

τ 7−→∑

n>1

an

ne2iπnτ

The image of the cusp 0 ∈ X0(p) is a rational torsion point, hence ϕ(0) ∈ E(Q)tors. In fact,we have:

φ(0) = L(E, 1) = 2∑

n>1

an

ne−2πn

√p ∈ Z

ω1

2.

So ϕ(0) = k(

u4,−u

8

)

with k = 0 or k = 1.

Proposition 4.1. — If we assume the truth of Birch and Swinnerton-Dyer conjecture for Ethen

ϕ(0) =(u

4,−u

8

)

.

Proof. — Indeed, if we assume the Birch and Swinnerton-Dyer conjecture is valid, then wehave:

L(E, 1) =ω1 · cp

|E(Q)tors|2|X(E/Q)| =

ω1

2|X(E/Q)|

The result follows from the fact the |X(E/Q)| is odd (if finite) by a theorem of Stein andWatkins [SW04].

The index of Γ0(p) in SL2(Z) is p + 1. As a set of representative of SL2(Z) modulo Γ0(p) wetake the matrices:

S0 =

(

1 00 1

)

and Sj =

(

0 −11 j

)

for j = 1, 2, . . . , p.

Let τ0 ∈ H so that j(τ0) is the j-invariant of E, then the analytic interpretation of theself-points (E,C) are the p + 1 points τ0, τ1,...,τp with:

τj = Sjτ0 =−1

τ + jfor j = 1, 2, . . . , p.

The Hecke operator Tp acts on modular forms of weight 2 on Γ0(p), by definition we have:

Tpf(τ) =1

p

p∑

j=1

f

(

τ + j

p

)

.

Since the function f is a Hecke eigenform with eigenvalue ap = 1, we also have Tpf(τ) =apf(τ) = f(τ). Taking the primitive, we deduce:

ϕ(τ) =

p∑

j=1

ϕ

(

τ + j

p

)

.

The constant term in the integration is 0 as it can been seen taking τ = i∞. Furthermore, fis also an eigenform of the Fricke involution so, ϕ Wp(τ) = −ϕ(τ) + ϕ(0). Hence:

ϕ(τ) = −p

∑

j=1

ϕ

(

Wp

(

τ + j

p

))

+ pϕ(0)



Remark that in any case pϕ(0) = ϕ(0) and that Wp

(

τ+jp

)

= Sjτ . So, we have:

Proposition 4.2. — For τ ∈ X0(p), we have:

ϕ(τ) +

p∑

j=1

ϕ(Sjτ) = ϕ(0).

In particular, if we take τ = τ0, we obtain an other approach in the proof of the second pointof theorem 2.2:

Corollary 4.3. — We have:∑

C

PC = ϕ(0).

Note that the proof we have just given for the trK/Q PC is clearly analytic compared to theone given in[DW09]).

4.4. Estimates for ω1 and ω2. — Recall that p = u2 + 64 with u ≡ 3 (mod 4). Hence,we have p > 73 and:

|u|√p

=

√

1 − 64

p= 1 − 32

p+ O

(

1

p2

)

.

We define x+ and x−

by:

x+ =

√

1

2

(

1 +|u|p

)

and x−

=

√

1

2

(

1 − |u|p

)

.

Proposition 4.4. — With the notations above (in particular p > 73), we have:

1 − 7

p6 agm (1, x+) 6 1

Proof. — Let x = x+, we clearly have that x 6 agm(1, x) 61

2(x + 1). We obtain the

proposition by a straight forward study of x in function of p > 73.

For estimating agm(1, x−), we will need the following lemma:

Lemma 4.5. — For x ∈]0, 1], we have:

− log x +3

2log 2 6

π

2

1

agm(1, x)6 − log x +

5

2log 2

Proof. — We let

g(x) =

∫ π/2

0

dt√

cos2 t + x2 sin2 t



so that we have agm(1, x) = π2· 1

g(x). The change of variables t′ = cos t/ sin t gives

g(x) =

∫ ∞

0

dt√

(t2 + 1)(t2 + x2).

Now, we split the integral∫ ∞0

as the sum∫

√x

0+

∫ ∞√x. The change of variables t′ = x/t, for

x > 0, in the latter integral shows that we have

g(x) = 2

∫

√x

0

dt√

(t2 + 1)(t2 + x2).

This gives us the inequalities

2√1 + x

∫

√x

0

dt√t2 + x2

6 g(x) 6 2

∫

√x

0

dt√t2 + x2

and, since we have 2∫

√x

0

dt√t2+x2

= − log x + 2 log(1 +√

1 + x), the lemma follows from

(

− log x + 2 log(1 +√

1 + x)) 1√

1 + x6 g(x) 6 − log x + 2 log(1 +

√1 + x)

and from a study of the functions on the right and on the left of this inequality.

Proposition 4.6. — We have:

1

π

(

log p + log8

25

)

61

agm(1, x−)

61

π(log p + log 2)) .

Proof. — We have log x−

= 1

2log

(

1

2

(

1 −√

1 − 64/p))

and an easy calculation proves that,

for p > 73:16

p6

1

2

(

1 −√

1 − 64/p)

625

p.

Hence,

−1

2log

25

p6 − log x

−6 −1

2log

16

p.

Then, we use lemma 4.5 to conclude.

Corollary 4.7. — Let p = u2 + 64 be a Neumann-Setzer prime with u ≡ 3 (mod 4). Ifu > 0, we have:

2π

p1/4

1

1 − 4

p

6 ω1 62π

p1/4

1

1 − 7

p

.

Whereas, if u < 0:

2

p1/4

(

log p + log8

25

)

6 ω1 62

p1/4(log p + log 2)

Proof. — We apply the two propositions above with the definitions of ω1 and ω2 (see equations(1) and (2)).



Let remark that in any case (u > 0 or u < 0):

(3) ω1 >2π

p1/4

4.5. Proof that the self-points are non-torsion. — We can now prove that the selfpoints are non-torsion using the complex modular parametrization.It follows from E(Q(C))tors = E(Q)tors that if PC were a torsion point then it would be arational torsion point. So, in order to prove that PC is not a torsion point it is sufficient toprove that φ(τ0) 6= 0 (mod 1

2Λ) for a certain τ0 ∈ H such that j(τ0) is the j-invariant of E.

For that we let τ0 = ω2/ω1 if u > 0 and τ0 = (ω2 −ω1)/(2ω2 −ω1) if u < 0. In fact, for u < 0,we have:

τ0 =

(

1 −12 −1

)

ω2

ω1

.

Since the matrix above belongs to SL2(Z), we see that in each case the image of yC = (E,C) ∈X0(p) in C/Λ (for a certain C depending on the sign of u) is given by φ(τ0). It is easy to seethat we have:

τ0 =1

2+ i · agm(1, x+)

2 agm(1, x−)∈ H.

Hence, φ(τ0) is real and we just need to prove that φ(τ0) 6= 0 (mod ω1

2).

Theorem 4.8. — With the notations above, we have the following estimates:

−6

p6 φ(τ0) 6 − 1

10p.

In particular, φ(τ0) 6= 0, ω1/2 (mod ω1) and PC is a non-torsion point.

Proof. — We let t = ℑ(τ0) and q = e2iπτ0 = −e−2πt. From propositions 4.4 and 4.6, we have:

1

2π

(

1 − 7

p

)

log8p

256 t 6

1

2πlog 2p

Hence,

1

2p6 e−2πt

6

(

25

8p

)

1−7/p

.

But, we have(

25

8p

)1−7/p6 5/p whenever p > 73. Finally, we obtain that q < 0 and:

(4)1

2p6 |q| 6

5

p.

We have an

n 6 1 (see [GJP+09]) so φ(τ0) = q + ε where:

(5) |ε| 6∑

n>2

|q|n =|q|2

1 − |q| 625

p(p − 5).

Then (4) and (5) give the inequality for φ(τ0). We deduce that φ(τ0) 6= 0 and that |φ(τ0)| <

π/p1/4 6 ω1/2 by equation (3).



In fact, we have proved that:φ(τ0) = q + O

(

1/p2)

where the constant is absolute and where q ≍ 1

p . We will use that in the next section.

4.6. Isogeneous self-points. — Let F be the Neumann-Setzer curve that is isogeneous toE (see section 3). Then, one can consider the point zD = (F,D) ∈ X0(p) where D ⊂ F isa sub-group of order p. The image QD = ϕ(zD) are also interesting point in E(K). We canuse exactly the same method as before to proof that QD is a non-torsion point for all D. Thequestion of the independence of the p + 1 points QD and the p + 1 points PC is natural. Webelieve that the only relations between those points are given by:

trK/Q PC = trK/Q QD = ϕ(0)

This would follow from the fact that if C and D are chosen so that they are defined over thesame field Q(C) then the points PC and QD are independent in E(Q(C)) (see [DW09]).

Let t0 be as in the previous section so that t0 correspond to

(E,C) =

(

C/Zτ0 ⊕ Z, 〈1p〉)

where 〈1

p〉 denotes the cyclic sub-group of order p generated by 1/p (mod Zτ0 ⊕ Z). For the

curve F , we can choose the point τ ′0

= 2τ0 corresponding to:

(F,D) =

(

C/Zτ ′0⊕ Z, 〈1

p〉)

where 〈1

p〉 denotes here the cyclic sub-group of order p generated by the point 1/p (mod Zτ ′0⊕

Z). The 2-isogeny between F and E correspond to the map induced by the identity:

C/Zτ ′0⊕ Z ≻ C/Zτ0 ⊕ Z

x (mod Zτ ′0⊕ Z) 7−→ x (mod Zτ0 ⊕ Z)

since Zτ ′0⊕ Z ⊂ Zτ0 ⊕ Z. This 2-isogeny being rational, the point (E,C) and (F,D) are

defined over the same field, E(Q(C)) = E(Q(D)).

We have q′ = e2πτ ′

0 = q2 with q = e2iπτ0 from the previous section. Using the same techniqueas in the previous section, we prove that:

φ(τ ′0) = q2 + O

(

1

p4

)

where the implied constant is absolute.

We believe that the points PC and QD are linearly independent in E(Q(C)) but we are notable to prove it. Nevertheless, if there exists a linear relationship of these points, it wouldinvolve rather large coefficients since:

Theorem 4.9. — With the notations above, suppose that there exist ℓ, m ∈ Z \ 0 suchthat:

ℓPC + mQD ∈ E(Q(C))tors

then max(|ℓ|, |m|) ≫ p3/4, where the implied constant is absolute.



Proof. — Recall that E(Q(C))tors = E(Q)tors, hence if ℓPC + mQD ∈ E(Q(C))tors, we wouldhave:

ℓφ(τ0) + mφ(τ ′0) ∈

ω1

2Z.

If ℓφ(τ0) + mφ(τ ′0) = 0 ∈ C then ℓ 6= 0 since φ(τ ′

0) does not correspond to a torsion point and

so:|m||ℓ| =

|φ(τ0)||φ(τ ′

0)| ≫ p ,

and |m| ≫ p.

If ℓφ(τ0) + mφ(τ ′0) = λω1/2 for some λ ∈ Z \ 0 then in this case:

|ℓ|1p

+ |m| 1

p2≫ |ℓφ(τ0) + mφ(τ ′

0)| >ω1

2>

2π

p1/4

so |ℓ| ≫ p3/4 or m ≫ p7/8.

5. Some other consequences

5.1. Growth of the modular degree of Neumann-Setzer curves. — There is an inter-esting consequence about the degree of the modular parametrization of the optimal Neumann-Setzer curve E. Indeed, using our estimates for ω1 and ω2, it is not difficult to see that wehave:

vol(E) = ω1 · ℑ(ω2) ∼2π log(p)√

pas p = u2 + 64 ≻ ∞.

Furthermore, since the Neumann-Setzer curves are semi-stable, the modular degree is givenby the following formula.

deg(ϕ) =p L(Sym2 f, 2)

2π vol(E).

Where L(Sym2 f, s) is the symmetric-square L-function associated to L(E, s) normalized sothat s = 3/2 is the point of symmetry in the functional equation (let’s remark that theconductor being square-free, there is no difference between the primitive and the imprimitivesymmetric-square).

Using the classical upper bound L(Sym2 f, 2) ≪ log(p)3 and the deeper lower boundL(Sym2 f, 2) ≫ 1/ log(p) obtained by Goldfeld, Hoffstein and Lieman [HL94], we deduce:

Theorem 5.1. — Suppose that there are infinitely many Neumann-Setzer primes p then forthe Neumann-Setzer curves E of conductor p we have as p ≻ ∞:

deg(ϕ) ≪ p3/2 log(p)2

deg(ϕ) ≫ p3/2/ log(p)2

Note that the degree of the modular parametrization is “large” because the j-invariants tendto infinity with p. Indeed, in [Del03]), it is proved that:



Theorem 5.2. — Let G be an infinite family of semi-stable elliptic curves G defined over Q

with conductor NG such that the j-invariant of G is bounded for all G ∈ G and such that Gare the strong Weil curves in their isogeny classes. If ϕG denotes the modular parametrizationof G, then as NG ≻ ∞:

deg(ϕG) ≪ N7/6

G (log NG)3

deg(ϕG) ≫ N7/6

G / log NG

In this context, the power 3/2 of theorem 5.1 should be compared with the power 3/2 in theprevious estimates.

5.1.1. Remark. — In fact, Watkins [Wat04] gave a very explicit version of the lower boundL(Sym2 f, 2) ≫ 1/ log p. (He normalized L(Sym2 f, s) so that 1/2 is the point of symmetry.)Using his work and our estimates we have for a Neumann-Setzer curve E of conductor p:

deg(ϕ) > 0.0006 · p3/2

log(p)2.

Hence the inequality deg(ϕ) < p + 1 never occurs for p > 1.8 · 1012 (but there probably existsa much smaller value of B such that deg(ϕ) > p + 1 for all p > B).

5.2. Explicit computations of the analytic order of the Tate-Shafarevich groups ofNeumann-Setzer curves. — Throughout this section, we assume the truth of the Birchand Swinnerton-Dyer conjecture for E. So that we have:

(6) |X(E)| =2L(E, 1)

ω1

Suppose that we want to compute numerically the values of |X(E)|. Then, we have tocompute numerically the series:

L(E, 1) = 2∑

n>1

an

ne−2πn/

√p.

And we need to truncate the series using sufficiently many coefficient. That means that wewrite:

(7)∑

n>1

an

ne−2πn/

√p =

∑

n<N0

an

ne−2πn/

√p + Error

and we need to take N0 sufficiently large to be are able to recognize |X(E)| from equation(6).

Theorem 5.3. — Let η > 0, there exists an explicit K > 0 such that for all Neumann-Setzercurves of conductor p > K it is sufficient to take

N0 >1 + η

4π

√p log p

in equation (7) in order to determine |X(E)|.



Proof. — Let η > 0, it is well known that there is an absolute constant K0 such that |an| 6

n1/2+η for all n > K0 (this is in fact true for all elliptic curves defined over Q; note that forη = 1/2 we can take K0 = 1).

We write S =∑

n<N0

an

n e−2πn/√

p and ε =∑

n>N0

an

n e−2πn/√

p. Hence we have:

1

ω1

(4S − 4|ε|) 6 |X(E)| 61

ω1

(4S + 4|ε|).

Since |X(E)| is an odd square, the length of the interval in the inequality above has to beless than 4 in order to determine |X(E)|. So, we need |ε| < ω1/2 hence, we need:

ε <π

p1/4.

Suppose that√

p > K0 then N0 >√

p and we have:

|ε| 6∑

n>N0

|an|n

e−2πn/√

p6

∑

n>N0

1

n1/2−ηe−2πn/

√p

61

p1−2η

4

(

e−2π/√

p)N0

(

1 − e−2π/√

p) 6

1

p1−2η

4

p1/2

4

(

e−2π/√

p)N0

.

(Note that p > 73). The values of N0 is sufficient for the theorem.

In particular, taking η = 1/2, we need to take N0 > 1

8

√p log p.

Using this, we have computed several values of |X(E)| of Neumann-Setzer curves. We givehere some numerical investigations related to these values. In particular, we consider thestudy of the frequencies of |X(E)| that are divisible by the primes q for q = 3, 5, 7, · · ·(trivially, the case q = 2 is a special one, and there is nothing to say about it). Computingsufficiently enough values of |X(E)|, we can compare these numerical frequencies with theheuristics on Tate-Shafarevich groups ([Del01], [Del07]). In this case, the heuristics predictthat, if q is (an odd) prime, the frequency of occurrences of q | |X(E)| should be given by:

f(q) = 1 −∏

k>1

(

1 − 1

q2k−1

)

=1

q+

1

q3− 1

q5+ · · · .

We first computed the values of |X(E)| for the 53996 Neumann-Setzer curves of conductorp 6 1012. The largest value |X(E)| = 1232 was obtained for p = 974419714193 (u = 987127).It is worth noting that |X(E)| = 1132 occured for u = 984355 (113 is prime). In fact, exceptfor q = 97 and q = 109, all the primes q 6 113 divides |X(E)| for at least one Neumann-Setzercurve E with p 6 1012.

We obtained the following results for the frequency of occurrences of q | |X(E)|:q 3 5 7 11

Frequency of q | |X(E)| 0.353 0.185 0.118 0.056f(q) ≈ 0.361 0.207 0.145 0.092



Expect for p = 3, the numerical values are not so close than the expected ones. Indeed, webelieve that Tate-Shafarevich groups acquire their expected behavior for rather large conduc-tor. To illustrate this, we also computed the orders of 10000 Tate-Shafarevich groups of thefirst Neumann-Setzer curves E having conductor p > 1015. We obtained the following table:

q 3 5 7 11Frequency of q | |X(E)| 0.368 0.198 0.140 0.084

We should mention that the largest value is |X(E)| = 2992 and that the average value forthese 10000 values of |X| is ≈ 1378.

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April 6, 2010

Christophe Delaunay, Université de Lyon, CNRS, Université Lyon 1, Institut Camille Jordan, 43, boulevarddu 11 novembre 1918, F-69622 Villeurbanne Cedex, France. • E-mail : [email protected]

Url : http://math.univ-lyon1.fr/∼delaunay

Christian Wuthrich, Shool of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD,United Kingdom. • E-mail : [email protected]


ANALYSIS OF THE CLASSICAL CYCLOTOMIC APPROACH

TO FERMAT′S LAST THEOREM

by

Georges Gras

Abstract. — We give again the proof of several classical results concerning the cyclotomicapproach to Fermat ′s last theorem using exclusively class field theory (essentially the reflectiontheorems), without any calculations. The fact that this is possible suggests a part of the logicalinefficiency of the historical investigations.We analyze the significance of the numerous computations of the literature, to show how theyare probably too local to get any proof of the theorem. However we use the derivation methodof Eichler as a prerequisite for our purpose, a method which is also local but more effective.Then we propose some modest ways of study in a more diophantine context using radicals; thispoint of view would require further nonalgebraic investigations.

Résumé. — Nous redonnons la preuve de plusieurs résultats classiques concernant l’approchecyclotomique du théorème de Fermat en utilisant exclusivement la théorie du corps de classes(notamment les théorèmes de réflexion), sans aucun calcul. Le fait que ceci soit possible suggèreune part d’inefficacité logique des investigations historiques.Nous analysons la signification de nombreux calculs de la littérature, afin de montrer en quoi ilssont probablement trop locaux pour donner une preuve du théorème. Cependant nous utilisonsla méthode de dérivation d’Eichler comme préalable à notre démarche, méthode aussi locale,mais plus effective.Ensuite, nous proposons quelques modestes voies d’étude, dans un contexte plus diophantien,utilisant des radicaux, point de vue qui nécessiterait d’établir de nouvelles propriétés non algé-briques.

Introduction and Generalities

The classical approaches to Fermat′s last theorem (FLT) are essentially of a p-adic nature in

the pth cyclotomic field; thus these studies turn to be arithmetic modulo p, in which case the

distinction between first and second case is necessary but unnatural as Wiles′s proof suggests.

2000 Mathematics Subject Classification. — 11D41, 11R18, 11R37, 11R29.Key words and phrases. — Fermat ′s last theorem, Class field theory, Cyclotomic fields, Reflection theo-rems, Radicals, Gauss sums.

The author thanks Christian Maire for his interest and comments concerning this didactic paper, RolandQuême for an observation on Wieferich′s criterion, and the Referee for his valuable help and for the correctionsof english.

86 Analysis of the classical cyclotomic approach to Fermat′s last theorem

Even if the starting point is of a global nature (pth powers of ideals, classes, units, logarithmic

derivative of Eichler,. . . ), the conclusion of the study is mostly local (congruences modulo p)

as we can see for instance in Ribenboim and Washington′s books [R, Wa].

We don′t know (for instance in the first case of FLT) if p-adic investigations (Kummer′scongruences, Mirimanoff or Thaine′s congruences, Wieferich or Wendt′s criteria,. . . ) are able,

from a logical point of view, to succeed in proving it. We think that probably not and we think

that all these dramatically numerous necessary conditions can, in some sense, be satisfied in a

very rare “ numerical setting ”, as for the question of Vandiver′s conjecture for which we have

given a probabilistic study in [Gr1, II.5.4.9.2]: the number of favourable primes less than p

(for a counterexample) can be of the form c . log(log(p)), c < 1.

This is to be relativized with the result of Soulé [S] showing (after that of Kurihara [Ku] for

n = 3) that for odd n, the real components Cℓωp−n of the p-class group (1) are trivial for any

large p. This result and the well-known relative case indicate that the probabilities are not

uniform in the following way:

For small values of odd n, the real components Cℓωp−n are trivial (deep result of [Ku, S]) and

for small values of even m, the relative components Cℓωp−m are trivial (because of the evident

nondivisibility by p of the first Bernoulli numbers B2, . . . , Bm0); so that the real compo-

nents Cℓωp−3 , . . . , Cℓωp−n0 , for a small odd n0, and the relative components Cℓωp−2, . . . , Cℓωp−m0 ,

for a small even m0, are trivial, which implies, by reflection, that the real components

Cℓω2, . . . , Cℓωm0 are trivial and the cyclotomic units ηω2 , . . . , ηωm0 are not local pth powers

at p. (2)

In the particular speculative case of the existence of a solution in the first case of Fermat′sequation, from results of Krasner [Kr], [G2], and many authors, for small values of odd n′,the last Bernoulli numbers Bp−n′ must be divisible by p, say Bp−3, . . . , Bp−n′

0for a small odd

n′0, giving the nontriviality of the relative components Cℓω3, . . . , Cℓω

n′

0and the fact that the

cyclotomic units ηωp−3 , . . . , η

ωp−n′

0are local pth powers (but not global pth powers because of

the previous result of Soulé, at least up to min (n0, n′0)), which creates a significant defect for

the probabilities.

As we see from the classical literature, strong diophantine or analytic arguments are absent,

even when the p-rank of the class group is involved since this p-rank is used as a formal

variable. Moreover the second case is rarely studied.

Of course a great part of the point of view developped here is not really new (many papers of

the early twentieth century, contain overviews of our point of view) but we intend to organize

the arguments in a more conceptual and accessible way, mainly to avoid Bernoulli′s numbers

considerations, and to suggest forthcoming studies in a more diophantine or analytic context

by using radicals instead of ideal classes.

(1)Standard definitions with the character of Teichmüller ω and the corresponding eigenspaces Cℓωi , also

denoted Cℓ(i), i = 1, . . . , p− 1; see Not. 2.7, and Th. 2.8, Subsec. 2.3.(2)The equivalence between Cℓωp−k 6= 1 and ηωk being a local pth power (k even) is given by the theory of

p-adic L-functions or the reflection theorem; see Example 2.9.


Georges Gras 87

We will see on this occasion that class field theory, in its various aspects, allows us to find

again all classical technical properties, without dreadful computations.

Some papers already go partially in this direction (e.g. Anglès [A2, A3], Granville [G1, G2],

Helou [He1, He2], Terjanian [Te], Thaine [Th1, Th2, Th3], and many others).

Finally, we must mention that all these studies strongly depend on the base field (here Q)

since it is shown in [A2] that many results or conjectures fail for the Fermat equation over a

number field k 6= Q.

In Section 1 we recall some basic facts for the convenience of the reader; they can also be

found for instance in Washington′s book [Wa].

In Section 2 we recall some very useful properties of class field theory (notion of p-primarity

which avoids painful computations, reflection theorems in the general setting developped in

[Gr1, II.5.4]) and we introduce the radical W associated to a solution in any case of the Fermat

equation.

Then we explain the insufficiency of the local study of FLT, and we put the bases of a global

approach with W which does not separate the first and second cases of FLT. We also examine

the influence of a solution of the Fermat equation on other arithmetic invariants.

In Section 3, for the first case of FLT, we study p-adically the radical W , introduced in

Section 2, and show how Mirimanoff′s polynomials are related to this radical, without use of

Bernoulli′s numbers; moreover we modify these polynomials by introducing the characters of

the Galois group, which illuminates the class field theory context.

From this, we show that the classical Kummer and Mirimanoff congruences are directly the

expression of reflection theorems.

To be complete, we revisit some p-adic studies, as those of Eichler [E1, E2], covering works of

Brückner [Br1, Br2] and Skula [Sk1, Sk2].

We then return to the well-known fact that Wieferich′s criterion is a consequence of reciprocity

law and, in an Appendix, we give a proof suggested by Quême; for this simpler proof, we

interpret, with current technics, some works of Fueter–Takagi (1922) and Inkeri (1948) (see

[R, IX.4]) which do not use reciprocity law.

Finally we give a standard proof of the Germain–Wendt theorem, and introduce some

(perhaps new) ideas to compare Mirimanoff′s polynomials and Gauss′s sums, and to study

“ Mirimanoff′s sums ” defined as sums of roots of unity.

In Section 4, we give some conclusions and prospectives in various directions.

We are aware of the futility of this attempt, but we believe that it can be helpful (or disap-

pointing) for those who wish to pursue this kind of methodologies.

1. Classical results depending on a solution of Fermat′s equation

Let p be a prime number, p > 2. Let a, b, c in Z\0 be pairwise relatively prime integers,

such that ap + bp + cp = 0. In the second case of FLT, we suppose that p | c.



We have the identity:

ap + bp = (a+ b)NK/Q (a+ b ζ) = −cp,where ζ is a primitive pth root of unity, K = Q(ζ), and NK/Q is the norm map in K/Q.

Let p be the unique prime ideal (1 − ζ) Z[ζ] of K dividing p. We have pp−1 = pZ[ζ].

Lemma 1.1. — Let ν be the p-adic valuation of c. If ν ≥ 1, then a + b = pνp−1cp0 and

NK/Q(a + b ζ) = p cp1, with p ∤ c0 c1 and pνc0 c1 = −c. If ν = 0 then a + b = cp0 and

NK/Q(a+ b ζ) = cp1 with c0 c1 = −c.

Proof. — If p | c, there exists i, 0 ≤ i ≤ p − 1, such that a+ b ζi ∈ p; thus a+ b ζj ∈ p for all

j = 0, . . . , p− 1 since a+ b ζj ≡ a+ b ζi mod p for any j.

So p | a+ b and, since p ∤ b, the p-adic valuations of a+ b and b (ζ − 1) are µ(p− 1) for some

µ ≥ 1 and 1, respectively.

Since p > 2, the p-adic valuation of a+ b ζ = a+ b+ b (ζ − 1) is equal to 1 as well as for the

conjugates a+ b ζi, i = 1, . . . , p − 1. The p-valuation of NK/Q(a + b ζ) is thus equal to p − 1

and that of a+ b is µ(p− 1) = (νp− 1)(p − 1), and the lemma follows.

Lemma 1.2. — Let ℓ 6= p be a prime number dividing c. Then ℓ |NK/Q(a+ b ζ) if and only

if ℓ ∤ a+ b (i.e., g.c.d. (c0, c1) = 1). Any ℓ |NK/Q(a+ b ζ) is totally split in K/Q.

Proof. — If ℓ |NK/Q(a + b ζ) we may suppose that a + b ζ ∈ l for a suitable l | ℓ so that ζ is

congruent modulo l to a rational, l is totally split in K/Q, thus ℓ is congruent to 1 modulo p.

The case ℓ ∤ a+ b is clear. If ℓ | a+ b and if l | a+ b ζ for l | ℓ, we get b (ζ − 1) ∈ l (absurd since

ℓ ∤ b.). Thus ℓ ∤ NK/Q(a+ b ζ).

Corollary 1.3. — (i) We have (a+ b ζ) Z[ζ] = p cp1 if p | c, where c1 is an integral ideal prime

to p, and (a+ b ζ) Z[ζ] = cp1 if not. We have NK/Q(c1) = c1.

(ii) Moreover c1 =∏

ℓ|c1 lνℓ , νℓ > 0, where l is, for each ℓ | c1, a suitable (unique) prime ideal

above ℓ.

Proof. — We have only to prove that if l | a + b ζ, then for any conjugate li (by mean of the

automorphism ζ −→ ζi, i 6= 1), we have li ∤ a+b ζ; indeed, if not we would have b (ζ−i−ζ) ∈ l

(absurd). Thus the ideal(

a+b ζ1−ζ

)

Z[ζ] or (a+ bζ) Z[ζ] is characterized by its norm cp1 and is a

pth power.

Remark 1.4. — (i) By permutation we have the following, with evident notations:

a+ b = pνp−1cp0 or cp0, NK/Q(a+ b ζ) = p cp1 or cp1, with − c = c0 c1,

b+ c = ap0, NK/Q(b+ c ζ) = ap

1, with − a = a0 a1,

c+ a = bp0, NK/Q(c+ a ζ) = bp1, with − b = b0 b1,

g.c.d. (a0, a1) = g.c.d. (b0, b1) = g.c.d. (c0, c1) = 1,


Georges Gras 89

(a+ b ζ) Z[ζ] = p cp1 or c

p1, with NK/Q(c1) = c1,

(b+ c ζ) Z[ζ] = ap1, with NK/Q(a1) = a1,

(c+ a ζ) Z[ζ] = bp1, with NK/Q(b1) = b1.

(ii) All the prime numbers dividing a1b1c1 are totally split in K/Q; thus any (positive) divisor

of a1b1c1 is congruent to 1 modulo p.

These computations and the proofs of FLT in particular cases suggest the following conjecture.

Conjecture 1.5. — Let p be a prime number, p > 3, and K = Q(ζ), where ζ is a primitive

pth root of unity. Put p := (1 − ζ) Z[ζ].

Then for x, y ∈ Z\0, with g.c.d. (x, y) = 1, the equation (x + y ζ) Z[ζ] = p zp or zp

(depending on whether x + y ≡ 0 mod (p) or not), where z is an ideal of K prime to p, has

no solution except the trivial cases: x+ y ζ = ±(1 − ζ) and ±(1 + ζ).

In other words, considering the two relations (a+ b ζ) Z[ζ] = pcp1 (or c

p1) and a+ b = pνp−1cp0

(or cp0), equivalent to the existence of a solution of the Fermat equation, we assert that the

second is unnecessary, the first one being equivalent to N(a + b ζ) = p cp1 (or cp1). It is likely

that this conjecture has already been stated, but we have found no reference.

2. Algebraic Kummer theory and reflection theorems

This Section is valid for the two cases of FLT.

2.1. p-primarity – local pth powers. — The following Theorem 2.2 will be essential to

clarify some aspects of ramification in Kummer cyclic extensions of degree p of K. Let Kp be

the p-completion of the field K (see [Gr1, I.6.3] for the classical notion of p-primarity due to

Hasse).

Lemma 2.1. — Let α ∈ K× be prime to p and such that αZ[ζ] is the pth power of an ideal

of K. (3)

The number α is p-primary (i.e., K( p√α )/K is unramified at p) if and only if it is a local

pth power (i.e., α ∈ K×pp ). This happens if and only if α is congruent to a pth power modulo

pp = (p) p.

Proof. — One direction is trivial. Suppose that K( p√α )/K is unramified at p; since α is

a pseudo-unit, this extension is unramified as a global extension and is contained in the p-

Hilbert class field H of K. The Frobenius automorphism of p in H/K depends on the class

of p which is trivial since p = (1− ζ); so p splits totally in H/K, thus in K( p√α )/K, proving

the first part of the proposition. The final congruential condition of p-primarity is well known

(see e.g. [Gr1, Ch. I, § 6, (b)]).

(3)Such numbers are called pseudo–units since units are a particular case; we will use this word to simplify.



Warning: the general condition of p-primarity in K is “ α congruent to a pth power modulo

pp = (p) p ”, but the general condition to be a local pth power at p inK is “ α congruent to a pth

power modulo pp+1 = (p) p2 ”. The fact that “ α is a pseudo-unit of K implies the equivalence ”

is nontrivial and specific of the pseudo-units of the pth cyclotomic field (such studies are given

in [Th3], for special pseudo-units, by means of explicit polynomial computations).

We have the following consequence, due to Kummer for units, which can be generalized to

pseudo-units.

Theorem 2.2. — Every pseudo-unit η of K, congruent to a rational (respectively to a pth

power) modulo p, is p-primary, thus a local pth power at p. If moreover the p-class group of

K is trivial, η is a global pth power.

Proof. — We have, for a suitable rational ρ, ηp−1 ≡ ρp−1 ≡ 1 mod (p) in Z(p)[ζ], where Z(p)

is the localization of Z at p.

Put ηp−1 = 1+p δ, δ ∈ Z(p)[ζ], and (η) = np; taking the norm of the relation (ηp−1) = n(p−1)p

we get NK/Q(ηp−1) = n(p−1)p with np−1 ≡ 1 mod (p), hence 1 ≡ 1 + pTrK/Q(δ) mod (p2)

giving TrK/Q(δ) ≡ 0 mod (p), thus δ ∈ p, proving the first part of the theorem (see Lem. 2.1).

If η ≡ up mod (p), u =∑

ui ζi ∈ Z(p)[ζ], then up ≡ ∑

upi =: ρ ∈ Z(p) modulo p; reciprocally,

η ≡ ρ mod (p) implies η ≡ ρp mod (p).

The extension K( p√η ) is thus unramified; so if the p-class group of K is trivial, this extension

must be trivial, which finishes the proof.

When the p-class group of K is trivial, K is said to be p-regular (in the Kummer sense), which

is here equivalent to its p-rationality; this property implies in general the above result for units.

See [MN], [JN], [GJ] for these notions in general, and [AN] where the Kummer property is

generalized. See Subsections 2.5, (a) and (b) for the study of the invariants T (K) and R2(K)

whose triviality characterizes the p-rationality and the p-regularity (in the K-theory sense),

respectively.

2.2. Introduction of some radicals. — We begin by the following remarks, from a solu-

tion (a, b, c) of the Fermat equation, which are the key of the present study.

Remark 2.3. — (i) We note that we have (a + b ζ)Z[ζ] = pcp1 or c

p1 (see Cor. 1.3, (i),

or Rem. 1.4, (i)). This means that the Kummer cyclic extensions (of degree p or 1)

K( p√

a+ b ζi )/K, i = 1, . . . , p − 1, are p-ramified (i.e. unramified outside p). In the

same way, K( p√

b+ c ζj )/K, K( p√

c+ a ζk )/K, j, k = 1, . . . , p − 1, are p-ramified cyclic

extensions.

(ii) When p | c, the extensions K( p√

b+ c ζj )/K, j = 1, . . . , p − 1, are unramified: indeed we

have b+ c ζj ≡ b mod (p), hence the conclusion with Theorem 2.2.


Georges Gras 91

But we know that these extensions must split at p which implies that necessarily c ≡ 0

mod (p2). (4)

We have c + a ζk = ζk (a + c ζ−k) with a + c ζ−k ≡ a mod (p); thus in the compositum

K( p√ζ , p

√

c+ a ζk ) (where K( p√ζ )/K is also p-ramified) we obtain the unramified extensions

K(

p√

a+ c ζk′)/

K, k′ = 1, . . . , p − 1, and similarly with c+ b ζj.

(iii) If p | c, then from Corollary 1.3, (i), the pseudo-unitsa+ b ζi

1 − ζiare such that

a+ b ζi

1 − ζi=

a+ b

1 − ζi− b ≡ −b mod (p) since a + b is of p-valuation ν p − 1 ≥ 2. Theorem 2.2 implies that

thea+ b ζi

1 − ζiare local pth powers at p and that the extensions K

(

p

√

a+b ζi

1−ζi

)/

K are unramified.

Notation 2.4. — Let Ep be the group of p-units of K. Then Ep = 〈 ζ, 1 − ζ 〉 ⊕ E+, where

E+ is the group of units of the maximal real subfield K+ de K. Put E+ = 〈 εi 〉i=1,...,p−32

, and

for i, j, k = 1, . . . , p − 1, put:

Ω := 〈 a+ b ζi, b+ c ζj, c+ a ζk 〉,Γ := 〈 ζ, 1 − ζ, ε1, . . . , ε p−3

2, a+ b ζi, b+ c ζj, c+ a ζk 〉 = Ep ⊕ Ω,

Wc := 〈 a+ b ζi 〉i .K×p/K×p,

Wa := 〈 b+ c ζj 〉j .K×p/K×p,

Wb := 〈 c+ a ζk 〉k .K×p/K×p,

W := Γ .K×p/K×p.

If p | c (second case of FLT), we introduce the group:

Ωprim := 〈 a+b ζi

1−ζi , b+ c ζj, a+ c ζk 〉, for which Γ = Ep ⊕ Ωprim.

Remark 2.5. — (i) It is easy to see from Corollary 1.3, (ii), that the 3(p−1)+ p+12

elements

ζ, 1 − ζ, ε1, . . . , ε p−32, a + b ζi, b + c ζj, c + a ζk, i, j, k = 1, . . . , p − 1, are multiplicatively

independent and, due to their particular form, the idea is that they are largely independent

in K×/K×p (this is the main diophantine argument).

Unfortunately, this is probably very difficult to prove since it looks like Vandiver′s conjecture

(which applies to the cyclotomic p-units, generated by 1 − ζ and its conjugates, which are

not independent in K×/K×p as soon as Vandiver′s conjecture is false). But in fact we will

see below that the required condition is not the total independence of the above numbers in

K×/K×p because of analytic formulas.

(ii) It is evident that ζ, 1−ζ, ε1, . . . , ε p−32

are independent in K×/K×p since it is by definition

a Z-basis of Ep.

(4) We have b+ c ζ = (b+ c)`

1 + cb+c

(ζ − 1)´

where b+ c = ap0. Let 1 + c

b+c(ζ − 1) = (1+u (ζ − 1))p locally; if

u ≡ u0 mod p, with u0 ∈ Z, then ζ−u0 (1 + u (ζ − 1)) ≡ 1 mod p2, giving 1 + cb+c

(ζ − 1) ≡ 1 mod (p) p2, thus

c ≡ 0 mod (p) p, hence modulo p2.



(iii) We have W = Γ .K×p/K×p and Ep .K×p/K×p ≃ Ep/E

pp ; then:

Γ .K×p/Ep .K×p ≃ Γ/Γ ∩ (Ep .K

×p) ≃ Ω/Ω ∩ (Ep .K×p)

whose order is the degree[

K( p√

Γ ) : K( p√

Ep )]

.

(iv) If p | c, then K( p√

Ωprim )/K is unramified and K( p√

Γ )/K( p√

Ep ) is unramified hence

p-split of degree (Ωprim : Ωprim ∩ (Ep .K×p)) (nonramification and decomposition propagate

by extension), which will be interpreted in Subsection 2.3.

Denote by K( p√W ) the extension K( p

√Γ ). We conclude (Rem. 2.3) that the extension

K( p√W )/K is a Plp-ramified p-elementary abelian extension of K (i.e., abelian of exponent

p), where Plp is the set of places of K above p (here reduced to the singleton p).

2.3. Use of class field theory: abelian Plp-ramification. — Let HPlp be the maximal

Plp-ramified abelian pro-p-extension of K, and let CℓPlp be the generalised p-class group of K

(i.e., the direct limit of the p-ray class groups modulo rays groups of conductor a power of p);

we have:

Gal (HPlp/K) ≃ CℓPlp .

From the general reflection formula proved in [Gr1, II.5.4.1, (iii)] we obtain: (5)

rkp(CℓPlp) − rkp(CℓPlp) = |Plp | + p− 1 − p− 1

2=

p+ 1

2.

Recall that in this formula, CℓPlp (the Plp-class group) is the quotient of the p-class group Cℓby the subgroup generated by the classes of the prime ideals above p, which gives, as we have

seen, CℓPlp = Cℓ.From the above, since K( p

√W ) ⊆ HPlp , we get:

rkp(Cℓ) = rkp(CℓPlp) −p+ 1

2≥ rkp(W ) − p+ 1

2.

Now we can prove the following from a solution (a, b, c) of the Fermat equation:

Theorem 2.6. — Let W be the radical generated, in K×/K×p, by the group of p-units Ep

and the numbers a+ b ζi, b+ c ζj , c+ a ζk, i, j, k = 1, . . . , p− 1. (6)

Then we have the inequalities rkp(W ) ≤ p+12

+ rkp(Cℓ) ≤ p.

If moreover p is regular (i.e., if Cℓ is trivial) then W = Ep/Epp .

Proof. — From many authors (see e.g. [G3] for more history), we know that the relative class

number h−, i.e., the order of the relative class group C− := Ker(

NK/K+ : C −→ C+ := CK+

)

,

is such that log(h−) < p4log(p) which proves that rkp(Cℓ−) ≤ p−1

4. From classical Hecke–

Leopoldt reflection theorem, we get rkp(Cℓ+) ≤ rkp(Cℓ−) giving the (very bad) inequality

rkp(Cℓ) ≤ p−12

, and the first part of the theorem.

(5)For any abelian group A we denote by rkp(A) the Fp-dimension of A/Ap.(6)In the second case of FLT with p | c, a+ b ζ is not a pseudo-unit, but a+b ζ

1−ζ, b+ c ζ, c+ a ζ are pseudo-units;

thus W is generated by 1 − ζ and pseudo-units.


Georges Gras 93

If p is regular we get rkp(W ) ≤ p+12

; since W contains Ep/Epp which is of p-rank p+1

2we have

the equality, proving the theorem.

In the regular case we obtain the following (see Not. 2.4):

(i) First case of FLT. From Remark 2.5, (iii), we obtain Ω ⊂ E .K×p since in the first case

the elements of Ω are pseudo-units. Then in that case, all the elements a+ b ζi, b+ c ζj, and

c + a ζk are of the form ε . αp, ε ∈ E, α ∈ Z[ζ]. Of course, one can take for ε a cyclotomic

unit since the group of cyclotomic units is of prime to p index in E.

(ii) Second case of FLT. From Remark 2.5, (iv), and Theorem 2.2, we obtain Ωprim ⊂ K×p;

so in the second case (with p | c), all the elements a+b ζi

1−ζi , b+ c ζj , and a+ c ζk are global pth

powers,which can perhaps simplify the usual proof.

From this we obtain easily the classical proofs by Kummer of FLT as those given in [W,

Th. 1.1 and Th. 9.3] or in [Hel, Chap. 1, § 8.4].

However, Eichler′s theorem [E1, E2] (i.e., rkp(Cℓ−) ≤ [√p+ 1 − 1.5 ] implies the first case of

FLT), that we will discuss and prove later (Th. 3.14), may be considered as a wide general-

ization of the regular case, but limited to the first case of FLT (see also [W, Th. 6.23] or [R,

IX.7] for similar proofs).

In the general case, the unlikely equality rkp(Cℓ+) = rkp(Cℓ−) used for the proof of Theorem

2.6 supposes the following facts (see [Gr1, II.5.4.9.2]) for which we introduce the characters

of the Galois group:

Notation 2.7. — (i) Let g = Gal (K/Q) and let ω be the character of Teichmüller of g

(i.e., the character with values in µp−1(Qp) such that for the sk ∈ g defined by sk(ζ) = ζk,

k = 1, . . . , p− 1, ω(sk) is the unique (p− 1)th root of unity in Qp, congruent to k modulo p).

We will also write ω(k) := ω(sk).

(ii) Any irreducible p-adic character of g is of the form χ := ωm, for m ∈ 1, . . . , p − 1; we

denote by χ0 the unit character (m = p− 1).

If χ is any p-adic character of g, we put χ∗ := ωχ−1 (reflection character).

(iii) The idempotent corresponding to χ is:

eχ := 1p−1

∑

s∈gχ(s−1) s = 1

p−1

p−1∑

k=1χ−1(k) sk ∈ Zp[g].

The action of eχ on a Zp[g]-module is well-defined; for a Z[g]-module M , we use instead the

Zp[g]-module M ⊗ZZp or the Zp[g]-module M ⊗

ZFp ≃ M/Mp; by abuse of notation we write

Mχ := Meχ for the χ-component of M in the above sense.

For instance, we denote by rkp(Cℓχ) the p-rank of the χ-component Cℓχ of the p-class group Cℓ(

Cℓχ is thus the maximal submodule of Cℓ on which g acts via cs = cχ(s) for all s ∈ g and any

class c ∈ Cℓχ)

.

For the group E of units, Eχ := Eeχ must be interpreted in E ⊗ZZp or E/Ep depending on

the context.

(iv) Let Kχ be the subfield of K fixed by Ker(χ).



To be self-contained, we recall here the main classical results which will be of constant use.

Theorem 2.8 (Prerequisites). — (i) (Kummer duality; see [Gr1, Rem. II.5.4.3]). Let H [p]

be the p-elementary p-Hilbert class field of K, A := Gal(H [p]/K), and R the radical of H [p]

(i.e., A ≃ Cℓ/Cℓp and H [p] = K( p√R )).

For any character χ of g and for χ∗ := ω χ−1 we have the canonical isomorphism of g-modules:

Gal(K( p√

Rχ∗ )/K) ≃ Aχ ·

Then we have Rχ∗ ⊂ Kχ∗ and K( p√

Rχ∗ )/K splits over Kχ ·(ii) (Reflection theorems; see [Gr1, 5.4.9.2, “ Analysis of a result of Hecke ”]). For any even

character χ 6= χ0 and for χ∗ := ω χ−1 we have:

rkp((Y/Yprim)χ∗) = rkp(Cℓχ∗) − rkp(Cℓχ) = 1 − rkp((Y/Yprim)χ),

where Y is the group of pseudo-units of K (elements equal to the pth power of an ideal prime

to p), and where Yprim is the subgroup of p-primary pseudo-units (i.e., local pth powers at p).

(iii) (Main theorem on cyclotomic fields of Thaine–Ribet–Mazur–Wiles–Kolyvagin; see [W,

§ 15.4]). For any even character χ 6= χ0 and for χ∗ := ω χ−1 we have:

• | Cℓχ | = |(

〈 εχ 〉 : 〈 ηχ 〉)

|−1p , where εχ is a generator of Eχ and ηχ = (1 − ζ)eχ.

• | Cℓχ∗ | = | bχ∗ |−1p , where bχ∗ := 1

p

p−1∑

k=1(χ∗)−1(k) k.

The use of the deep result (iii) is not really necessary in this paper but it clarifies the reasonings

since we are only interested by the logical aspects of the influence of a solution of Fermat′sequation on these invariants and not by an optimization of the statements.

Example 2.9. — If for an even χ 6= χ0, the group Cℓχ∗ is nontrivial, there exists a nontrivial

χ∗-pseudo-unit αχ∗ (i.e., αχ∗ /∈ K×p).

If αχ∗ is p-primary then from (i) this defines a χ-unramified cyclic extension of degree p of Kχ;

so that Cℓχ 6= 1 and(

〈 εχ 〉 : 〈 ηχ 〉)

≡ 0 mod (p) from (iii) (counterexample to the Vandiver

conjecture).

If αχ∗ is not p-primary then from (ii) we get rkp((Y/Yprim)χ∗) = 1 and rkp((Y/Yprim)χ) = 0

which implies that all the χ-pseudo-units are p-primary, especially εχ, hence ηχ ∈ 〈 εχ 〉 is also

a local pth power at p. We have obtained a class field theory version of a result given by the

following properties of p-adic L-functions:

Lp(0, χ) ≡ Lp(1, χ) mod (p) [W, Cor. 5.13] ,

Lp(0, χ) = −bχ∗ [W, Th. 5.11] ,

Lp(1, χ) = τ(χ)p

p−1∑

k=1χ−1(k)log(1 − ζk) = τ(χ)

plog(ηp−1

χ ) [W, Th. 5.18] ,

where the Gauss sum τ(χ) is of p-valuation ≤ p−2, giving easily bχ∗ ≡ 0 mod (p) if and only

if ηχ is a local pth power at p (see Subsec. 3.3 and 3.4).


Georges Gras 95

Then from the above, concerning the equality rkp(Cℓ+) = rkp(Cℓ−), we would have, for each

even χ such that rkp(Cℓχ∗) ≥ 1, the alternative rkp(Cℓχ∗) ≥ 2, or rkp(Cℓχ∗) = 1 and in the

writing pCℓχ∗ = 〈 cℓ(aχ∗) 〉 then apχ∗ =: (α) with α p-primary; all this is of course very strong

because of the probabilistic value of rkp(Cℓ+) discussed in “ Introduction and Generalities ”.

We will return to reflection theorem in the proof of Theorems 3.7 and 3.9.

If we refer to [W, § 6.5], the value of rkp(Cℓ) is conjecturally O( log(p)

log(log(p))

)

. With such a result,

the inequality of Theorem 2.6 would be:

rkp(W ) ≤ p+ 1

2+ O

(

log(p)

log(log(p))

)

,

noting that the principal term p+12

comes from the p-units; this means, from Remark 2.5, (iii),

that most of the elements of Ω (see Not. 2.4) are of the form ε . αp, ε ∈ Ep, α ∈ Z[ζ]. In case

Vandiver′s conjecture is satisfied, Theorem 2.6 reduces to:

rkp(W ) ≤ p+ 1

2+

p− 1

4, instead of ≤ p.

It is implausible that the p-rank of the radical W , generated by the images in K×/K×p of

the 3(p− 1) + p+12

multiplicatively independent elements of Γ, could be less than p.

2.4. Comparison of the local and global approaches. — Now we intend to show that

any restriction to the local case leads to the following fact, where Kp is the completion of K

at p:

rkp

(

Gal(

Kp(p√W )/Kp)

)

≤ p;

in other words, the four radicals Wa, Wb, Wc, Ep/Epp become largely dependent by p-

completion of the base field.

More precisely, we have Kp(p√W ) = Kp( p

√

Wp ), where Wp = Γ .K×pp /K×p

p is the local radical

generated by the image in K×p /K

×pp of the 3(p − 1) + p+1

2elements ζ, 1 − ζ, ε1, . . . , ε p−3

2,

a+ b ζi, b+ c ζj, c+ a ζk, i, j, k = 1, . . . , p− 1.

For instance, if p | c, Wp is the local radical generated by Ep (see Rem. 2.5, (iv)).

Since p splits completely in H and is totally ramified in HPlp/H, by local class field theory the

p-rank of Gal(

HPlp/H)

is less than or equal to the p-rank of the inertia group of the maximal

p-ramified abelian pro-p-extension Mp of Kp = Hp, equal to the p-rank of the subgroup of

units of K×p , thus equal to p.

Since Kp( p√

Wp ) = Hp( p√

Wp ) ⊆Mp, this yields as expected:

rkp(Wp) = rkp

(

Gal(

Kp(p√

Wp )/Kp

)

)

≤ p.

Returning to the global situation and using Theorem 2.6, we obtain directly that:

rkp(Wp) ≤ rkp(W ) ≤ p+ 1

2+ rkp(Cℓ) ≤ p,

which is surprising since the global inequality is obtained via an approximate analytic formula.



So in the local situation we only have the following informations:

rkp(Wp) ≤ p+ 1

2+ rkp(Cℓ),

knowing that (in a “numerical” point of view) Wp does not contain more than p independent

elements in K×p /K

×pp , to be compared with the global situation:

rkp(W ) ≤ p+ 1

2+ rkp(Cℓ),

knowing that the p-rank of W in K×/K×p is only limited by 3(p − 1) + p+12

.

In the two directions (local or global), a contradiction (i.e., a proof of FLT) would be obtained

by proving the following inequalities:

(i) In the local case:

rkp(Wp) >p+ 1

2+ rkp(Cℓ),

under the fact that rkp(Wp) is p−δ(p), where the defect δ(p), in the first case of FLT, depends

essentially of the local properties of Mirimanoff′s polynomials (see Th. 3.5 and Th. 3.9), which

gives the sufficient condition to be proved:

δ(p) < p− p+ 1

2− rkp(Cℓ) =

p− 1

2− rkp(Cℓ),

which is unusable with the analytic inequality rkp(Cℓ) ≤ p−12

equivalent to δ(p) = 0.(7)

In the second case of FLT, such a proof is also impossible since, as we have seen, rkp(Wp) ≤rkp(Ep) = p+1

2.

(ii) In the global case, for the two cases of FLT:

rkp(W ) >p+ 1

2+ rkp(Cℓ),

under the fact that rkp(W ) is 3(p− 1) + p+12

−∆(p), where the defect ∆(p) depends on deep

diophantine properties, which gives the sufficient condition to be proved:

∆(p) < 3(p− 1) +p+ 1

2− p+ 1

2− rkp(Cℓ) = 3 (p − 1) − rkp(Cℓ),

realized as soon as ∆(p) < 5 p−12

with the analytic inequality rkp(Cℓ) ≤ p−12

, which may be

provable.

Remark 2.10. — (i) In the previous analysis, one may object that in an evident way, global

radicals and class groups give equivalent informations (in spite of the fact that here we consider

generalized classes), but we insist on the fact that these radicals, hence the corresponding

classes, are of a very special nature (see for instance Conjecture 1.5, specific of this particular

case).

(ii) If we replace the fundamental units εi by the cyclotomic units, we obtain the radical˜W = 〈 ζ, 1− ζn, a+ b ζi, b+ c ζj, c+a ζk 〉 .K×p/K×p, n, i, j, k = 1, . . . , p− 1, all the elements

being of the special form x+ y ζq.

(7)Note that Mirimanoff′s congruences tend to yield a large δ(p).


Georges Gras 97

The radical ˜W is of p-rank 3(p − 1) + p+12

− ˜∆(p), which requires to prove that ˜∆(p) <

3(p − 1) − rkp(Cℓ), with ˜∆(p) ≥ ∆(p) because of possible cyclotomic units being pth powers

of units (defect of Vandiver′s conjecture), which seems to be acceptable, even if ˜∆(p) is not

so good, to perform ˜∆(p) < 5 p−12

.

2.5. Links with other invariants. — Since analytic aspects are important to get good

upper bounds, it is useful to connect (or replace) the classical class group with other invariants.

Moreover, a solution of Fermat′s equation has important consequences on any arithmetic

invariant, as the following ones.

(a) Case of the torsion subgroup of Gal (HPlp/K).

Recall that Gal (HPlp/K) ≃ CℓPlp is isomorphic to Zp+12

p ⊕T , where T is the (finite) p-torsion

subgroup. Thus we get rkp(W ) ≤ rkp(CℓPlp) = p+12

+rkp(T ), giving rkp(T ) ≥ rkp(W )− p+ 1

2.

If G is the Galois group of the maximal Plp-ramified pro-p-extension of K, then the group Gis defined by d generators and r relations, where:

d = rkp(H1(G,Z/pZ)) = rkp(CℓPlp) =

p+ 1

2+ rkp(T ),

r = rkp(H2(G,Z/pZ)),

with the duality H2(G,Z/pZ)∗ ≃ pT (see for instance [Gr1, App., Th. 2.2]), giving:

rkp(H2(G,Z/pZ)) ≥ rkp(W ) − p+ 1

2= 3(p − 1) − ∆(p).

One may expect that there exist some constraints on such cohomology groups.

The field K is said to be p-rationnal (see [MN]) if T = 1, which is equivalent to Cℓ = 1 (K is

p-regular in the Kummer sense).

From the reflection theorem (see [Gr2, Th. 10.10]), we have for any χ with χ∗ = ω χ−1: (8)

rkp(Tχ) = rkp(Cℓχ∗).

From the interpretation of the reflection principle for the groups Cℓχ recalled in the Theorem

2.8, (ii) (see also [Gr1, II.5.4.9.2]), we obtain a similar result between the groups Tχ and Tχ∗:

rkp((Y/Yprim)χ∗) = rkp(Tχ) − rkp(Tχ∗) = 1 − rkp((Y/Yprim)χ),

for any even χ, where Y is the group of pseudo-units and Yprim the subgroup of p-primary

pseudo-units.

Hence, for the group T , the “ Vandiver conjecture ” is T − = 1.

Let us mention the two relations (equalities up to a p-adic unit):

| T + | = | Cℓ+ | . Reg+

Disc+, | T − | =

| Cℓ− |`

Zp log(I−) : Zp log(P−)´

,

(8)For a direct proof, use the fact that the relative component Cℓ−Plpis the sum of T − and of the Galois group

of the compositum of the relative Zp-extensions giving the representation Zp[g]−; the real part Cℓ+Plp

is the

sum of T + and of Zp with trivial character; so [Gr1, Th. II.5.4.5] gives the formula.



where Reg+ is the p-adic regulator, Disc+ the discriminant, of K+, I the group of ideals prime

to p, P the subgroup of principal ideals, of K; if c is the complex conjugation and a an ideal

of K, let n be such that an 1−c2 = (α), then log(a

1−c2 ) := 1

nlog(α) where log is the Iwasawa

logarithm for which log(p) = 0 (note that for the minus part, the units do not enter in the

use of log; see [Gr1, Cor. III.2.6.1, Rem. III.2.6.5] for more details and references).

As for the class group, the existence of a solution of Fermat′s equation has a great influence

on the group T , for instance on the study of the index(

Zp log(I−) : Zp log(P−))

regarding

the relations (x+ y ζ) = p zp1 or z

p1 giving:

log(

z1−c2

1

)

:=1 − c

2

1

plog(x+ y ζ) =

1 − c

2

1

plog

(

1 +y

x+ y(ζ − 1)

)

.

Mention also the following reasoning giving another interpretation of a result of Iwasawa [Iw],

which may have some interest (9):

For an even χ, since Zp log(P−) = log(U−) where U is the group of principal units of Kp, we

obtain easily:

|Tχ∗| =|Cℓχ∗ |

(

eχ∗ .Zplog(I) : eχ∗ . log(U)) ·

The main theorem on cyclotomic fields (see Th. 2.8, (iii)) gives |Cℓχ∗| = |bχ∗ |−1p (the p-part of

the corresponding generalized Bernoulli number bχ∗ ∈ Zp).

We know that for any prime ideal l of K, l 6= p, we have:

l pS = G(l)p Z[ζ],

where S := 1p

∑p−1k=1 k s

−1k is the Stickelberger element (10) and G(l) the Gauss sum:

G(l) := −∑

t∈Fl

ψ(t) ζtr(t)ℓ ,

where Fl is the residue field, ψ the canonical character of order p of F×l , ζℓ a primitive ℓth

root of unity, and tr the trace in the residual extension Fl/Fℓ. Thus taking log we obtain for

all even χ:

eχ∗ . S . log(l) = eχ∗ . bχ∗ . log(l) = eχ∗ . log(G(l)).

Then |bχ∗ |−1p eχ∗ .Zplog(l) = eχ∗ .Zplog(G(l)), thus:

|Tχ∗ | =|bχ∗ |−1

p(

1

|bχ∗ |−1peχ∗ .Zplog (G) : eχ∗ . log (U)

),

where G is the group generated by all the Gauss sums G(l).

(9)From a talk given in 1982 in the University Laval, Québec; published in the mathematical series, No20

(1984), of the department of mathematics.(10)We have eχ∗ . S = bχ∗ . eχ∗ ; this explains that we use a different definition from that of [W] for the

generalized Bernoulli numbers.


Georges Gras 99

So, the Vandiver conjecture for χ even (Cℓχ = Tχ∗ = 1) is equivalent to the fact that

eχ∗ .Zp log (G) = eχ∗ . log(U), and the whole Vandiver conjecture is equivalent to the fact

that the images of the Gauss sums in U generate the minus part of this Zp-module.

(b) Case of the regular and wild kernels.

Recall the fundamental diagram of K-theory, in which WK2(K) is called the wild kernel and

R2(K) the regular kernel in the ordinary sense. We specify the diagram recalled in [Gr1,

II.7.6] to the case of the cyclotomic field K (h is the Hilbert symbol and hreg the regular

Hilbert symbol, which is explicit):

1 // WK2(K) // K2(K)h

//

⊕

v∈Plncµ(Kv) π

//

µ(K)

// 1

1 // R2(K) // K2(K)hreg

//

⊕

v∈Plncµ(Kv)

reg// 1

since (R2 : WK2) = 1 for K = Q(ζ) (use [Gr1, II.7.6.1]).

For R2 we have a Kummer interpretation, coming from results of Tate [Ta], which is given by

the exact sequence:

1 −→ µp ⊗ N2 −−−→ µp ⊗WPlp

f−−−→ pR2 −→ 1,

where WPlp is the initial radical of HPlp/K, f being defined by f(ζ ⊗ α) := ζ , α for all

α ∈ WPlp , and where N2 := α ∈ K×, ζ , α = 1/K×p (Tate′s kernel) is such that (as

g-modules):

µp ⊗ N2 ≃ (µp ⊗ µp) ⊕ µp−12

p .

We then have rkp(R2) = rkp(WPlp) − p+12

= rkp(Cℓ) (see [Gr1, II.7.7.2.2]). More precisely,

using characters, we have here another principle of reflection, since we must associate χ with

χ := ω−1χ = (χ∗)−1, giving for all χ:

rkp(R2, χ) = rkp(Cℓω−1χ) = rkp(Tω2χ−1).

As for the group T , we get for any even χ:

rkp((Y/Yprim)χ∗) = rkp(R2, ω2χ−1) − rkp(R2, ωχ) = 1 − rkp((Y/Yprim)χ),

and “ Vandiver′s conjecture ” for R2 is R−2 = 1.

This can be deduced from the above exact sequence by proving that the groups 〈 ζ 〉⊗ZpCℓ and

pR2 are isomorphic g-modules, which is coherent with the above reflection. Another proof

uses the isomorphism proved by Jaulent [J] between WK2/(WK2)p and 〈 ζ 〉 ⊗

Zp

˜Cℓ, where ˜Cℓ is

the logarithmic p-class group, and the isomorphism ˜Cℓ ≃ Cℓ for K (see [Gr1, Exer. III.7.1]).

The field K is said to be p-regular (in the K-theory sense) if the p-Sylow of the regular kernel

R2 is trivial (see [JN, GJ]); here it is the case if and only if Cℓ = 1.



We have here a complete parallelism between regular kernel and class group (with another

Galois action), which may be interesting by studying for instance the map f on the elements

x+ y ζ of the radical W ⊆WPlp , and so on.

We know that R+2 := R2(K

+) is given by the value at −1 of the Dedekind zeta function ζK+

of K+; more precisely, after the proof of Birch–Tate conjecture by Wiles (on this subject, see

e.g. Greither [Gre]) we get:

|R+2 | =

24 p

2p−1

2

∣

∣ ζK+(−1)∣

∣

(see Washington′s book [Wa, Ch. IV] to compute the analytic expression of |R+2 |). For the

minus part |R−2 |, we don′t know convenient analytic formula as for | T − |; we only have the

isomorphism pR−2 ≃ (〈 ζ 〉 ⊗

ZpCℓ)−.

3. Some classical local considerations revisited (first case of FLT)

To study the p-rank of the radical W we begin with the partial radical Wc, in the first case

of FLT, or the radical generated by Wc and the units.

Thus in this Section we suppose that p ∤ c; so we will have similar results by permutations

of a, b, c with no more global informations as explained in Section 2; moreover, since

a + b ζ = ζ (b + a ζ−1), the radical W contains the conjugates of b + a ζ−1 and we can

add the transpositions of the set a, b, c, so that the reasonings (in the first case of FLT)

are valid for any (x, y) ∈ (a, b), (b, a), (b, c), (c, b), (c, a), (a, c).

3.1. Logarithmic derivative: Mirimanoff ′s polynomials. — We need, once for all, a

convenient characterization of p-primarity; the best way is to use the method of derivation of

Eichler. Everything depends on this.

From a solution (a, b, c) in the first case of Fermat′s equation, we study the relation:

p−1∏

i=1(a+ b ζi)λi = α p, λi ∈ 0, . . . , p− 1, α ∈ Z[ζ]. (11)

Since a+ b is a pth power (Lem. 1.1), it is equivalent to consider, for e := ba+b

:

p−1∏

i=1(1 + e (ζi − 1))λi = β p, λi ∈ 0, . . . , p − 1, β ∈ Z(p)[ζ].

This relation is equivalent to the polynomial relation:

F (X) :=p−1∏

i=1(1 + e (Xi − 1))λi = G(X)p +A(X)Φp(X), G, A ∈ Z(p)[X],

where Φp(X) is the pth cyclotomic polynomial.

(11)Without any change, we can study the same relation in Zp[ζ] instead of Z[ζ]; in that case, we will obtain

a N.S.C. (see Th. 3.5).


Georges Gras 101

Lemma 3.1. — We can choose G(X) modulo Φp(X) such that:

F (X) =p−1∏

i=1(1 + e (Xi − 1))λi = H(X)p +B(X) (Xp − 1), H, B ∈ Z(p)[X].

Proof. — Since F (1) = 1, G(1)p + A(1) p = 1, thus G(1)p ≡ G(1) ≡ 1 mod (p) and G(1) =

1 + Λ p, Λ ∈ Z(p). Put G1(X) := G(X) − ΛΦp(X); this yields to G1(1) = G(1) − Λ p = 1.

We have F (X) = G1(X)p +A1(X)Φp(X) for some A1(X).

We obtain F (1) = 1 = G1(1)p + A1(1) p = 1 + A1(1) p, in other words A1(1) = 0. Thus

A1(X) = (X − 1)B(X). We then put H(X) := G1(X).

By logarithmic derivation, since e 6≡ 0 mod (p) in the first case of FLT and since F (X) is

invertible modulo (p,Xp − 1), this gives:

p−1∑

i=1

λi iXi−1

1 + e (Xi − 1)∈ (p,Xp − 1)Z(p)[[X]]. (1)

Remark 3.2. — From this formula we deduce (taking X = 1) the necessary condition∑p−1

i=1 λi i ≡ 0 mod (p), which gives one nontrivial relation between the λi. This relation

is due to an obstruction on the ω-component (see Rem. 3.4 ).

The interest of Lemma 3.1 is that (p,Xp − 1)′ ⊆ (p,Xp − 1).

The series1

1 + e (Xi − 1)=

∑

j≥0(−1)j ej (Xi − 1)j are convergent for the (X − 1)-adic topology

and, since (Xi − 1)p ∈ (p,Xp − 1) = (p, (X − 1)p), we obtain, after multiplication by X, the

equivalent condition:

p−1∑

i=1λi iX

ip−1∑

j=0(−1)j ej (Xi − 1)j ∈ (p, (X − 1)p).

Thus, using (Xi − 1)j =∑

k≥0(−1)j−k

(j

k

)

Xik, with(j

k

)

= 0 for k > j, this yields:

∑

k≥0

p−1∑

i=1λi iX

i(k+1) . (−1)kp−1∑

j=k

(j

k

)

ej ∈ (p, (X − 1)p).

Since j ≤ p− 1 and(j

k

)

= 0 for k > j, we can limit k to the value p− 1; for k = p− 1 we get

the termp−1∑

i=1λi iX

ip ep−1 ≡p−1∑

i=1λi i ≡ 0 mod (p,Xp − 1).

Then, under the condition∑p−1

i=1 λi i ≡ 0 mod (p), we can suppose that k varies from 0 to

p− 2. Put:

ϕk+1(X) :=p−1∑

i=1λi iX

i(k+1), k = 0, . . . , p − 2.

We obtain the following condition (2), equivalent to (1) under the condition∑p−1

i=1 λi i ≡0 mod (p):

p−2∑

k=0ϕk+1(X) . Ak ∈ (p, (X − 1)p), with Ak := (−1)k

p−1∑

j=k

(j

k

)

ej . (2)



Lemma 3.3. — We have Ak ≡ (−1)k ek

k!D(k)(e) ≡

(−ba

)kmod (p), k = 0, . . . , p − 2, where

D(Y ) := 1 + Y + . . . + Y p−1.

Proof. — We have A0 = D(e) = ep−1e−1

≡ 1 mod (p) since e 6≡ 1 mod (p) (otherwise a ≡0 mod (p)). The first general relation giving Ak is immediate by induction, using

(j

k

)

=j!

k! (j−k)!and D(k)(Y ) = k!

0!+ (k+1)!

1!Y + . . .+ (k+p−1−k)!

(p−1−k)!Y p−1−k.

Since D(Y ) = 1 + Y + . . . + Y p−1 = Y p−1Y −1

≡ (Y − 1)p−1 mod pZ[Y ], we have D(k)(e) ≡(p− 1) . . . (p− k) . (e − 1)p−1−k ≡ (−1)k k! (e − 1)−k mod (p).

Then Ak ≡ (−1)k ek

k!D(k)(e) ≡

(

ee−1

)k=

(−ba

)kmod (p), hence the result.

We intend to use this formula in the case of the action of the idempotents eχ ∈ Zp[g], χ = ωm

(where g = Gal (K/Q)) on the previous pseudo-unit 1 + e (ζ − 1) (see Not. 2.7).

The formulation of the condition F (X) = H(X)p +B(X)(Xp − 1) corresponds to the choice

λi ≡ 1p−1

ω−m(i) modulo pZp[ζ]; the necessary condition∑p−1

i=1 λi i ≡ 0 mod (p) (see Rem. 3.2)

is satisfied for any m ∈ 1, . . . , p− 1, except m = 1 (i.e., χ = ω).

For m 6= 1 we obtain from the above:

ϕk+1(ζ) = 1p−1

p−1∑

i=1ω−m(i) i ζi(k+1) ≡ 1

p−1

p−1∑

i=1ω1−m(i) ζi(k+1) mod (p),

p−2∑

k=0ϕk+1(ζ) . Ak =

p−1∑

k=1ϕk(ζ) . Ak−1 ≡ 1

p−1

p−1∑

k=1

(p−1∑

i=1ω1−m(i) ζik

)

. Ak−1 mod (p).

We have obtained the necessary condition (put j := i k modulo p):

−p−1∑

k=1

p−1∑

j=1ω1−m(jk−1) ζj . Ak−1 =

(p−1∑

k=1ωm−1(k) . Ak−1

)(

−p−1∑

j=1ω1−m(j) ζj

)

≡ 0 mod (p),

where:

−p−1∑

j=1ω1−m(j) ζj =: τ(ω1−m),

is the Gauss sum of ω1−m, for which:

τ(ω1−m) . τ (ωm−1) = p, where τ(ϕ) := −p−1∑

k=1ϕ(k)ζ−k = ϕ(−1) τ(ϕ)

for any character ϕ.

But τ(ω1−m), as element of Zp[ζ], is of p-valuation m− 1, m ∈ 1, . . . , p− 1 (see Prop. 3.17

in Subsec. 3.4). The final necessary condition is thus, for m 6= 1:

p−1∑

k=1ωm−1(k) . Ak−1 ≡ 0 mod pZp[ζ]. (3)


Georges Gras 103

Remark 3.4. — For m = 1 (i.e., χ = ω) a direct computation gives:

(1 + e (ζ − 1))eω =p−1∏

i=1

(

1 + e (ζi − 1))

ω−1(i)p−1 ≡

p−1∏

i=1

(

1 + eω−1(i) (ζi − 1)) 1

p−1

≡p−1∏

i=1

(

1 + e (ζ − 1))

1p−1 ≡ 1 + e (ζ − 1) mod p2,

using ζi − 1 = ζi−1ζ−1

(ζ − 1) ≡ i (ζ − 1) mod p2.

Theorem 3.5. — Let (a, b, c) be a solution in the first case of Fermat ′s equation; put e = ba+b

.

Let χ = ωm be a p-adic character of g distinct from ω.

Then the pseudo-unit (a+ b ζ)eχ or (1 + e (ζ − 1))eχ is a pth power in Kp if and only if:

p−1∑

k=1ωm−1(k)

(−ba

)k ≡ 0 mod (p).

Proof. — We have to prove the sufficiency of the condition. We note that this congruential

condition is (for χ 6= ω) only equivalent to F ′(X) ∈ (p,Φp(X)) = (p, (X − 1)p−1) in Zp[X],

since (X − 1)Φp(X) = Xp − 1 ≡ (X − 1)p mod (p) (see (1), (2), (3)).

Suppose that the condition F ′(X) ∈ (p, (X − 1)p−1) is satisfied for the coefficients λi =1

p−1χ−1(i) in F (X) =

∏p−1i=1 (1 + e (Xi − 1))λi .

Write F (X) =∑p−1

n=0 un (X − 1)n + U(X) (X − 1)p in Zp[X]; since:

F ′(X) =p−1∑

n=1nun(X − 1)n−1+ pU(X) (X − 1)p−1 + U ′(X) (X − 1)p

is in (p, (X − 1)p−1), this yields to un ≡ 0 mod (p) for n = 1, . . . , p− 1.

Then F (ζ) ≡ u0 mod (p); from Theorem 2.2, F (ζ) being a pseudo-unit congruent to a rational

modulo p is a local pth power. Which proves the theorem obtained by Thaine [Th3] using

generalized binomial computations.

Remark 3.6. — (i) We have obtained that in our viewpoint using radicals, the p-primarity of

the pseudo-unit (1+ e (ζ−1))eχ , χ 6= ω, is directly characterized by means of the polynomial:

Mm(Z) :=p−1∑

k=1ωm−1(k)Zk.

As the reader can see, this polynomial is a variant of the classical polynomial of Mirimanoff

˜Mm(Z) :=p−1∑

k=1km−1Zk and is congruent modulo p to it (see [R, VIII.1] for more information;

see [A1] for the use of Mirimanoff′s polynomials in Iwasawa theory over K; see [Th2, I] for

the definition of polynomial congruences equivalent to Mirimanoff′s congruences and giving a

direct proof of some Wieferich′s criteria).

(ii) We see that Mm(Z) comes from the Gauss sum τ(ωm−1) that we have encountered before

(put Z = ζ), and this has probably a deep signification (see Subsec. 3.4 for some insights).



This shows that this indexation is not convenient; we observe that Mm(Z) must be denoted

Mχ∗(Z), where χ∗ = ωχ−1 = ω1−m, and more generally Mϕ(Z) :=∑p−1

k=1 ϕ−1(k)Zk for any

character ϕ.

Thus, to summarize:

Mχ∗(Z) :=p−1∑

k=1(χ∗)−1(k)Zk

Mχ∗(ζ) =p−1∑

k=1(χ∗)−1(k) ζk = −τ((χ∗)−1);

for convenience, we will use the two notations, the rule being Mωh = Mp−h.

We see also that by all permutations of a, b, c, the p-primarity of the corresponding pseudo-

units (x+ y ζ)eχ , χ 6= ω (i.e., χ∗ 6= χ0), is equivalent to the congruence:

Mχ∗

(

−y

x

)

=p−1∑

k=1(χ∗)−1(k)

(

−y

x

)k

≡ 0 mod (p).

This notation which associates χ (for (x + y ζ)eχ) and χ∗ (for Mχ∗

(−yx

)

) anticipates the use

of reflection theorems.

(iii) The advantage of this definition of Mirimanoff′s polynomials, indexed by the characters

of g, is that they may be related to characters of some subfields of K, giving a more precise

information (use Th. 2.8, (i)), and the knowledge of the p-class groups of the subfields may

have suitable consequences for the properties of these polynomials (e.g. χ = ωp−12 , χ∗ = ω

p+12 ).

Theorem 3.7 (algebraic form of Kummer′s congruences). — Let (a, b, c) be a solu-

tion in the first case of Fermat ′s equation.

If for an odd character χ 6= ω, Mχ∗

(−ba

)

6≡ 0 mod (p) (where χ∗ = ωχ−1), then the χ-

component Cℓχ := Cℓeχ of the p-class group is nontrivial.

Proof. — We have (1 + e (ζ − 1))eχ /∈ K×p since this pseudo-unit is not a local pth power

at p. Put (1 + e (ζ − 1))eχ Z(p)[ζ] = zp; if the ideal z is principal, say z = (z), then:

(1 + e (ζ − 1))eχ = ε zp, where ε ∈ Eχ := Eeχ ;

since χ is odd and distinct from ω (the character of 〈 ζ 〉), ε = 1, giving a global pth power

for (1 + e (ζ − 1))eχ (contradiction). Thus cℓ(z) ∈ Cℓχ is nontrivial.

From the main theorem on cyclotomic fields (see Th. 2.8, (iii))), the p-valuation of | Cℓχ | is

that of the generalized Bernoulli number:

bχ := 1p

p−1∑

k=1χ−1(k)k;

so bχ ≡ 0 mod (p) or, equivalentely since χ = ωm, m 6= 1 odd, the ordinary Bernoulli number

Bp−m is congruent to 0 modulo p (see [W, Cor. 5.15]).


Georges Gras 105

Actually Stickelberger′s theorem is sufficient to get bχ ≡ 0 mod (p); if we want the reciprocal

of “ Herbrand′s theorem ”, we can use [Ri], [Th4] to get that bχ ≡ 0 mod (p) is equivalent to

Cℓχ 6= 1.

We find again in a more precise way the classical situation of Kummer′s congruences which

are:

bχ · Mχ∗

(−ba

)

≡ 0 mod (p).

If Mχ∗

(−ba

)

6≡ 0 mod (p), then Cℓχ 6= 1 and for χ∗ (even and nontrivial), we know by reflection

(see Exa. 2.9) that the χ∗-cyclotomic unit ηχ∗ := (1 − ζ)eχ∗ is a local pth power at p. It is a

global pth power if and only if Cℓχ∗ is nontrivial (Vandiver′s conjecture false at χ∗).

Remark 3.8. — If χ 6= χ0 is even, if Mχ∗

(−ba

)

6≡ 0 mod (p), and if the ideal z is principal (in

the wtiting (1 + e (ζ − 1))eχ Z[ζ] = zp), we only obtain the relation (1 + e (ζ − 1))eχ = εχ zp,

where εχ ∈ Eχ is not a local pth power at p.

The basic example for this is Cℓχ∗ = 1, thus Cℓχ = 1 (Vandiver′s conjecture true at χ); we

then have bχ∗ 6≡ 0 mod (p) thus Mχ

(−ba

)

≡ 0 mod (p), which implies that (1 + e (ζ − 1))eχ∗ is

a global pth power since Cℓχ = 1.

If z is nonprincipal, then Cℓχ 6= 1 (counterexample to Vandiver′s conjecture), Cℓχ∗ 6= 1, bχ∗ ≡ 0

mod (p), and Mχ

(−ba

)

is a priori arbitrary (see Rem. 3.11 for improvements of these reason-

ings).

Theorem 3.9 (algebraic form of Mirimanoff′s congruences: the reflection theorem)

Let χ 6= χ0 be even, and let χ∗ = ωχ−1 (χ∗ is odd distinct from ω).

Then we have Mχ∗

(−yx

)

.Mχ

(−yx

)

≡ 0 mod (p) for any of the six pairs (x, y) corresponding to

a solution in the first case of Fermat′s equation.

Proof. — To prove this congruence, we suppose that both Mχ∗

(−yx

)

and Mχ

(−yx

)

are not

congruent to 0 modulo p to obtain a contradiction.

From the Theorem 2.8, (ii), or [Gr1, II.5.4.9.2], the analysis of the reflection theorem in K

leads to the following equalities (χ even):

rkp((Y/Yprim)χ∗) = rkp(Cℓχ∗) − rkp(Cℓχ) = 1 − rkp((Y/Yprim)χ),

where Y is the group of pseudo-units of K, and where Yprim is the subgroup of p-primary

pseudo-units.

The condition Mχ

(−yx

)

6≡ 0 mod (p) is thus equivalent to (x + y ζ)eχ∗ ∈ Y \Yprim giving

rkp((Y/Yprim)χ∗) = 1, and similarly the condition Mχ∗

(−yx

)

6≡ 0 mod(p) is equivalent to

(x+ y ζ)eχ ∈ Y \Yprim, giving rkp((Y/Yprim)χ) = 1 (contradiction).

Corollary 3.10. — Let χ 6= χ0 (i.e., χ∗ 6= ω) be even. Suppose that Cℓχ is trivial (Vandiver′sconjecture true at χ).

If Mχ

(−yx

)

6≡ 0 mod (p), then Mχ∗

(−yx

)

≡ 0 mod (p) and the fundamental χ-unit εχ is p-

primary as well as the χ-cyclotomic unit ηχ := (1 − ζ)eχ.



Proof. — From Mχ

(−yx

)

6≡ 0 mod (p) and Theorem 3.7 we get that Cℓχ∗ is of p-rank ≥ 1,

hence equal to 1 since Cℓχ = 1; so by Kummer duality (see Th. 2.8, (i)), the radical of the

corresponding unramified χ∗-extension of K is given by the fundamental χ-unit εχ which is

thus p-primary. By hypothesis, Eχ is also generated by the χ-cyclotomic unit ηχ. This is the

result obtained in [Th2, II] via congruential computations.

So, Mirimanoff′s congruences, obtained by ugly computations, are nothing but the reflection

principle in class field theory.

Remark 3.11. — Let χ be even distinct from χ0.

(i) If Mχ

(−yx

)

6≡ 0 mod (p), then from the proof of Theorem 3.9 we have rkp((Y/Yprim)χ∗) = 1,

rkp((Y/Yprim)χ) = 0 (all the χ-pseudo-units are p-primary, especially εχ), and for the class

group we get rkp(Cℓχ) + 1 = rkp(Cℓχ∗), which means that the χ∗-class group is nontrivial.

Then (x + y ζ)eχ is p-primary (which is coherent with Mχ∗

(−yx

)

≡ 0 mod (p)) but can be a

global pth power.

(ii) If Mχ∗

(−yx

)

6≡ 0 mod (p), rkp((Y/Yprim)χ) = 1, rkp((Y/Yprim)χ∗) = 0 (all the χ∗-pseudo-

units are p-primary), and rkp(Cℓχ∗) = rkp(Cℓχ).

(iii) If Mχ∗

(−yx

)

≡Mχ

(−yx

)

≡ 0 mod (p), then (x+ y ζ)eχ and (x+ y ζ)eχ∗ are p-primary, but

we dont know if they are global pth powers or not; if for instance (x + y ζ)eχ = zp then the

ideal ceχ

1 is principal. If (x + y ζ)eχ is not of the form εχ zp, c

eχ

1 is not principal (the χ-class

group is nontrivial), and (x+y ζ)eχ defines the radical of a χ∗-unramified extension of K (the

χ∗-class group is of course nontrivial).

If (x + y ζ)eχ∗ is not a pth power, ceχ∗

1 is nonprincipal (because Eχ∗ = 1) and defines the

radical of a χ-unramified extension of K, giving Cℓχ 6= 1 (Vandiver′s conjecture false at χ),

hence also Cℓχ∗ 6= 1.

(iv) If Cℓχ∗ is trivial, then the unit εχ is not p-primary and all the χ∗-pseudo-units are p-

primary (hence global pth powers); then we get Mχ

(−yx

)

≡ 0 mod (p).

(v) For χ = χ0, we know that Cℓχ∗ = Cℓω is trivial; in this case, Mχ0

(−yx

)

=∑p−1

k=1

(−yx

)k

takes always the value 0 for −yx

6≡ 1 mod (p).

For χ = χ0, Cℓχ is trivial and in this case we obtain the supplementary Mirimanoff congruence:

Mχ∗

(

−y

x

)

= Mω

(

−y

x

)

=p−1∑

k=1ω−1(k)

(

−y

x

)k

≡ 0 mod (p)

since it corresponds to the p-primarity of NK/Q(x+ y ζ) = zp1 .

3.2. Derivation technics: the method of Eichler. — We begin with a particular case

of this method to analyze a global approach to the computation of the p-rank of the radicals

Wa, Wb, Wc, and W .


Georges Gras 107

We consider the necessary condition of the previous subsection, concerning the first case of

FLT, to have∏p−1

i=1 (1 + e (ζi − 1))λi ∈ K×p, for e := ba+b

:

p−1∑

i=1

λi iXi−1

1 + e(Xi − 1)∈ (p,Xp − 1).

The trick is to suppose that the support S of the set of integers λi (that is the set of indices

i such that λi 6≡ 0 mod (p)) is not too big in the expression:∑

i∈Sλi iX

i−1∏

j∈S,j 6=i(1 − e+ eXj) ∈ (p,Xp − 1),

so that there is no reduction by Xp − 1 in the computation of the products:

Xi−1∏

j∈S,j 6=i(1 − e+ eXj), for i ∈ S.

For this, the condition is that i− 1 +∑

j∈S,j 6=ij < p, equivalent to

∑

i∈Si ≤ p. If we suppose that

S ⊆ 1, 2, . . . , ρ := [√

2p − 0.5], the condition is satisfied.

We thus have the congruence:∑

i∈Sλi iX

i−1∏

j∈S,j 6=i(1 − e+ eXj) ≡ 0 mod pZ(p)[X].

The (unique) term of minimal degree is obtained for the minimal value i0 of i ∈ S and gives

λi0 . (1 − e)ρ−1 ≡ 0 mod (p), then λi0 ≡ 0 mod (p) (contradiction). We have obtained:

Theorem 3.12. — Let (a, b, c) be a solution in the first case of Fermat′s equation.

Then each of the three radicals Wa = 〈 b + c ζj 〉.K×p/K×p, Wb = 〈 c + a ζk 〉.K×p/K×p,

Wc = 〈 a+ b ζi 〉.K×p/K×p, j, k, i = 1, . . . , p − 1, is of p-rank at least ρ := [√

2p − 0.5].

Same conclusion replacing K by Kp (local radicals).

But as is always the case, the conclusion of the proof is of a local nature.

Remark 3.13. — (i) The monogenic Fp[g]-module Wc generated by a + b ζ defines a sub-

representation of the regular one; thus there exist at least ρ distinct characters χ such that

(a+ b ζ)eχ is not a global (or local) pth power.

(ii) Let (xi, yi) ∈ (a, b), (b, c), (c, a), i = 1, . . . , ρ; then by the same method it is easy to

prove that the pseudo-units xi + yi ζi are independent in K×/K×p, giving by conjugation

many subradicals in W of p-rank ρ.

Now we give a variant of the theorem of Eichler from a solution (a, b, c) in the first case of the

Fermat equation. We study the relation, where e := ba+b

(still for the support S of the λi):

∏

i∈S

(

a+ bζ−i

a+ bζi

)λi

=∏

i∈S

(

(1 + e (ζ−i − 1))

(1 + e (ζi − 1))

)λi

= βp, β ∈ Z(p)[ζ].

Put (a+ b ζi) Z[ζ] = cpi and (a+ b ζ−i) Z[ζ] = c

pi . From the above relation we deduce:

∏

i∈S

(

ci

ci

)λi

= (β) Z(p)[ζ].



Reciprocally, any relation of principality∏

i∈S

(

ci

ci

)λi

= (β′) Z(p)[ζ] gives:

∏

i∈S

(

(1 + e (ζ−i − 1))

(1 + e (ζi − 1))

)λi

= ζh ε+ β′p, ε+ ∈ E+, h ≥ 0;

we suppose that∑

i∈S λi i ≡ 0 mod (p); this implies easily h = 0. Then the relative norm

NK/K+(ε+ β′p) must be 1, so that (ε+)2 NK/K+(β′p) = 1 giving ε+ = (η+)p for a real unit

η+; this yields the first relation with β = η+ β′.

Write (1 + e (ζ−i − 1))λi = ζ−λi i(e+ (1− e) ζi))λi . The first relation is thus equivalent to the

relation (reutilizing by abuse the same notations for F , H, B in Z(p)[[X]]):

F (X) :=∏

i∈S(e+ (1 − e)Xi)λi (1 + e(Xi − 1))−λi = H(X)p +B(X)(Xp − 1),

giving by logarithmic derivation, F being invertible modulo (p,Xp − 1):

(1 − e)∑

i∈S

λi iXi−1

e+ (1 − e)Xi− e

∑

i∈S

λi iXi−1

1 + e(Xi − 1)∈ (p,Xp − 1),

and finally:

(1 − 2e)∑

i∈S

λi i Xi−1

(e+ (1 − e)Xi)(1 + e(Xi − 1))∈ (p,Xp − 1).

If 2e ≡ 1 mod (p) we get a ≡ b mod (p) and by circular permutations, the analogous congru-

ences would give a ≡ b ≡ c mod (p), thus 0 ≡ a + b+ c ≡ 3a mod (p) (absurd for p > 3); so

we may suppose that 2e 6≡ 1 mod (p).

As before we obtain:∑

i∈Sλi iX

i−1∏

j∈S,j 6=i(e+ (1 − e)Xj)(1 − e+ eXj) ∈ (p,Xp − 1).

If 2∑

j∈Sj ≤ p+ 1 there is no reduction modulo Xp − 1 in the computation of this expression.

Then, for S ⊆ 1, . . . , [√p+ 1 − 0.5], the (unique) term of minimal degree is obtained for

the minimum i0 of S, giving immediately λi0 ≡ 0 mod (p) (contradiction).

Since the classes cℓ(ci . ci−1) are relative classes, we have proved (taking in account that we

have imposed a relation on the λi):

Theorem 3.14 (Eichler′s theorem). — Let p > 2 be prime. If the p-rank of the relative

class group of K satisfies rkp(Cℓ−) ≤ ρ′ := [√p+ 1 − 1.5] then the first case of FLT holds for

the prime p. (12)

(12)Since cℓ`

ci

ci

´

= si.cℓ`

c1c1

´

, i ∈ S, the monogenic g-module generated by cℓ`

c1c1

´

contains the cℓ`

ci

ci

´

and is

contained in the regular representation Fp[g]; this means that at least ρ′ different characters χ give a nontrivial

Cℓχ and the statement is true with the index of irregularity i(p) instead of rkp(Cℓ−).


Georges Gras 109

3.3. Some other p-adic technics. — Now we consider the Dwork uniformizing parameter

in Kp which has the following characteristic properties (see e.g. [Gr1, Exer. II.1.8.3]):

(i) p−1 = −p,(ii) sk() = ω(k), k = 1, . . . , p− 1.

In the following lemma we suppose that 1+ e (ζ−1) is a pseudo-unit, so that p-primarity and

local pth power property are equivalent (see Lem. 2.1, Th. 2.2). We compute in Zp[ζ] = Zp[].

Lemma 3.15. — Let χ = ωm, m ∈ 1, . . . , p − 1; then for e 6≡ 0 mod (p) we have the

relation (1 + e (ζ − 1))eχ = 1 +mϕχ, where ϕχ ∈ Zp[].

Then (1 + e (ζ − 1))eχ is a local pth power if and only if ϕχ ≡ 0 mod ().

Proof. — Suppose that (1 + e (ζ − 1))eχ = 1 +nv, where v is a unit of Kp and n ≥ 1; put

v ≡ v0 mod (), v0 ∈ Z\pZ.

Applying eχ we have:

(1 + e (ζ − 1))eχ ≡ (1 +nv0)eχ ≡ 1 + eχ(nv0)

≡ 1 + 1p−1

p−1∑

j=1ω−m(j) sj(

nv0) ≡ 1 + 1p−1

p−1∑

j=1ω−m(j)ωn(j)nv0

≡ 1 + nv0p−1

p−1∑

j=1ωn−m(j) ≡ 1 +nv mod (n+1),

which is absurd except if n ≡ m mod (p − 1). Thus (1 + e (ζ − 1))eχ = 1 +m ϕχ.

Ifm = p−1, we know that the norm of such a pseudo-unit is of the form np with n ≡ 1 mod (p),

hence (1 + e (ζ − 1))eχ0 ≡ 1 mod (p2), proving the lemma in this case; suppose m < p− 1.

The pth power condition is ϕχ ≡ 0 mod (p−m−1) (apply Th. 2.2), with p−m− 1 > 0.

Suppose that ϕχ ≡ 0 mod (); then we get (1 + e (ζ − 1))eχ = 1 +m+1ϕ′χ, for ϕ′

χ ∈ Zp[].

Then applying again the idempotent eχ, the first part of the proof gives ϕ′χ ≡ 0 mod (),

then inductively the result up to (1 + e (ζ − 1))eχ ∈ 1 + (m+p−1).

The value m = 1 does not work here since we know that (1+e (ζ−1))eω ≡ 1+e mod (2)

(see Rem. 3.4) and since we have supposed p ∤ e.

Corollary 3.16. — Write log (1 + e (ζ − 1)) = e22 + . . . + ep−1

p−1, ei ∈ Zp. Then the

set of characters χ = ωm, m ∈ 2, . . . , p − 1, such that (1 + e (ζ − 1))eχ is p-primary, is

m ∈ 2, . . . , p− 1, em ≡ 0 mod (p).

Proof. — Left to the reader.

We see that the condition depends on a single congruence to 0 modulo , whose probability

may be 1p, giving another aspect of the rarity of such a condition for many values of m (at

least p−12

from Mirimanoff ′s congruences).



3.4. p-adic Gauss sums and Mirimanoff ′s polynomials. — We use the context of the

previous Subsection 3.3, especially the Dwork uniformizing parameter ∈ Qp(ζ) such that

p−1 = −p and sk() = ω(k) for k = 1, . . . , p − 1.

Let χ = ωm, here indexed by m ∈ 0, . . . , p − 2. We note that (additively):

eχ . ζ =1

p− 1

p−1∑

k=1χ−1(k)ζk =

−1

p− 1τ(χ−1).

Now put:

ζ =−1

p− 1

(

u0 + u1 + . . .+ up−2p−2

)

,

where uk ∈ Zp, with u0 ≡ 1 mod (p). We know that eχ .j = 0 if j 6≡ m modulo (p− 1) and

eχ .m = m, so that eχ . ζ = −1

p−1umm and τ(χ−1) = umm, for allm ∈ 0, . . . , p−2. (13)

Then, since for τ(χ) := χ(−1) τ(χ), we have τ(χ−1) τ(χ) = p for χ 6= χ0, we obtain for m 6= 0:

umm (−1)m up−1−mp−1−m = (−1)m um up−1−m (−p) = p,

giving the relation um up−1−m = (−1)m+1, for m 6= 0. For the unit character, τ(χ0) = 1 and

we find u0 = 1.

We have obtained a classical result:

Proposition 3.17. — Let χ = ωm, m ∈ 0, . . . , p − 2, τ(χ−1) := −∑p−1k=1 χ

−1(k) ζk the

Gauss sum of χ−1; put τ(χ) := −∑p−1k=1 χ(k) ζ−k = χ(−1) τ(χ).

Then we have τ(χ−1) = umm, τ(χ) = χ(−1)up−1−mp−1−m, which implies the relation

um up−1−m = (−1)m+1, for all m ∈ 1, . . . , p− 2, and u0 = 1.

The modified Mirimanoff polynomial is, for χ∗ = ωχ−1 (see Rem. 3.6, (ii)):

Mχ∗(Z) :=p−1∑

k=1(χ∗)−1(k)Zk,

and in the first case of FLT we must compute Mχ∗(−xy

) modulo (p) for the usual (x, y)

depending of a solution and its permutations.

We suppose now that ω takes its values in the field F of (p− 1)th roots of unity. We consider

the ideal p0 | p of F such that ω(k) ≡ k mod p0 for all k. All the computations take place in

the compositum FK in which we denote by P the (unique) prime ideal above p0.

The condition of p-primarity of (a + b ζ)eχ , for χ = ωm, m ∈ 1, . . . , p − 1, χ 6= ω (see

Subsec. 3.1) becomes, in FK with χ∗ = ω1−m:

Mχ∗

(

−b

a

)

. τ(χ∗) ≡ 0 mod Pp−1,

(13)In these computations, we must write the unit character ω0 instead of ωp−1 because of the expression of

ζ since p−1 = −p and τ (ω0) = 1.


Georges Gras 111

where Mχ∗

(−ba

)

∈ F and τ(χ∗) := −∑p−1k=1 χ

∗(k)ζk ∈ FK is of P-valuation m − 1, giving

Mχ∗

(−ba

)

≡ 0 mod p0, and where we have:

Mχ∗(ζ) = −τ((χ∗)−1).

The above properties of Gauss sums lead to the following, where we only suppose that a and

b are coprime integers.

Put (a+ b ζ) Z[ζ] = C1, thus ζ ≡ −ab

mod C1 seen in FK. This gives:

−Mχ∗

(

−a

b

)

= −p−1∑

k=1(χ∗)−1(k)

(

−a

b

)k

≡ τ((χ∗)−1) mod C1.

Thus in the same way (using ζ−1 ≡ −ba

mod C1):

−M(χ∗)−1

(

−b

a

)

≡ τ(χ∗) mod C1,

which yields, in F , for any χ 6= ω (i.e., χ∗ 6= χ0):

Mχ∗

(

−a

b

)

.M(χ∗)−1

(

−b

a

)

≡ τ((χ∗)−1) . τ (χ∗) ≡ p mod C1.

Let σ be an element of Gal (FK/K); for any (p − 1)th root of unity ξ, σ(ξ) = ξt with a

suitable t prime to p − 1, so that the action of σ on the powers of ω preserves the relation

ϕ .ϕ−1 = χ0 between the characters, and preserves the ideal C1 which is in K; thus the

expressions Mχ∗

(−ab

)

.M(χ∗)−1

(−ba

)

are conjugated by Galois so that the p-adic study (14) of

the products Mωd

(−ab

)

.Mω−d

(−ba

)

, d | p− 1, is sufficient.

The congruence modulo C1 in FK is now in F , thus it is actually modulo the ideal NK/Q(C1)

seen in F . Since it is the norm of a+ b ζ, it is the homogeneous form in a, b:

Φp(a, b) := ap−1 − ap−2b+ . . .− a bp−2 + bp−1.

Put Mχ∗

(−ab

)

.M(χ∗)−1

(−ba

)

− p = Φp(a, b) .Ψχ(a, b)

ap−2 bp−2, then Ψχ(a, b) is an homogeneous form

of degree p− 3.

We have, for any character ϕ, Mϕ(Z) = ϕ(−1)Zp Mϕ(Z−1), which givesMϕ(Z)Mϕ−1(Z−1) =

Mϕ(Z−1)Mϕ−1(Z), hence proves the symmetry between a and b, and the invariance of

Mχ∗

(−ab

)

.M(χ∗)−1

(−ba

)

by complex conjugation in F/Q. So these expressions have coeffi-

cients in the maximal real subfield F+ of F .

To summarize, we have obtained:

Proposition 3.18. — Let x, y be indeterminates and put Mϕ(Z) :=∑p−1

k=1 ϕ−1(k)Zk for

any character ϕ. Then for all χ 6= ω, we have the relation:

Mχ∗

(

−x

y

)

.M(χ∗)−1

(

−y

x

)

= p+ Φp(x, y) .Ψχ(x, y)

xp−2 yp−2,

where Ψχ(x, y) is a symmetrical homogeneous form of degree p− 3 with coefficients in F+.

(14)More precisely the knowledge of the p′

0-valuations, for all the prime ideals p′

0 of F above p.



Now we suppose that (x, y, z) is a solution in the first case of the Fermat equation. Recall

that the condition of p-primarity of (x + y ζ)eχ which was modulo Pp−1 in FK is now,

because of the total ramification in FK/F , modulo the prime ideal p0 of F under P, and is

Mχ∗

(−yx

)

≡ 0 mod p0.

From the above we obtain that:

Mχ∗

(

−x

y

)

.M(χ∗)−1

(

−y

x

)

≡ 0 mod p0

is equivalent to Ψχ(x, y) ≡ 0 mod p0.

For instance, for p = 5 we have (noting that F+ = Q and p0 = (5)):

Mω−1

(

−x

y

)

.Mω

(

−y

x

)

= 5 + Φ5(x, y) .x2 + x y + y2

x3 y3,

Mω2

(

−x

y

)

.Mω2

(

−y

x

)

= 5 − Φ5(x, y) .x2 + 3x y + y2

x3 y3.

Of course these forms Ψ do not represent 0 in F5.

Remark 3.19. — (i) Notice that these congruences have nothing to do with Mirimanoff′scongruences despite the fact that as soon as one of the factors Mχ∗

(−xy

)

, M(χ∗)−1

(−yx

)

is

congruent to 0 modulo p0, this is the case of the expression Ψχ(x, y) and reciprocally.

More precisely, Mχ∗

(−xy

)

≡ 0 mod p0 is equivalent to (y+x ζ)eχ p-primary, hence to (x+y ζ)eχ

p-primary (since χ 6= ω), thus to Mχ∗

(−yx

)

≡ 0 mod p0.

Similarly, M(χ∗)−1

(−yx

)

≡ 0 mod p0 is equivalent to the p-primarity of the two pseudo-units

(x+ y ζ)eeχ and (y + x ζ)eeχ , then to M(χ∗)−1

(−xy

)

≡ 0 mod p0, where χ := ω2 χ−1, which may

have some interest (see in Subsec. 2.5, (b), the reflection between R2,χ and Tω2χ−1).

(ii) It would be interesting to perform the same study with the Davenport–Hasse relations

between Gauss sums, for two characters:∏

χ, χd=χ0

τ(χ .ψ) = ψ−d(d) . τ(ψd) .∏

χ, χd=χ0

τ(χ),

for any divisor d of p− 1, and with the Jacobi sums given by the relation:

τ (χ) τ (ψ)

τ (χψ)= −

p−1∑

k=1χ(k)ψ(1 − k).

3.5. Mirimanoff ′s sums. — We still consider the context of the previous Subsection 3.4,

for which ω takes its values in the field F of (p − 1)th roots of unity. We fix the prime ideal

p0 of F above p in the following way: fix a primitive (p − 1)th root of unity ξ0 ∈ F and a

primitive (p− 1)th root r0 ∈ Z modulo p; then we decrete that ξ0 ≡ r0 mod p0.

Since for any character ϕ of g := Gal (K/Q), Mϕ(Z) =∑p−1

k=1 ϕ−1(k)Zk, if we put, for a

solution (x, y, z) in the first case of Fermat′s equation:

−y

x≡ rt

0 ≡ ξt0 =: ξ mod p0,


Georges Gras 113

we have in F the congruence:

Mϕ

(

−y

x

)

≡Mϕ(ξ) mod p0;

hence the congruences Mϕ

(−yx

)

≡ 0 mod (p) in Qp and Mϕ(ξ) ≡ 0 mod p0 in F are equivalent.

We propose to call the sums of roots of unity:

µϕ(ξ) :=p−1∑

k=1ϕ−1(k) ξk ∈ F,

the Mirimanoff sums attached to the character ϕ and the (p − 1)th root of unity ξ.

It is clear that the algebraic numbers:

µϕ(ξ) . µϕ∗(ξ), ϕ 6= χ0, ω and µϕ(ξ) . µϕ−1(ξ−1), ϕ 6= χ0,

give the easy way to study the congruences of Mirimanoff and the congruences given in

Proposition 3.18.

Unfortunately, the root ξ is uneffective and the properties of the sums µϕ(ξ) depend largely

of the order of ξ (i.e., the order of −xy

modulo p); hence we must envisage all the possibilities.

Warning: in the factor ϕ−1(k), k is considered modulo p, but in the factor ξk, k is considered

modulo p− 1, under the condition that k ∈ 1, . . . , p− 1.In a more numerical setting, put ϕ = ωh and ξ = ξt

0; then, writing k ≡ rj0 mod (p), we get:

µϕ(ξ) =: µh(t) =p−1∑

k=1ω−h(k) ξ t k

0 =p−1∑

j=1ξ−h j0 ξ

t [rj0]p

0

=p−1∑

j=1ξ−h j + t [r

j0]p

0 , h, t ∈ 1, . . . , p− 1,

where [rj0]p is the unique residue modulo p of rj

0 in the set 1, . . . , p − 1.Then let Φp−1 be the (p−1)th cyclotomic polynomial, of degree ν := φ(p−1); after reduction

modulo Φp−1, we obtain: µh(t) = q0 + q1ξ0 + . . . + qν−1ξν−10 , qi ∈ Z, which can be studied

modulo p0 in an easy way.

Naturally, these sums are completely analogous to Mirimanoff′s polynomials specialized at

suitable classes modulo p, but we hope that the formulation in terms of sums of roots of unity

is likely of a better understanding.

3.6. Wieferich′s criterion: a local consequence of the reciprocity law. — As indi-

cated in Ribenboim′s book, the Wieferich criterion may be deduced from the law of reciprocity

(this has been done first by Furtwängler from Eisenstein′s reciprocity law [R, IX.3]). For this

purpose, an explicit formula of Hasse may also be used [R, IX.5].

Here we propose a more basic proof using the p-conductor of a Kummer extension in the

following way, where(••)

pis the pth power residue symbol, with values in 〈 ζ 〉.



Theorem 3.20 (Wieferich′s criterion). — Let ℓ be a prime number, ℓ 6= p, and suppose

that x+ y ζ is a pseudo-unit (i.e., (x+ y ζ) is the pth power of an ideal of K prime to p).

(i) Then(

ζx y + ζ−y x

ℓ

)

p= 1.

(ii) If ℓ | y with p ∤ y and if (x, y, z) is a solution of Fermat′s equation (15), then ℓ p−1 ≡1 mod (p2).

Proof. — The expression of α := ζx y + ζ−y x is such that α is still a pseudo-unit, and

α ≡ x+ y ≡ (x+ y)p mod (1 − ζ)2.

The general law of reciprocity (see e.g. [Gr1, Th. II.7.4.4]) yields to:

(

α

ℓ

)

p

(

ℓ

α

)−1

p= (ℓ, α)p

where (•, •)p is the Hilbert′s symbol at the place p. This symbol is equal to 1 if and only if ℓ is

a local norm in the Kummer extension Kp( p√α )/Kp; the conductor of this extension divides

pp−1 since α is congruent to a pth power modulo p2 (see the general conductor formula in

[Gr1, Prop. II.1.6.3]). Since ℓ p−1 ≡ 1 mod (p) the normic condition is satisfied for ℓ.

But the symbol(

ℓα

)−1

pis tivial since (α) is the pth power of an ideal; thus:

(

ζx y + ζ−y x

ℓ

)

p= 1.

If ℓ | y, we have ζx y+ζ−y x ≡ ζ−y x mod (ℓ) and 1 =(

ζ−y xℓ

)

p=

(

ζℓ

)−y

p

(

xℓ

)

p; but x = zp

0 −y ≡zp0 mod (ℓ) giving

(

xℓ

)

p= 1 and

(

ζℓ

)

p= 1 since p ∤ y.

If (ℓ) = l1 . . . ld in K, then∏d

i=1

(

ζli

)

= 1, but we have(

ζl1

)k= sk

(

ζl1

)

=(

ζk

lk

)

=(

ζlk

)k, so that

(

ζlk

)

does not depend on k, giving(

ζl1

)

= 1; thus the multiplicative group of the residue field

of l1 contains an element of order p2, proving the point (ii) of the theorem.

Then the discovery of Wieferich′s criteria consists in proving that small prime numbers ℓ (e.g.

ℓ = 2) divide a b c (see [GM], [Th2], for a study of Fermat′s quotients in relation with FLT);

in the second case, the hypothesis ℓ | y, p ∤ y may be inaccurate, so the Wieferich criterion is

uneffective in the second case.

It is clear that the prime numbers ℓ ≡ 1 mod (p), such that Fermat′s equation up + vp +1 = 0

has no nontrivial solutions in the finite field Fℓ, are divisors of a b c (where (a, b, c) is a global

solution in any case of Fermat′s equation); then experimental computations show that many

such primes do exist. One may conjecture that their number tends to infinity with p, which

gives many uneffective Wieferich′s criteria.

In this direction we have the following interesting approach.

(15)So that x+ y = zp0 as usual; the second case of FLT being equivalent here to p | x.


Georges Gras 115

3.7. Wendt ′s criterion: a non modulo p local–global result. — Let ℓ be a prime

number of the form 1 + n p, n ≥ 2, and let l be an ideal above ℓ in K. We consider the

algebraic number θn :=∏n

i, j=1(ξi + ξj + 1), where the ξk, k = 1, . . . , n, are the nth roots of

unity.

We have θn ∈ Z\0; this number has been used for instance in the following papers : [LS]

(for a similar purpose as us) and [A-HB], [F] to prove that the first case of FLT holds for

infinitely many primes p.

See [R, IV.4] for its explicit computation via Wendt′s determinant. If ℓ ∤ θn this means that

Fermat′s equation in the residue field Fℓ of l has no nontrivial solutions; thus if a, b, c is a

solution in Z of Fermat′s equation, necessarily ℓ divides one of these numbers, say ℓ | c.Now we state the following result (in the spirit of Germain′s theorem).

Theorem 3.21 (Wendt′s criterion). — Let ℓ = 1+n p be a prime number which does not

divide the natural integer θn. Moreover, we suppose that p is not a pth power modulo ℓ.

Then the first case of FLT holds for p.

Proof. — Suppose that ℓ | c for a solution in the first case of Fermat′s equation. We have

a+ b = cp0, NK/Q(a+ b ζ) = cp1 with −c = c0 c1 (see Rem. 1.4, (i)).

If ℓ | c0 then b ≡ −a mod (ℓ), giving:

cp1 = NK/Q(a+ b ζ) ≡ ap−1p−1∏

i=1(1 − ζi) = ap−1 p mod (ℓ)

(a contradiction since a+ c ≡ a ≡ bp0 mod (ℓ), giving that p is a local pth power at ℓ).

So ℓ | c1; from Lemma 1.2, ℓ ∤ c0 giving, from a + b = cp0, a + c = bp0, and b + c = ap0, the

relation 0 = a + b− cp0 ≡ bp0 + ap0 + (−c0)p mod (ℓ) which defines a non trivial solution in Fℓ

(absurd).

The conclusion of the theorem is the same if we replace the hypothesis “ p is not a pth power

modulo ℓ ”, by “ p ∤ n ” since in that case, Wieferich′s criterion is not satisfied for ℓ.

Appendix. Wieferich′s criterion without reciprocity law

(from a proof rediscovered by Roland Quême). (16)

We use the same notations as in Subsection 3.6. See also Notations 2.7.

Let ℓ 6= p be a prime number. We suppose that by choosing suitable x, y among a, b, c,

we have ℓ | y and p ∤ x + y in the writing (x + y ζ) Z[ζ] = zp1 (valid in any case of Fermat′s

equation). Consider eω ∈ Z[g] modulo p.

We know that cℓ(z1)eω = 1 (another application of the reflection theorem; see [Gr1, II.5.4.6.3]),

so that (x+ y ζ)eω = εω δpω, εω ∈ Eω = 〈ζ〉, δω ∈ K×; hence εω = ζh for h ≥ 0.

(16) Adress: Roland Quême, 13 Avenue du château d’eau, 31490 Brax, Url: http://roland.queme.free.fr/,

email: [email protected]



Thus this yields:

(x+ y ζ)eω ∈ ζh .K×p,

hence the relation( (x+y ζ)eω

ℓ

)

p=

(

ζℓ

)h

pwhere (x+y ζ)eω ≡ xeω (a pth power) modulo ℓ, proving

that:(

ζ

ℓ

)h

p= 1.

But (x+ y ζ)eω ∈ ζh .K×p is equivalent to (1 + yx+y

(ζ − 1))eω ∈ ζh .K×p; using Remark 3.4

( (1 + yx+y

(ζ − 1))eω ≡ 1 + yx+y

(ζ − 1) mod p2) we get immediately h ≡ yx+y

mod (p).

If moreover y 6≡ 0 mod (p) (e.g. first case of FLT, or second case with x ≡ 0 mod (p)) we

obtain the result on Wieferich′s criterion in the same way as in Subsection 3.6, without any

use of the reciprocity law.

4. Conclusion

We have shown that much of the classical literature on FLT has been concerned with very

basic facts of class field theory, often rediscovered by means of painful congruential com-

putations; but recall that class field theory is essentially algebraic as soon as, for instance,

Čebotarev′s density theorem is not used (among other analytic tools), and that, algebrically,

all is “ possible ”. So it appears that this approach is relatively poor, despite the power of class

field theory to enunciate technical properties.

Moreover, most of the arguments are local, especially local at p. (17)

The fact that the relative class group takes place in these studies does not change our point of

view since it is utilized without serious analytic arguments (except the unusable upperbound

log (h−) < p4log(p) and the ingenious but elementary derivation technic of Eichler). Moreover

the analytic class number formula for the relative class group is not really analytic since it is,

roughly speaking, equivalent to Stickelberger′s theorem and is, in some sense, algebraic (the

main theorem on cyclotomic fields gives a better knowledge of the class field theory aspects,

but it is not really necessary).

It is likely that the most serious cyclotomic approaches are the study of “ Mirimanoff′s sums ”,

since at least half of them must be zero modulo p0, and that of Wendt′s criterion since it is

connected with the theory of prime numbers; but all this only concerns the first case of FLT,

which is unnatural.

Still in the first case, from the well-known class field theory exact sequence of Zp-modules:

1 −→ U/E −−−→ Gal(HPlp/K) −−−→ Cℓ −→ 1,

(17)Recall that a pseudo-unit α of K is in K×p if and only if α ∈ K×pq for all q ∈ p, l1, . . . lr, where the prime

ideals l1, . . . lr generate the p-class group of K (see [Gr1, Exer. II.6.3.8]); but this criterion is not effective.


Georges Gras 117

where HPlp is the maximal abelian p-ramified pro-p-extension of K, U the group of principal

units of Kp, E the closure in U of the group of global units ε ≡ 1 mod p, we get for any even

character χ 6= χ0:

1 −→ Uχ/Eχ −−−→ Tχ −−−→ Cℓχ −→ 1,

where all groups are p-torsion groups since χ 6= χ0 is even. For p large enough, the result

of Kurihara–Soulé is Cℓωp−3 = 1; suppose that it is possible to extend it to Tωp−3 = 1

(taking “ p-ramification ” instead of “ nonramification ”), then Eωp−3 = Uωp−3 which means

that the fundamental ωp−3-unit εωp−3 is not a local pth power and that the fundamental

ωp−3-cyclotomic unit ηωp−3 (equal to εωp−3 since Cℓωp−3 = 1) is not a local pth power, which

is equivalent to bχ∗ = bω3 6≡ 0 mod (p), in other words to Bp−3 6≡ 0 mod (p), which would

contradict the first case of FLT (at least for p large enough).

We believe more in the possibility of a nonalgebraic study of the radical generated by ζ, 1− ζ,a+b ζ, b+c ζ, c+a ζ and their conjugates, which would be independent of the considered case

of FLT, and which is not equivalent to a general study of the group pCℓ because as a matter

of fact we are concerned with very specific p-classes, the same remark being valid for the

utilization of other arithmetical invariants of K. As the Referee mentions, all these invariants

are isomorphic or dual to adequate Tate twists of the cohomology group H2(G,Z/pZ) (where

G is the Galois group of the maximal p-ramified pro-p-extension of K) which relativizes the

interest, but we don′t know if the use of the pseudo-units x+ y ζ in these contexts leads, in

practice, to the same “ numerical ” criteria and to the same diophantine approach.

It is indeed surprising that, to our knowledge, there is no important diophantine results on

the mixed radical W , using simultaneously a, b, c, and possibly the cyclotomic numbers, which

constitutes a particular case of the study of the polynomial identity, in the polynomial ring

Z[X]:n∏

i=1(ui + viX

di)λi = H(X)p +B(X) (Xp − 1), 0 ≤ di, λi ≤ p− 1.

References

[A1] B. Anglès, On some p-adic power series attached to the arithmetic of Q(ζp), J. Number Theory122 (2007), 1, 221–246.

[A2] B. Anglès, Norm residue symbol and the first case of Fermat’s equation, J. Number Theory 91,2(2001), 297–311.

[A3] B. Anglès, Units and norm residue symbol, Acta Arith. 98, 1 (2001), 33–51.

[A-HB] L.M. Adleman and D.R. Heath-Brown, The first case of Fermat′s last theorem, Invent. Math.79 (1985), 409–416.

[AN] J. Assim and T. Nguyen Quang Do, On the Kummer–Leopoldt constant of a number field,Manuscripta Math. 115, 1 (2004), 55–72.

[Br1] H. Brückner, Zum ersten Fall der Fermatschen Vermutung, J. Reine Angew. Math. 274/275(1975), 21–26.



[Br2] H. Brückner, Zum Beweis des ersten Falles der Fermatschen Vermutung für pseudoregulärePrimzahlen ℓ (Bemerkungen zur vorstehenden Arbeit von L. Skula.), J. Reine Angew. Math. 253(1972), 15–18.

[E1] M. Eichler, Zum 1. Fall der Fermatschen Vermutung. Eine Bemerkung zu zwei Arbeiten von L.Skula und H. Brückner, J. Reine Angew. Math. 260 (1973), 214.

[E2] M. Eichler, Eine Bemerkung zur Fermatschen Vermutung, Acta Arith. 11 (1965), 129–131; Er-rata. Ibid. ohne Seitenzahl, p. 261.

[F] E. Fouvry, Théorème de Brun-Titchmarsh; application au théorème de Fermat, Invent. Math. 79(1985), 383–407.

[G1] A. Granville, The Kummer–Wieferich–Skula approach to the first case of Fermat′s Last Theorem,Gouvêa, Fernando (ed.) et al., Advances in number theory, The proceedings of the third conferenceof the Canadian Number Theory Association, Oxford: Clarendon Press 1993, 479–497.

[G2] A. Granville, On Krasner′s criteria for the first case of Fermat′s last theorem, Manuscr. Math.56 (1986), 67–70.

[G3] A. Granville, On the size of the first factor of the class number of a cyclotomic field, Invent.Math. 100 (1990), 321–338.

[GM] A. Granville and M.B. Monagan, The first case of Fermat′s last theorem is true for all primeexponents up to 714, 591, 416, 091, 389., Trans. Am. Math. Soc. 306, 1 (1988), 329–359.

[Gr1] G. Gras, Class Field Theory: from theory to practice, SMM second corrected printing 2005.[Gr2] G. Gras, Théorèmes de réflexion, J. Théorie des Nombres de Bordeaux 10, 2 (1998), 399–499.[GJ] G. Gras et J-F. Jaulent, Sur les corps de nombres réguliers, Math. Z. 202 (1989), 343–365.[Gre] C. Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier 42, 3(1992), 449–499.

[Hel] Y. Hellegouarch, Invitation aux mathématiques de Fermat–Wiles, Masson, Paris 1997.[He1] C. Helou, Norm residue symbol and cyclotomic units, Acta Arith. 73 (1995), 147–188.[He2] C. Helou, Proof of a conjecture of Terjanian for regular primes, C. R. Math. Rep. Acad. Sci.Canada 18 (1996), 5, 193–198.

[Iw] K. Iwasawa, A note on Jacobi sums, Symposia Mathematica 15, Academic Press (1975), 447–459.[J] J-F. Jaulent, Sur le noyau sauvage des corps de nombres, Acta Arith. 67 (1994), 335–348.[JN] J-F. Jaulent et T. Nguyen Quang Do, Corps p-rationnels, corps p-réguliers et ramification re-streinte, J. Théorie des Nombres de Bordeaux 5 (1993), 343–365.

[Kr] M. Krasner, Sur le premier cas du théorème de Fermat, C. R. Acad. Sci., Paris 199 (1934),256–258.

[Ku] M. Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of Z, Compos.Math. 81, 2 (1992), 223–236.

[LS] H.W. Lenstra jun. and P. Stevenhagen, Class field theory and the first case of Fermat′s last theo-rem, Cornell, Gary (ed.) et al., Modular forms and Fermat′s last theorem. Papers from a conference,Boston 1995, New York, Springer (1997), 499–503.

[MN] A. Movahhedi et T. Nguyen Quang Do, Sur l’arithmétique des corps de nombres p-rationnels,Sém. Th. Nombres Paris (1987/1988), Prog. in Math. 89 (1990), 155–200.

[R] P. Ribenboim, 13 Lectures on Fermat′s Last Theorem, Springer, New York 1979.[Ri] K. Ribet, A modular construction of unramified p-extensions of Qp, Invent. Math. 34 (1976),151–162.

[Sk1] L. Skula, Some historical aspects of the Fermat problem, Pokroky Mat. Fyz. Astron. 39, 6(1994), 318–330.


Georges Gras 119

[Sk2] L. Skula, Eine Bemerkung zu dem ersten Fall der Fermatschen Vermutung, J. Reine Angew.Math. 253 (1972), 1–14.

[S] C. Soulé, Perfect forms and the Vandiver conjecture, J. Reine Angew. Math. 517 (1999), 209–221.[Ta] J. Tate, Relations between K2 and Galois cohomology, Invent. Math. 36 (1976), 257–274.[Te] G. Terjanian, Sur la loi de réciprocité des puissances ℓ-èmes, Acta Arith. 54, 2 (1989), 8-125.[Th1] F. Thaine, On Fermat′s last theorem and the arithmetic of Z[ζp + ζ−1

p], J. Number Theory 29,

3 (1988), 297–299.[Th2] F. Thaine, On the first case of Fermat′s last theorem, J. Number Theory 20 (1985), 128–142.[Th3] F. Thaine, Polynomials generalizing binomial coefficients and their application to the study ofFermat′s last theorem, J. Number Theory 15 (1982), 304–317.

[Th4] F. Thaine, On the ideal class groups of real abelian number fields, Ann. Math. (2) 128, 1 (1988),1–18.

[Wa] L.C. Washington, Introduction to cyclotomic fields, Graduate Texts in Math. 83, Springer-Verlag1982, enlarged second edition 1997.

March 14, 2010

Georges Gras, Villa la Gardette, chemin Château Gagnière, F-38520 Le Bourg d’OisansE-mail : [email protected] • Url : http://monsite.orange.fr/maths.g.mn.gras/


ASYMPTOTICALLY GOOD FAMILIES

by

Farshid Hajir

Abstract. — We define the general concept of asymptotically good families, and describe them

in the context of curves and codes over finite fields, number fields, and regular graphs.

Résumé. — Nous définissons le concept de familles asymptotiquement exactes et décrivons

celui-ci pour les courbes et les codes sur les corps finis, les corps de nombres et les graphes

réguliers.

1. Introduction

There is a class of optimization problems in various branches of mathematics, including num-

ber theory, algebraic geometry, coding theory, and graph theory, that can be classified under

one rubric, namely that of “asymptotically good families.” In this short article, I sketch some

common characteristics for several specific problems and describe a general framework for all

of them. The larger aim is to encourage and enlarge study of the deep and fruitful analogies

that exist between them, and especially to stimulate further cross-fertilization of ideas and

methods.

A very well-established analogy of this type is that between number fields and function fields of

curves: it has motivated much of the advances in arithmetic and algebraic geometry. Analogies

and other types of connections between codes and graphs have proven to be very fruitful to

both fields as well. Starting in the 1980s, the construction of codes using methods of algebraic

geometry revitalized both coding theory and the study of varieties with extremal properties.

Less well-explored are connections between the theory of number fields and those of graphs

and codes: it is hoped that studying these connections in the context of asymptotically good

families may enrich all these domains of study, for example by bringing to the attention of

researchers in any one of these fields methods and ideas which are natural in one or more of

the others.

The author was supported by a grant from the NSA.

122 Asymptotically Good Families

Especially since it’s quickly done, it will be instructive to give, right from the outset, a

description of the formalization of the concept of asymptotically good families, prior to dis-

cussing the instances which led to the more abstract definition. To begin, we require a context

C = (O,T , τ, α), where O,T are sets and τ, α are maps τ : O → T and α : O → R≥0. Here,

O is the set of objects of interest, T is a paramater space of types of the objects, and α is

the critical invariant measuring the “quality” of the object. The parameter space is usually

a familiar and countable set; for convenience, we will assume that τ is surjective. It goes

without saying that our normalization is such that “good” objects are those of high quality.

What interests us in particular is not any single object of high quality (a “gem”), but an

infinite necklace on which we may hang a sequence of gems. More precisely, a family F in

O is a sequence F1, F2, · · · of pairwise distinct elements of O. We say that F = (Fi)

is isotypic of type t if every member of F has type t, i.e. τ(Fi) = t for all i. We extend

α to families by putting α(F) = lim infi→∞ α(Fi), for F = (F1, F2, · · · ) and say that F is

asymptotically good if α(F) > 0. In the contexts we have in mind, it is typically difficult to

construct asymptotically good families, or at least to do so explicitly.

With these preliminaries in place, we can now define the main object of interest attached to

a context C = (O,T , τ, α), namely the asymptotic envelope function A : T → R≥0 given by

A(t) := supF of type t

α(F),

where the limit is taken over all isotypic families of type t. Thus, the map A is induces by τ

and α as in the following diagram.

Oτ

α

!!CCCC

CCCC

TA

// R≥0

It is clear that the asymptotic envelope function is a measure not of the quality of individual

objects, but rather of the quality of infinite non-repeating strings of those of a fixed type. We

will say that functions L,U : T → R≥0 are lower, respectively, upper bounds for A in the

context C if

L(t) ≤ A(t) ≤ U(t) for all t ∈ T.

In most cases we will discuss, we will be able to estimate A(t) by upper and lower bounds

but of course would like to have an explicit formula for A(t) itself. Typically, the theory

provides a natural and “decent” upper bound, meaning one that is believed to be sharp.

Interestingly, the source of this upper bound is usually a zeta function known or at least

suspected to satisfy an appropriate Riemann hypothesis. Obtaining lower bounds L(t) involves

the creation of examples with extremal properties, usually from objects carrying inordinately

many symmetries – it is not surprising that automorphic forms are a typical source. What

has been at times a revelation is that automorphic forms are at the root of good lower bounds


Farshid Hajir 123

even in contexts that do not at first glance appear to be related to number theory or algebraic

geometry.

2. Some Examples

Now let us introduce the contexts Cff (function fields), Cnf (number fields), Clc (linear codes),

and Crg (regular graphs), by specifying their types, critical invariants etc. and describing the

known lower and upper bounds for their asymptotic envelopes. There are many other contexts

that fit the general framework, for example that of tightly packed lattices in Euclidean space,

but we will treat them elsewhere. Naturally, the reader is encouraged to be on the lookout

for other contexts which fit into the rubric of asymptotically good families!

2.1. Function Fields of Curves over Finite Fields. — To introduce the context Cff let

O = Off be the set of all extensions K/F(x) where |F| and [K : F(x)] are both finite. In other

words, our objects are function fields of smooth projective geometrically irreducible curves

over a finite field, i.e. transcendence degree 1 fields over finite fields, but note that our curves

come equipped with a particular map to the projective line. The space of types Tff = Q is the

set of all prime powers, i.e. of integers q = pm where p is a prime and m is a positive integer,

and τ(K/F(x)) = τff(K/F(x)) := |F|. Last but not least, we define the critical invariant α by

α(K/Fq(x)) = αff(K/Fq(x)) :=|N1(K/Fq)|

g(K),

where g(K) is the genus of K (or of the curve X corresponding to K) and N1(K/Fq) = |X(Fq)|is the number of degree 1 primes of K/Fq

, or, what is the same, the number of Fq-rational

points of X. Roughly speaking, the idea is to find curves with many points, as measured

against the genus of the curve. The upper bound for the critical invariant α(K) for an

individual K comes from the Hasse-Weil bound:

N1(K/Fq(x)) ≤ q + 1 + 2g(K)√

q.

It is a reflection of the fact that the zeta function of K satisfies the Riemann Hypothesis.

When applied to families, this already gives the bound Aff(q) ≤ 2√

q. Taking this much

further, Serre, Ihara and Drinfeld-Vladut obtained a succession of improvements yielding,

for an asymptotically good family of curves of fixed type q, Uff(q) =√

q − 1. Via a class-

field tower construction involving a graph argument, Serre (see [S] and [EHKPWZ]) gave a

general lower bound for Aff(q): Lff(q) = C log(q) for a positive absolute constant C. When

m is even, and q = pm ≥ 49, a much better lower bound is obtained by using modular curves,

actually reaching the Drinfeld-Valdut upper bound, thus proving that A(p2k) = pk − 1 if

pk ≥ 7.

2.2. Number Fields. — For the context of number fields Cnf , the set of objects Onf consists

of fields K of finite degree n(K) over Q. The type of a number field is defined to be τ(K) =

r1(K)/n(K); it is the proportion of the embeddings of K into C with image contained in R.



The space of possible types in this context is T = [0, 1] ∩ Q. As the critical invariant, we

choose the recriprocal logarithmic root discriminant:

αnf(K) :=n(K)

log |disc(K)| ,

where disc(K) is the absolute discriminant of K and n(K) = [K : Q] is its absolute degree;

for the field Q, we put α(Q) = 0. Under the Generalized Riemann Hypothesis (GRH), we

have a bound due to Stark, Odlyzko and Serre, namely

Anf(t) ≤ U∗(t) := (log(8π) + γ + πt/2)−1.

The “*” is to remind us that this holds under the additional assumption of GRH. There is

an unconditional upper bound as well; see [Od] for more details on these bounds. As for

lower bounds, the only source of good families in Cnf we currently know are nested fields

K0 ( K1 ( · · · which are ramified at finitely many places and shallowly ramified (they

exist by a theorem of Golod and Shafarevich). As a result we do not have an explicit lower

bound L(t), though by [HM], we have A(0) ≥ 1/ log(83) and A(1) ≥ 1/ log(955). Most

researchers believe the upper bound U∗(t) is sharp. Note that U∗(0) ≈ 1/ log(44.7) and

U∗(1) ≈ 1/ log(215.3).

2.3. Linear Codes. — Now consider the context Clc,, with Olc being the set of all linear

codes over finite fields; a general reference is [TV]. Recall that a linear code of length n and

dimension k over Fq is a k-dimensional linear subspace of Fnq . As in the case of Cff , we define

the type of a linear code C/Fq to be q = pm. We equip Fnq with the Hamming metric,(1) and

let d be the minimum distance between two distinct codewords (elements of C). A code can

be used for communicating through a noisy channel in a way that allows for the correction of

errors that may occur through transmission, at the cost of transmitting at a lower efficiency

rate. We define the quality of a linear code C of dimension k, length n and minimum distance

d to be α(C) = kd/n2. The ratios R(C) = k/n and δ(C) = d/n are known as the rate and

relative distance of C; they both belong to the unit interval. The closer the rate is to 1, the

more efficient the code is, while the closer the relative distance is to 1, the greater its capacity

for error detection and correction. Since the quality of a code is the product of its rate and

its relative distance, a family of codes over Fq is asymptotically good if and only if the rates

and relative distances of its members stay bounded away from 0: this ensures that the codes

are efficient and carry good error-correction capabilities. Thanks to the multiplicity of ways

for deforming one code into another one with slightly different parameters, we have a “higher

resolution” picture of the distribution of asymptotically good families in this context. Namely,

consider the set X consisting of limit points of the set of all (δ(C), R(C)) ∈ [0, 1]2 as C runs

over all linear codes over Fq. Then there is a function ρq(δ) such that for all (δ0, R0) ∈ [0, 1]2,

(δ0, R0) belongs to X if and only if R0 ≤ ρq(δ0). We have explicit upper and lower bounds

ρGVq (δ) ≤ ρq(δ) ≤ ρJPL

q (δ),

(1)the Hamming distance between two vectors is the number of positions in which they differ


Farshid Hajir 125

which we will not specify here, see [TV]. The lower bound is known as the Gilbert-Varshamov

bound, and the upper one as the JPL Bound (its authors worked at the Jet Propulsion

Laboratory). From these explicit functions, one can extract explicit lower and upper bounds

Llc(q) and Ulc(q) for Alc(q).

2.4. Regular Graphs. — Our last context Crg has as its objects Org, the set of connected

finite regular graphs; a general introduction is given in the survey article [HLW], to which we

refer for the results discussed below. Recall that a t-regular graph is a graph whose vertices

all have degree t, i.e. have t edges emanating from them. We define the type of a graph

G = (V,E) ∈ Org to be the degree t of any one of its vertices v ∈ V and suppose t ≥ 3,

thus Trg = Z≥3. Self-loops and multiple edges are allowed for our graphs. If S ⊆ V , we let

∂S be the set of edges from S to its complement V \ S. We will discuss two types of critical

invariants for t-regular graphs. First, we may work with αer(G) = h(G) where h(G) is the

edge expansion ratio of G, defined by

h(G) = minS⊆V,|S|≤|V |/2

|∂S||S| .

Alternatively, the adjacency matrix of G (with rows and columns indexed by V having u, v-

entry equal to the number of edges from u to v) is a real symmetric n by n matrix. Writing its

eigenvalues as t = λ0 ≥ λ1 ≥ . . . ≥ λn ≥ −t, we let λ(G) = λ1(G) be its “second” eigenvalue.

Let us define the “spectral gap” of a t-regular graph G to be αsg(G) = t−λ(G). This quantity

is closely related to h(G) via the Dodziuk/Alon-Milman theorem:

t − λ(G)

2≤ h(G) ≤

√2t

√

t − λ(G).

Consequently, a family of t-regular graphs is asymptotically good with respect to the critical

invariant αer if and only if it is good with respect to αsg. Such a family is called a family of

expander graphs. They have many applications in cryptography as well as coding theory, not

to mention other branches of mathematics. Let us work with αsg, the spectral gap from now

on. By a theorem of Alon-Boppana, we have the upper bound

Asg(t) ≤ t − 2√

t − 1,

i.e. we can take U sg(t) = t − 2√

t − 1. As in the previous cases, to obtain a lower bound, we

must construct families of t-regular graphs of large spectral gap.

The best we can hope for is such a family which meets the upper bound t−2√

t − 1. With this

in mind, we say that a graph is t-Ramanujan if it is t-regular and satisfies αsg(G) ≥ t−2√

t − 1.

(Usually this is stated in the equivalent formulation: λ(G) ≤ 2√

t − 1). Thus, if t ≥ 3 is an

integer such that a family of t-Ramanujan graphs exist, then Asg(t) = t − 2√

t − 1. Thanks

to the work of Lubotzky, Phillips, Sarnak, Margulis, Morgenstern ..., it is known that if t− 1

is a prime power, then families of t-Ramanujan graphs exist. The known constructions are

all “automorphic” at root. We also note that a regular graph is t-Ramanujan if and only if its

Ihara zeta function satisfies the Riemann Hypothesis (Cor. 4.5.9 of Lubotzky [L]).



3. Some Open Questions

The analogies sketched above go quite a bit deeper in certain situations. Namely, for some

of the ordered pairs of contexts introduced above, there are known constructions which map

an asymptotically good family F = (F1, F2, . . .) of objects in C to an asymptotically good

family F ′ in C ′, together with an estimate for α(F ′) in terms of α(F).

For example, if C = Cff and C ′ = Clc, then the Goppa construction of algebraic-geometric

codes gave a totally unexpected improvement on the Gilbert-Varshamov lower bound (for

q ≥ 49). It also led indirectly to the determination of Aff(pm) for all even m. By contrast,

the mapping of certain types of good families from Cnf to Clc by Guruswami [Gu] is not very

well-studied. Another highly important such mapping is from expander graphs to linear codes,

giving the first construction of asymptotically good codes that can be coded and decoded in

linear time.

It’s clear that among the four contexts introduced here, the one about which we know the

least is number fields. It would be highly interesting to find an “automorphic” construction

of asymptotically good families of number fields, or a method for producing them from an

asmptoticaly good family of codes or graphs. Currently, the only known method in the

number field context is the Golod-Shafarevich criterion. Is it possible to adapt the probablistic

methods that have proved so fruitful in other contexts to this setting?

Note also that for number fields, we do not yet have an explicit lower bound Lnf(t) ≤ Anf(t).

It’s reasonable to expect a bound Lnf(t) = ((t−1) log(83)+ t log(955))−1, i.e. to fit a “convex”

function to the two boundary points that we have. However, this seems to be quite out of

reach at the moment, because it involves problems of signatures of units which are quite

mysterious.

The major open problem in the context of regular graphs is clearly: For which t ≥ 3 do

families of t-Ramanujan graphs exist? Hoory, Linial and Wigderson conjecture that families

of t-Ramanujan graphs exist for all t ≥ 3 (Conjecture 5.13 of [HLW]). Thus, they conjecture

that

Asg(t) =? t − 2√

t − 1 for all t ≥ 3.

The best evidence for this conjecture is probably the theorem of Friedman to the effect that

for any ǫ > 0, fixed t and n tending to infinity, the probability that a random t-regular graph

on n vertices has λ(G) ≤ 2√

t − 1 + ǫ is 1 − on(1).

Both in the case of Cff and Crg, since we know that the upper bound is sharp for a substantial

subset of parameters t ∈ T , it is tempting to believe that it is so for all values of t. There

is a simple, but deep, example in which that turns out not to be the case: namely for the

Shannon capacity of cyclic graphs. To define the context Csc, let Osc = Cnt |n ≥ 1, t ≥ 3

be the set of all n-fold self-products of the cyclic graph Ct on t vertices (say a regular t-gon).

As in the case of regular graphs, we have T = Tsc = Z≥3 and τ(Cnt ) = t. To introduce the

critical invariant, recall that For a graph G, the independence number of G, int(G), is the

size of a maximal subset of its vertices not joined by any edges. The critical invariant, the


Farshid Hajir 127

Shannon capacity, is defined by

αsc(Cnt ) :=

log2(int(Cnt ))

n.

We refer the reader to [AZ] for the information-theoretic motivation of this definition. Shan-

non showed that Asc(t) ≤ t/2 for all t. It is then very easy to show that for even integers t,

Asc(t) = t/2 but Shannon discovered that the computation of Asc(t) for odd t ≥ 5 is highly

non-trivial, and he was unable to determine even A(5). Note that as in the case of Cff , there is

a simple formula for Asc(t) for exactly half of the types, and one could guess that Asc(t) = t/2

for all t ≥ 3. However, it turns out that for odd t ≥ 5,

(t − 1)/2 < Asc(t) ≤t

1 + (cos(π/t))−1< t/2.

It is known by a celebrated theorem of Lovasz that the middle inequality above is sharp for

t = 5, but the value of Asc(t) is not known for larger odd t; see [AZ] for more details.

Just as the existence of t-Ramanujan graphs for t 6= pe +1 is unknown, leaving the possibility

that for such t there is a strict inequality Arg(t) < Urg(t) = t − 2√

t − 1, for q = pm with m

odd, we have

C log(pm) ≤ Aff(pm) ≤ √pm − 1,

with the upper bound sharp for even m but not known to be so for odd m. It would be of

great interest to find a single prime p for which we can determine whether Aff(p) =√

p− 1 is

true or false.

References

[AZ] Aigner, Martin; Ziegler Gunter M. Proofs from The Book. Including illustrations by Karl H.Hofmann. Third edition. Springer-Verlag, Berlin, 2004. viii+239 pp.

[EHKPWZ] Elkies, Noam D.; Howe, Everett W.; Kresch, Andrew; Poonen, Bjorn; Wetherell, JosephL.; Zieve, Michael E. Curves of every genus with many points. II. Asymptotically good families. DukeMath. J. 122 (2004), no. 2, 399–422.

[Gu] Guruswami, Venkatesan Constructions of codes from number fields. IEEE Trans. Inform. Theory49 (2003), no. 3, 594–603.

[HM] Hajir, F.; Maire, C. Tamely ramified towers and discriminant bounds for number fields II, J.Symb. Comp. 33 (2002), no. 4, 415–423.

[HLW] Hoory, Shlomo; Linial, Nathan; Wigderson, Avi Expander graphs and their applications Bull.Amer. Math. Soc. 43 (2006), 439-561.

[L] Lubotzky, Alexander Discrete groups, expanding graphs and invariant measures. With an ap-pendix by Jonathan D. Rogawski. Progress in Mathematics, 125. Birkhauser Verlag, Basel, 1994.xii+195 pp.

[Od] Odlyzko; A.M. Bounds for discriminants and related estimates for class numbers, regulators andzeros of zeta functions: a survey of recent results, Sém. de Théorie des Nombres, Bordeaux 2 (1990),119-141.

[S] Serre, J.P. Rational points on curves over finite fields, unpublished lecture notes by F. Q. Gouvêa,Harvard University, 1985.



[TV] Tsfasman, M. A.; Vladut, S. G. Algebraic-geometric codes. Translated from the Russian by theauthors. Mathematics and its Applications (Soviet Series), 58. Kluwer Academic Publishers Group,Dordrecht, 1991. xxiv+667 pp.

May 3, 2010

Farshid Hajir, Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003,

USA. • E-mail : [email protected]


EQUIVARIANT L-FUNCTIONS AT s = 0 AND s = 1

by

David Solomon

Abstract. — For an abelian extensions of number fields, we review some basic theory and

formulate the Stark Conjecture in terms of the ‘equivariant’ L-function at s = 0. After surveying

the known cases, we describe some refinements and extensions due to Rubin, Brumer et al. and

results concerning Fitting ideals of class groups. Finally, we summarise some recent work on

minus parts at s = 1.

Résumé (Les Fonctions L Équivariantes en s = 0 et en s = 1). — Après quelques

rappels, nous énonçons la Conjecture de Stark pour une extension abélienne de corps de nombres,

formulée en termes de la fonction L ‘équivariante’ en s = 0. Nous survolons les cas connus

et expliquons certaines conjectures plus fines dues à Rubin, Brumer et al. ainsi que quelques

résultats concernant l’idéal de Fitting du groupe des classes. Enfin, nous résumons certains

travaux récents concernant les parties moins en s = 1.

1. Introduction

This article is is an expanded version of the notes from four lectures given by the author at

the conference ‘Fonctions L et Arithmétique’ in Besançon, in June 2009. It surveys work on

several different conjectures concerning the special values of L-functions attached to characters

of Galois extensions K/k of number fields.

In our presentation – and largely in historical fact – the development of such conjectures

begins with the seminal work of Stark in [St]. We shall, however, consider only the case in

which G = Gal(K/k) is abelian, for which the theory is currently richest. In this context,

we shall work with the equivariant L-function ΘS = ΘS,K/k(s) attached to K/k and a set S

of places of k subject to certain conditions. This simply assembles the usual S-truncated L-

functions for all irreducible characters of G into a single function taking values in the complex

group-ring C[G]. Consequently, the conjectures can be formulated in terms of certain elements

and ideals of group-rings R[G] and ‘arithmetic’ R[G]-modules attached to K etc. Here, R

2000 Mathematics Subject Classification. — 11R20, 11R27, 11R29, 11R37, 11R42, 11S31.

Key words and phrases. — number field, L-function, zeta-function, abelian extension, Stark’s Conjecture,

Brumer’s Conjecture, S-unit, class group, class field theory, explicit reciprocity law.

130 Equivariant L-Functions at s = 0 and s = 1

is a commutative ring, variously Q (the rationals), Z (the integers), Qp or Zp (the p-adic

rationals or integers, for a prime p). The commutativity of R[G] allows us to use the algebra

of determinants, annihilators, Fitting ideals etc.

In Sections 2 and 3 we briefly review the definitions and theory of the L-functions concerned,

starting with those attached to ray-class characters of k, moving on to characters of G via

class-field theory and hence to the equivariant L-function ΘS mentioned above. More details

of the basic theory can be found in standard books on Algebraic Number Theory such as [La].

See also [Ta, Ch. 0] or [Ma] for some of the more advanced facts. In Section 4 we motivate

and then state the Stark’s basic conjecture in the abelian case, in a formulation due to Rubin.

This concerns the leading Taylor coefficient of ΘS(s) at the point s = 0. (Despite the title,

the latter half of [St] also focusses on s = 0, as does the majority of subsequent work.)

Section 5 briefly reviews the current state of research on the basic abelian conjecture and

its ‘integral’ refinements. Following on from the latter, Section 6 explains the link – via the

Brumer-Stark Conjecture – with Brumer’s conjecture on the annihilation of (the minus-part

of) the class group of K in the case where K is CM and k totally real. The latter is a

conjectural generalisation of Stickelberger’s Theorem. Section 7 tells the story of recent work

attempting to refine the Brumer Conjecture using Fitting ideals of class groups, much of it

due to Greither and Kurihara. The last two sections deal with recent work of the author

concerning the minus-part of ΘS(s) at the point s = 1. The lack of a suitable ‘equivariant

functional equation’ means that there is no simple logical connection with the above-mentioned

work at s = 0. Instead, a fundamental role is played by a certain p-adic logarithmic map

sp which is introduced in Section 8. Finally, in Section 9 we explain two conjectures made

in [So1] and [So2]: the Integrality Conjecture concerning the image Sp of sp in Qp[G] and

the Congruence Conjecture. The latter is a sort of conjectural explicit reciprocity law that

makes a link with the Stark Conjecture in the plus-part at s = 0. We also pose a rather more

tentative ‘Question’ which aims to relate Sp to the issues discussed in Section 7.

This survey will suit readers with little previous knowledge of the subject but leaves much

out. In particular, we shall not touch on the analogous conjectures at integer values of s

different from 0 and 1, nor on Serre’s or Gross’ p-adic conjectures or their extensions. The

function-field case and that of non-abelian G are also hardly mentioned. For more detailed

and/or extensive accounts the reader may consult the sources cited in the text or the four

earlier survey articles in [BPSS] by Dummitt, Flach, Greither and Popescu.

1.1. Basic Notations and Conventions. — In addition to the notations already intro-

duced, N, R and C will denote the natural, real and complex numbers respectively. We shall

denote by Q the algebraic closure of Q in C and also fix an algebraic closure Qp of Qp for each

prime number p. A ‘number field’ L is always a finite extension of Q within Q. Its abelian

closure in Q is denoted Lab. Let F be any field, l ∈ N and p a prime number. We shall write

µ(F ) (resp. µl(F ), resp. µp∞(F ) ) for the group of all roots of unity (resp. all lth roots of

unity, resp. all p-power roots of unity) in F . When F is omitted it is understood to be Q. If


David Solomon 131

L is a number field, we shall write WL for the cardinality |µ(L)| of µ(L). Finally, we shall use

the notation χ0 for the trivial character of any finite abelian group.

2. Ray-Class L-Functions

2.1. Basic Definitions. — Let k be a number field with ring of integers Ok, r1 real places

and r2 complex places so that r1 + 2r2 = n := [k : Q]. For our purposes, a cycle for k will be

a formal product f = f0f∞ where f0 is any non-zero ideal of Ok and f∞ is the formal product

of any subset of the real places of k. Thus f can be written uniquely as an infinite product

with only finitely many non-trivial terms:

f = f0f∞ =∏

p

pnp(f)∏

v

vnv(f)

Here, p runs through the set of all non-zero, prime ideals of Ok and np(f) = ordp(f0) ∈ Z≥0.

Thus p|f0 if and only if np(f) ≥ 1. Similarly, v runs through real places of k and nv(f) ∈ 0, 1.By analogy, ‘v|f∞’ will indicate nv(f) = 1. Let I(k) denote the group of (non-zero) fractional

ideals of k under multiplication and P (k) its subgroup of principal fractional ideals. To each

cycle f for k there corresponds the subgroup If(k) of I(k) consisting of those fractional ideals

prime to f0, and a subgroup Pf(k) of P (k) consisting of the principal ideals possessing a

generator ‘congruent to 1 mod f’:

Pf(k) := (α) : α ∈ k×, ordp(α− 1) ≥ np ∀ p|f0, ιv(α) > 0 ∀ v|f∞where ιv : k → R is the embedding corresponding to the real place v. Clearly, Pf(k) is

contained in If(k) and the quotient If(k)/Pf(k) is, by definition, the ray-class group Clf(k) of

k modulo f. (We shall write [a]f for the image in Clf(k) of any a ∈ If(k).) It is finite and, of

course, abelian, so its characters may be identified with homomorphisms χ : If (k) −→ µ(C)

such that χ((α)) = 1 for all (α) ∈ Pf(k). To any such f and χ we associate a ray-class

L-function Lf(s, χ), initially defined on the set s ∈ C : ℜ(s) > 1 by

(1) Lf(s, χ) :=∑

aOka∈If(k)

χ(a)Na−s =∏

p prime

p∤f0

(

1 − χ(p)Np−s)−1

(By standard comparisons, the above sum and Euler product converge absolutely to the same

analytic function on this set.)

Example 2.1. — The Dedekind Zeta-Function. Suppose f is trivial i.e. f0 = Ok and f∞is the empty product, so that Clf(k) = Cl(k), the class group. If also χ is the trivial character

χ0 of Cl(k), then (1) gives Lf(s, χ) =∑

aOk, a6=(0)

Na−s which coincides with the Dedekind zeta

function ζk(s) of k.

Example 2.2. — Ray-Class L-functions for k = Q. Take f to be (f) = fZ for some

f ∈ Z≥1 and f∞ to be ∞, the unique real place of Q. There is then an isomorphism from

(Z/fZ)× to Clf(Q) sending a to [(a)]f, where a is any positive integer prime to f . Thus a



character χ of Clf(Q) coincides with a Dirichlet character modulo f and one checks easily

from (1) that Lf(s, χ) is just the corresponding Dirichlet L-function.

2.2. Primitivity and the Functional Equation. — We shall say that one cycle g divides

another f (written g|f) iff np(g) ≤ np(f) for all p (i.e. g0|f0) and nv(g) ≤ nv(f) for all v. In

this case If(k) ⊂ Ig(k) and Pf(k) ⊂ Pg(k) so there is a homomorphism

πf,g : Clf(k) −→ Clg(k)

[a]f 7−→ [a]g

Using weak approximation one can show firstly that πf,g is surjective and secondly that if h

also divides f then ker(πf,g) ker(πf,h) = ker(πf,(g,h)). (Here (g, h) denotes the cycle that is the

h.c.f. of g and h in the obvious sense). It follows that there exists a unique minimal cycle w.r.t.

divisibility, say fχ, such that χ factors through πf,fχ i.e. such that there exists a character χ

of Clfχ(k) with χ = χπf,fχ . We shall call fχ the conductor of χ and χ the primitive character

associated to χ.

Remark 2.3. — The Idelic Viewpoint. Let Id(k) be the idèle group of k and C(k) :=

Id(k)/k× the idèle-class group. For each cycle f one can use weak approximation to define a

(surjective) homomorphism C(k) → Clf(k). Thus each ray-class character χ modulo f gives

rise to an idèle-class character that is continuous and of finite order. All such characters arise

in this way. Moreover χ1 and χ2 give rise to the same idèle-class character if and only if

χ1 = χ2.

If χ is primitive (i.e. fχ = f, so χ = χ) we shall write simply L(s, χ) for the ‘primitive L-

function’ Lf(s, χ). If χ is imprimitive then Lf(s, χ) and L(s, χ) differ at most by finitely may

Euler factors. More precisely, one clearly has

(2) Lf(s, χ) =

(

∏

p|f

p∤fχ

(

1 − χ(p)Np−s)

)

L(s, χ).

So suppose χ is a primitive ray-class character modulo f. We summarise the well-known

‘analytic continuation’ and ‘functional equation’ for L(s, χ). Firstly, L(s, χ) extends to a

meromorphic function on C. This is analytic at s except possibly when s = 1 where it has a

simple pole iff χ = χ0 i.e.

(3) ords=1(Lf(s, χ)) = −δχ,χ0

in Kronecker’s notation. (It follows easily from (2) that exactly the same is true of Lf(s, χ),

even when χ is imprimitive.) Secondly, let a1(χ) (resp. a2(χ)) denote the number of real

places v such that v ∤ f∞ (resp. v|f∞) so that a1(χ) + a2(χ) = r1 and define a completed

L-function

Λ(s, χ) := (|dk|N f0)s/22r2(1−s)π−(ns+a2(χ))/2Γ(s/2)a1(χ)Γ((1 + s)/2)a2(χ)Γ(s)r2L(s, χ)


David Solomon 133

where dk denotes the discriminant of k. (The Γ-factors can be considered as Euler factors at

infinite places.) Then we have an identity of meromorphic functions:

(4) Λ(1 − s, χ) =i−a2(χ)τ(χ−1)

(N f0)1/2Λ(s, χ−1)

where χ−1 denotes the inverse character of χ (which has the same conductor) and τ(χ) denotes

the Gauss sum (see e.g. [Ma]. Note that |τ(χ)|2 = N f0 = (−1)a2(χ)τ(χ)τ(χ−1).)

3. The Galois Viewpoint

3.1. Set-Up. — Suppose now that K is a finite, Galois extension of the number field k such

that G := Gal(K/k) is abelian. Class-field theory associates to K/k a cycle f = fK/k with the

following properties:

(i) A real place v of k divides f∞ iff one (hence any) place above v in K is complex.

(ii) A non-zero prime ideal p of Ok divides f0 iff p ramifies in K.

(iii) There is a well-defined homomorphism (the Artin homomorphism)

Clf(k) −→ G

sending [p]f to the Frobenius element of G at p for each non-zero prime ideal p ∤ f0.

Note that fK/k is the unique minimal cycle for k for which the description in (iii) gives a

well-defined homomorphism Clf(k) → G. It is then unique and surjective and the image of

[a]f (for any ideal a prime to f0) will be denoted σa.

Let G denote the set of all complex irreducible characters of G, i.e. all homomorphisms

χ : G → C×. Composing any such χ with the Artin homomorphism gives rise to a ray-class

character ClfK/k(k) → C×, also denoted χ. Thus fχ divides fK/k and the two cycles are equal

iff χ is primitive mod fK/k. (In fact fK/k is always the l.c.m. of the set fχ : χ ∈ G.) We

have

LfK/k(s, χ) :=

∏

p prime

p∤fK/k,0

(

1 − χ(σp)Np−s)−1

=∏

p6∈Sram

(

1 − χ(σp)Np−s)−1

where Sram = Sram(K/k) denotes the set of (finite) ramified primes in K/k. It is sometimes

convenient to remove further Euler factors from the R.H.S. above. We denote by S∞ = S∞(k)

the set of all infinite places of k and by Smin = Smin(K/k) the set Sram ∪ S∞. For any finite

set S of places containing Smin and any χ ∈ G, we define the S-truncated L-function to be

(5) LS(s, χ) :=∏

p6∈S

(

1 − χ(σp)Np−s)−1

=

(

∏

p∈S

p∤fχ

(

1 − χ(p)Np−s)

)

L(s, χ)

which clearly has a meromorphic continuation to C. Expanding the first product gives

(6) LS(s, χ) =∑

aOk(a,S)=1

χ(σa)Na−s =∑

g∈G

χ(g)ζS(s, g) for ℜ(s) > 1



where (a, S) = 1 indicates that the ideal a is prime to every p ∈ S and the partial zeta-function

ζS(s, g) is the Dirichlet series∑

Na−s where a ranges through all such integral ideals with

σa = g. As before, it is convergent for ℜ(s) > 1 for any g ∈ G.

Remark 3.1. — Artin L-Functions. We consider briefly the more general situation where

K/k is Galois but G = Gal(K/k) is not necessarily abelian, where χ is the character of a

d-dimensional complex representation ρ : G → GL(V ) (but possibly d > 1) and where S

is any set of places of k containing S∞ (but not necessarily Sram). In this set-up the Artin

L-function may be defined for ℜ(s) > 1 by generalising the second member of (5):

LS,Artin(s, χ) :=∏

p6∈S

det(1 −Np−sAP)−1

Here, P is any prime of K above p with inertia group TP ⊂ G say, and AP denotes the

endomorphism of V ρ(TP ) induced by ρ(FrobP(K/k)) (the latter being defined only up to an

element of ρ(TP)). If G is abelian and d = 1 then ρ = χ ∈ G and it is not hard to show

that LS,Artin(s, χ) agrees with the third member in (5) (which, of course, makes sense even

if Sram 6⊂ S). If G is non-abelian, ρ does not in general give rise to-ray class characters over

k. However, Brauer induction and the formal properties of Artin L-functions allow us to re-

express them in terms of ray-class L-functions for extensions over various intermediate fields

k′ with K ⊃ k′ ⊃ k. For more details of Artin L-functions, the properties they enjoy and for

Stark’s conjectures in the nonabelian case, we refer to [Ta, Chs. 0,1].

3.2. The Equivariant L-Function. — The rest of this article will be concerned exclusively

with the case G abelian and S ⊃ Smin. We can now give three equivalent definitions our basic

object of study, the S-truncated equivariant L-function ΘS(s) = ΘS,K/k(s). Firstly, for K/k,

G and S as above we set

(7) ΘS(s) =∑

g∈G

ζS(s, g)g−1

This is a priori a function on s ∈ C : ℜ(s) > 1 with values in the group-ring C[G]. Each

χ ∈ G extends C-linearly to a ring homomorphism χ : C[G] → C and (6) gives

(8) χ(ΘS(s)) = LS(s, χ−1) for every χ ∈ G.

Character theory implies that equations (8) determine ΘS(s) uniquely so it follows from (5)

that ΘS(s) could also have been defined by the Euler product (in C[G])

(9) ΘS(s) =∏

p6∈S

(

1 − σ−1p Np−s

)−1

which makes sense and converges for ℜ(s) > 1. Finally, we can invert equations (8) explicitly

to write ΘS(s) in terms of L-functions. Let eχ be the idempotent of C[G] associated to χ ∈ G,


David Solomon 135

i.e. eχ := 1|G|

∑

g∈G χ(g)g−1. Character theory and (5) give

(10) ΘS(s) =∑

χ∈G

LS(s, χ−1)eχ =∑

χ∈G

(

∏

p∈S

p∤fχ

(

1 − χ−1(p)Np−s)

)

L(s, χ−1)eχ

The properties of L(s, χ) now show that ΘS extends to a meromorphic function on C that is

analytic except at s = 1 and satisfies (10). The functions ζS(s, g) for g ∈ G therefore possess

similar extensions and equation (3) shows that they all have simple poles at s = 1 with the

same residue, namely 1/|G| times that of LS(s, χ0). Note also that ΘS(s) is R[G]-valued

for s ∈ R>1, by (9). It follows from the meromorphic continuation that it restricts to an

R[G]-valued, analytic function on R \ 1.We note two important ‘functorial’ properties of ΘS which follow easily from (9), properties

of the Frobenius and analytic continuation. First, if S′ is a finite set of places containing S

then, clearly

(11) ΘS′,K/k(s) =∏

p∈S′

p 6∈S

(

1 − σ−1p Np−s

)

ΘS,K/k

Secondly, let K ′ be any intermediate field with K ⊃ K ′ ⊃ k and let πK/K ′ : C[G] →C[Gal(K ′/k)] be the natural ring homomorphism induced by the restriction homomorphism

G→ Gal(K ′/k). Then

(12) ΘS,K ′/k = πK/K ′ ΘS,K/k

as meromorphic functions on C.

Finally, we point out that there are at least two significant obstacles to obtaining a natural

‘functional equation’ for ΘS(s) by combining (10) with (4). Firstly, the Gauss sums in (4)

depend on χ. Secondly, the parenthesised ‘imprimitivity factor’ on the R.H.S. of (10) not

only depends on χ but may vanish at s = 0 for certain χ. A rather complicated ‘functional

equation’ may nevertheless be constructed along lines suggested in [So1, Rem. 2.3(iii)] by

involving also ΘT,K ′/k(s) for certain intermediate fields K ′ as above and subsets T of S.

4. ΘS at s = 0 and Stark’s Conjecture

4.1. Motivation. — Given K/k and S as above and a certain integer r ≥ 0 depending on

K/k, S, we shall give a formulation of the Basic Abelian Stark Conjecture concerning the rth

Taylor coefficient of ΘS(s) at s = 0. This is very similar to that of Conjecture A′ in [Ru] and

may be motivated by consideration of three elementary examples.

Example 4.1. — Cyclotomic Fields. We consider the case k = Q and K = Q(ζf ) where

ζf := exp(2πi/f) for some integer f > 2. We may assume w.l.o.g. that f 6≡ 2 (mod 4) so that

fQ(ζf )/Q = f := fZ∞ and and Smin(Q(ζf )/Q) equals Sf := p prime : p|f ∪ ∞ which we

take for S. Composing the isomorphism (Z/fZ)× → Clf(Q) of Example 2.2 with the Artin

isomorphism gives the usual isomorphism from (Z/fZ)× to G = Gf := Gal(Q(ζf )/Q) sending



a to ga where ga(ζf ) = ζaf for any integer a with (a, f) = 1, and if also a > 0 then σaZ = ga.

It follows that

ζSf(s, ga) =

∑

n≥1

n≡a mod f

n−s = f sζ(s, a/f) for 0 < a ≤ f , (a, f) = 1 and ℜ(s) > 1

where ζ(s, a/f) donotes the Hurwitz zeta-function (see [Wa, Ch. 4]). The computation of

ζ(0, a/f) in Theorem 4.2 of loc. cit. therefore gives

(13) ΘSf ,Q(ζf )/Q(0) = −∑

1≤a<f

(a,f)=1

(

a

f− 1

2

)

g−1a ∈ Q[Gf ]

Example 4.2. — Real Cyclotomic Fields. Take k = Q, f and other notation as above

but now let K = Q(ζf )+ = Q(ζf + ζ−1f ), the maximal real subfield of Q(ζf ). We take

S to be Sf as before. (Note that Sf ⊃ Smin(Q(ζf )+/Q) with equality unless f = 3 or

4 i.e. Q(ζf )+ = Q.) Since Gal(Q(ζf )/Q(ζf )+) = 1, g−1 we have an isomorphism from

(Z/fZ)×/±1 to G = G+f = Gal(Q(ζf )+/Q). Thus the map πQ(ζf ),Q(ζf )+ of (12) sends

both ga and gf−a 6= ga to the same element ga, say, of G+f . It follows easily from (13) that

ΘSf ,Q(ζf )+/Q(0) = 0, so we can write

(14) ΘSf ,Q(ζf )+/Q(s) = Θ′Sf ,Q(ζf )+/Q(0)s +O(s2) as s→ 0

where Θ′Sf ,Q(ζf )+/Q(0) ∈ R[G]. In fact, we have (see e.g. p. 203, paper IV of [St])

Θ′Sf ,Q(ζf )+/Q(0) = −1

2

∑

1≤a<f/2

(a,f)=1

log |((1 − ζaf )(1 − ζ−a

f ))|g−1a

= −1

2

∑

g∈G+f

log |g(εf )|g−1(15)

where εf := (1 − ζf )(1 − ζ−1f ) ∈ Q(ζf )+,×. It is well known that εf is a local unit at finite

places not dividing f (in fact at all finite places unless f is a prime power).

We remark that, thanks to (8) and (5), equations (13) and (15) can also be established

character-by-character, using the corresponding formulae for L(s, χ) at s = 0, where χ is an

odd or even primitive Dirichlet character of conductor f dividing fZ∞ (cf. Example 2.2).

There are, however, complications when f0 properly divides fZ.

Example 4.3. — The Case K = k. In this case C[G] identifies with C and for any S

containing S∞, equation (9) gives

Θk/k,S(s) =∏

p∈S\S∞

(1 −Np−s)ζk(s)


David Solomon 137

Starting from the Analytic Class Number formula for ζk(s) at s = 1 and the functional

equation, it follows (cf [Ta, Cor. I.2.2]) that

(16) Θk/k,S(s) = −hS,kRS,k

Wks|S|−1 +O(s|S|) as s→ 0

where hS,k is the cardinality of the S-class group ClS(k) of k and RS,k is the S-regulator of

k, defined as follows. Let US(k) be S-unit group of k (namely the elements of k× which are

local units at all finite places not in S). By Dirichlet’s Theorem, we can choose a Z-basis

ε1, . . . , ε|S|−1 of US(k)/µ(k) and if we choose also any |S| − 1 places v1, . . . , v|S|−1 in S, then

RS,k :=∣

∣

∣det(

log(||εi||vj))|S|−1

i,j=1

∣

∣

∣ 6= 0, where || · ||vjdenotes the normalised absolute value at

vj . Moreover RS,k is easily seen to be independent of the choices and ordering of the εi and

the vj.

Notice that in each of the above examples, there is an integer r ≥ 0 such that ΘS vanishes

to order at least r at s = 0 and, moreover, the coefficient of sr in the Taylor series (denoted

Θ(r)S (0) for simplicity) is a Q[G]-multiple of an r×r determinant of ‘G-equivariant logarithms’

of S-units of K (a phrase to be made precise below). Indeed, we can take r to be the

precise order of vanishing in each example – namely 0, 1 and |S| − 1 respectively – provided

we adopt the usual convention that the 0 × 0 determinant equals 1. Stark’s Conjecture

is a precise generalisation of this observation. To formulate it, we first need to calculate

rS(χ) := ords=0(LS(s, χ)) for each character χ ∈ G. Using (5), the functional equation (4),

the definition of Λ(s, χ), properties of Γ(s) and (3), we find

(17) rS(χ) = |p ∈ S : p ∤ fχ, χ(p) = 1| + a1(χ) + r2 − δχ,χ0

This can be restated more elegantly as follows. For any place v of k, finite or infinite, we

write Dv for the decomposition subgroup of G at v, thus

Dv :=

Dp(K/k) if v corresponds to a non-zero prime ideal p of Ok,

1, cv if v|f∞,K/k is real with complex conjugation cv ∈ G, and

1 otherwise.

We also say that v splits in K iff Dv = 1. Using equation (17) one shows easily:

Proposition 4.4. — If χ ∈ G then rS(χ) = |v ∈ S : Dv ⊂ ker(χ)| − δχ,χ0. In particular

rS(χ) = rS(χ′) whenever χ and χ′ have the same kernel ( i.e. they are Galois-conjugate over

Q) e.g. if χ′ = χ−1.

4.2. The Conjecture. — Let K/k and S be as above and let r ∈ Z≥0. Consider

Hypothesis H(K/k, S, r). — The following conditions are satisfied:

(i) There exist r distinct places v1, . . . , vr ∈ S which split in K.

(ii) |S| ≥ r + 1.



It is clear from Proposition 4.4 that H(K/k, S, r) implies rS(χ) ≥ r for every χ ∈ G and

hence, by equation (10), that there exists Θ(r)S,K/k(0) ∈ R[G] (unique) such that

(18) ΘS,K/k(s) = Θ(r)S,K/k(0)s

r +O(sr+1) as s→ 0

For any place w of K we define the above-mentioned G-equivariant logarithm:

Logw : K× −→ R[G]

x 7−→ ∑

g∈G log(||g(x)||w)g−1

Let S(K) denote the set of places of K lying above those in S. Because it is G-stable, the

group US(K)(K) – which by abuse of notation, we shall denote US(K) – may be regarded

as a finitely generated, multiplicative Z[G]-module (sometimes written additively). Given

w1, . . . , wr ∈ S(K) there is a unique Q[G]-linear map

Regw1,...,wr

S : Q ⊗Z∧r

Z[G] US(K) −→ R[G]

sending a ⊗ (ε1 ∧ . . . ∧ εr) to adet(

Logwi(εj)

)r

i,j=1for any a ∈ Q and ε1, . . . , εr ∈ US(K).

(Note that Z[G] is commutative as G is abelian, so∧r

Z[G] US(K) is a well-defined Z[G]-

module, written additively. If r = 0 we interpret it as Z[G] and RegS as the natural injection

Q⊗Z[G] → R[G] with image Q[G].) The following is essentially due to H. M. Stark, although

the formulation given is Rubin’s (see Remark 4.10 below).

Conjecture SC(K/k, S, r). — Basic Abelian Stark Conjecture at s = 0

Let K/k, G, S and r be as above and suppose that Hypothesis H(K/k, S, r) is satisfied.

Thus (18) holds and we may choose r distinct places v1, . . . , vr ∈ S splitting in K and a place

wi of K above vi for each i. Then

Θ(r)S,K/k(0) = Regw1,...,wr

S (η) for some η ∈ Q ⊗ ∧rZ[G] US(K)

We shall call any such η a ‘solution of SC(K/k, S, r) w.r.t. w1, . . . , wr’.

Remark 4.5. — More than r Split Places. If r + 1 places in S split in K – and

in particular if K = k – then SC(K/k, S, r) holds. Indeed, Proposition 4.4 implies that

χ(Θ(r)S,K/k(0)) = 0 for all χ 6= χ0 and hence by (12) that Θ

(r)S,K/k(0) = Θ

(r)S,k/k(0)eχ0

. It follows

from equation (16) of Example 4.3 that 0 is a solution of SC(K/k, S, r) unless |S| = r+ 1 in

which case a solution has the form η := −(hS,k/Wk)|G|−r ⊗ (ε1 ∧ . . . ∧ εr) where ε1, . . . , εr is

any Z-basis of US(k)/µ(k) satisfying det(

log(||εi||vj))|S|−1

i,j=1> 0.

Remark 4.6. — Dependence on the Places wi. The above shows that it suffices to

consider the Basic Conjecture in the case where S contains precisely r splitting places. The

places w1, . . . , wr are then determined up to replacing each wi by giwi for some gi ∈ G (which

changes any putative solution by the action of g1 . . . gr) and re-ordering (which affects only

its sign). Because the dependence of the Conjecture on the choice of w1, . . . , wr is so simple,

one often suppresses it and writes RegS instead of Regw1,...,wr

S .


David Solomon 139

Remark 4.7. — Variation of S and K. Suppose that K/k, S and r satisfy the con-

ditions of SC(K/k, S, r) and that η is a solution of the conjecture. If S′ is a finite set

of places containing S then, clearly, H(K/k, S′, r) is satisfied and it follows from (11) that

η′ :=∏

p∈S′

p 6∈S

(

1 − σ−1p

)

η is a solution of SC(K/k, S′, r). Similarly, if K ′ is an intermediate field

with K ⊃ K ′ ⊃ k and G′ denotes Gal(K ′/k), then H(K ′/k, S, r) is automatically satisfied.

It is then a simple exercise to deduce from (12) that NK/K ′η is a solution of SC(K ′/k, S, r)(w.r.t. the places of K ′ below w1, . . . , wr) where NK/K ′ denotes the map from Q⊗∧r

Z[G] US(K)

to Q ⊗ ∧rZ[G′] US(K ′) induced by the norm from US(K) to US(K ′).

Remark 4.8. — It is clearly possible to have rS(χ) ≥ r for all χ ∈ G even when H(K/k, S, r)

fails. In such a case equation (18) still holds and a corresponding variant of SC(K/k, S, r)

has been formulated and studied in [EP]

4.3. Uniqueness of the Solution. — Suppose that K/k, S and r satisfy the conditions

of SC(K/k, S, r), so in particular rS(χ) ≥ r for all χ ∈ G. The solution η (if one exists) is not

in general unique but it may be rendered so by insisting that it lie in a certain ‘eigenspace’

of Q ⊗ ∧rZ[G]US(K). To see this, consider a character χ ∈ G such that rS(χ) = rS(χ−1) > r.

Equations (18) and (10) and the definition of rS(χ−1) then imply eχΘ(r)S (0) = 0. Consequently,

(19) Θ(r)S (0) = 1.Θ

(r)S (0) =

(

∑

χ∈G

eχ

)

Θ(r)S (0) = eS,rΘ

(r)S (0)

where:

eS,r :=∑

χ∈G

rS(χ)≤r

eχ =∑

χ∈G

rS(χ)=r

eχ

Although a priori an element of C[G], Prop. 4.4 shows that eS,r actually lies in Q[G] and,

together with a little character theory, it even gives the formula

(20) eS,r =

∏

v∈S\v1,...,vr

(

1 − |Dv |−1NDv

)

if |S| > r + 1

(

1 − |Dv|−1NDv

)

+ eχ0,G if |S| = r + 1 and S \ v1, . . . , vr = vwhere v1, . . . , vr ∈ S split in K and for any finite group H we set NH =

∑

h∈H h ∈ Z[H].

Thus if η is a solution of SC(K/k, S, r) and lies in Q ⊗ ∧rZ[G] US(K) then so does eS,rη and

equation (19) gives

RegS(eS,rη) = eS,rRegS(η) = eS,rΘ(r)S (0) = Θ

(r)S (0)

so, in fact, eS,rη is another solution lying in the eS,r-component (or ‘eigenspace’) eS,r(Q ⊗∧r

Z[G] US(K)). Such a solution will be called ‘canonical’. On the other hand, one can use

Dirichlet’s Theorem to show that eS,r(Q ⊗ US(K)) is free of rank r over eS,rQ[G] and then

deduce that RegS is injective on eS,r(Q ⊗ ∧rZ[G] US(K)). We conclude:

Proposition 4.9. — SC(K/k, S, r) has a solution if and only if it has a unique canonical

solution.



The canonical solution of SC(K/k, S, r) will be denoted ηS,K/k or just ηS , if it exists.

Remark 4.10. — Relation with Conjectures of Rubin and Stark. Conjec-

ture SC(K/k, S, r) is equivalent Conjecture A′ of [Ru] whose formulation is very similar

but requires the choice of an auxiliary finite set T of finite places of k, subject to certain

conditions. The choice of T does not affect its veracity, only the value of a putative solution.

It becomes important only for Rubin’s refined, integral (as opposed to ‘basic’) abelian Stark

conjecture. This is his Conjecture B′ which requires that the solution of Conjecture A′ lie

in a certain Z[G]-lattice spanning the eigenspace over Q and depending on T . Some more

details of the relationships between these conjectures can be found in [So2, Remark 2.8], in

a special case.

Stark’s original conjecture was formulated in terms of Artin L-functions of characters of a

Galois extension which is not necessarily abelian. It appears as Conjecture I.5.1 in [Ta].

However, Propositions 2.3 and 2.4 of [Ru] show that in our set-up, Conjecture A′ – and

hence SC(K/k, S, r) – are equivalent to Stark’s conjecture for all characters χ ∈ G such that

rS(χ) = r.

5. Some Particular Cases of SC(K/k, S, r)

We briefly survey the known cases of the the basic Stark Conjecture and some of its integral

refinements. These will be grouped according to the value of r: 0, 1 or ≥ 2. Let K/k be an

abelian extension with group G and S ⊃ Smin be as above.

5.1. The Case r = 0. — For any suchK/k, S, the HypothesisH(K/k, S, 0) is automatically

satisfied and we shall see that SC(K/k, S, 0) follows from results of Siegel-Klingen. First, the

interpretation of RegS in the case r = 0 (explained above) means that SC(K/k, S, 0) is

equivalent to the statement that ΘS,K/k(0) lies in Q[G] or, indeed, in eS,0Q[G] by (19). We

can assume S = Smin by (11). Now, if S∞ contains a split place then the conjecture holds

(see Remark 4.5). Thus we can assume K is totally complex and k totally real, which forces

|S| > 1. Also, if v ∈ S∞ then Dv = 1, cv where cv is the complex conjugation associated to

v. Equation (20) shows that cveS,0 = −eS,0 so that cvΘS,K/k(0) = −ΘS,K/k(0) by (19). Thus

ΘS,K/k(0) is fixed by the subgroup H := 〈cv1cv2

: v1, v2 ∈ S∞〉 of G. Equation (12) therefore

allows us to replace K by KH , i.e. we can assume H is trivial and (still) that K is totally

complex. This means that K is of CM-type i.e. cv equals c ∈ G (of order 2) independently of

v ∈ S∞. In this set-up, it follows from work of Siegel [Si] and Klingen (or of Shintani [Sh1,

Cor. to Thm. 1]) that ζSmin(0, g) ∈ Q for all g ∈ G, and the conjecture follows.

For the rest of this subsection we shall continue to assume that k is totally real, K is CM and

S = Smin. We have seen that c acts by −1 on ΘS,K/k(0) ∈ Q[G] and therefore ΘS,K/k(0) ∈(1 − c)Q[G]. However, in the case of Example 4.1, it is evident from (13) that ΘSf ,Q(ζf )/Q(0)

actually lies in (1 − c)W−1Q(ζf )Z[Gf ]. (Note that WQ(ζf ) is the l.c.m. of 2 and f .) We can take

this further: let AnnZ[G](µ(K))Z[G] denote the annihilator of µ(K) as a Z[G]-module (which


David Solomon 141

clearly contains WK). In Example 4.1, AnnZ[Gf ](µ(Q(ζf ))) is easily seen to be generated over

Z by the elements b − gb for all integers b such that (b, 2f) = 1. Furthermore, it follows

from (13) that (b− gb)ΘSf ,Q(ζf )/Q(0) has coefficients in Z, so:

AnnZ[Gf ](µ(Q(ζf )))ΘSf ,Q(ζf )/Q(0) = 〈(b− gb)ΘSf ,Q(ζf )/Q(0) : b ∈ Z, (b, 2f) = 1〉Z⊂ Z[Gf ] ∩ (1 − c)Q[Gf ] = (1 − c)Z[Gf ](21)

The second member above is the Stickelberger Ideal of Z[Gf ]. (It differs very slightly from

that of [Wa, §6.2], for example, which is not quite contained in (1 − c)Z[Gf ].) For any K/k

as above, we define the Generalised Stickelberger Ideal

(22) Stick(K/k) := AnnZ[G](µ(K))ΘSmin,K/k(0)

From what we already know, this is a Z[G]-submodule of (1 − c)Q[G]. However a result of

Deligne-Ribet [DR] (and, independently, of Pi. Cassou-Noguès [C-N]) gives the following

generalisation of (21) which may be seen as an ‘integral’ refinement of SC(K/k, Smin, 0).

Theorem 5.1. —

With assumptions and notations as above, Stick(K/k) is contained in (1 − c)Z[G].

5.2. The Case r = 1. — Assume Hypothesis H(K/k, S, 1) is satisfied i.e. |S| ≥ 2 and there

exists v1 ∈ S splitting in K. Fix w1 above v1 in S(K). Any element η ∈ Q ⊗ Λ1Z[G]US(K) =

Q ⊗ US(k) may be written 1m ⊗ ε with ε ∈ US(K) and clearly,

η is a solution of SC(K/k, S, 1) ⇐⇒ Θ′S,K/k(0) =

1

mLogw1

(ε)

⇐⇒ ζ ′S,K/k(0, g) =1

mlog ||g(ε)||w1

∀ g ∈ G(23)

Remark 5.2. — Criterion for the Canonical Solution when r = 1

If |S| > 2 and η = 1m ⊗ ε is a solution of SC(K/k, S, 1) then it is the canonical solution ηS iff

||ε||w = 1 for all w ∈ S(K) not above v1. If |S| = 2 the italicised condition must be replaced

by ‘||ε||w is independent of w ∈ S(K) above v’ where S \ v1 = v. We leave the proofs of

these statements as an exercise.

Stark gave an integral refinement of SC(K/k, S, 1) as follows.

Conjecture RSC(K/k, S). — Refined Abelian Stark Conjecture for r = 1

Suppose K/k, and S satisfy Hypothesis H(K/k, S, 1). Then SC(K/k, S, 1) holds with canon-

ical solution ηS such that

(i) ηS = 1WK

⊗ εS for some εS ∈ US(K) (depending on choice of w1) and

(ii) the extension K(ε1/WK

S )/k is abelian.

The arguments of Remark 4.5 extend to prove the RSC(K/k, S) if S contains a split place

other than v1 e.g. if K = k. This is shown in Prop. 3.1 of [Ta, Ch. IV]. If v1 is an infinite

place, the remaining proven cases of the Refined Conjecture – and indeed of SC(K/k, S, 1) –



are as follows. For v1 finite, more will be said at the beginning of Section 6 and in both cases,

more details can be found in loc. cit.

(i) k = Q. Here v1 = ∞ and K is a real, absolutely abelian field (w.l.o.g. different from Q)

so that fK/Q = fZ for some f > 4, f 6≡ 2 (mod 4). The Kronecker-Weber Theorem implies

K ⊂ Q(ζf )+ = Q(ζf + ζ−1f ). For SC(K/Q, S, 1), we use Remark 4.7 to reduce to the special

case K = Q(ζf )+, S = Smin(Q(ζf )+/Q) = Sf for which equation (15) of Example 4.2 shows

that 12 ⊗ε−1

f is a solution of SC(Q(ζf )+/Q, Sf , 1) with w1 given by the inclusion Q(ζf )+ → R.

The Refined Conjecture RSC(K/Q, S) can also be reduced to this special case (use Prop. 3.4

and 3.5 of [Ta, Ch. IV]). Remark 5.2 implies that 12 ⊗ε−1

f is the canonical solution. Moreover,

WQ(ζf )+ = 2 and one checks that Q(√εf ) is contained in Q(ζ4f ) (and even in Q(ζ2f ) if 2|f)

so RSC(Q(ζf )+, Sf ) holds.

(ii) k imaginary quadratic. In this case v1 is the unique (complex) infinite place and

RSC(K/k, S) is proven in the paper IV of [St]. The reader can also refer to the abridged

account given in [Ta, IV.3.9]. Again one reduces to the case where K is a certain ray-class

field for which the canonical solution is given in terms of an elliptic unit and (23) is proven

via the Second Kronecker Limit Formula.

(iii) G is 2-elementary. If G ∼= (Z/2Z)t for some t then SC(K/k, S, r) holds quite generally

for any admissible S and r (see below). Taking r = 1, the Refined Conjecture RSC(K/k, S) is

proven in [Ta, IV.5.5] under the additional assumption that G is generated by the subgroups

Dv for v ∈ S. (This does not require v1 to be infinite.)

As far as the author is aware, the only other cases of SC(K/k, S, 1) proven to date (with v1infinite) are due to Shintani. In [Sh2], he proves a version of SC(K/k, S, 1) in certain cases

where k is real quadratic and K is a quadratic extension of an absolutely abelian field such

that only one real place v1 of k splits in K.

Remark 5.3. — Construction of Abelian Extensions.

Notice that condition (i) ofRSC(K/k, S) determines εS up to an element of US(K)tor = µ(K).

Furthermore, equation (23) now predicts

(24) ||εS ||g−1w1= exp(WKζ

′S,K/k(0, g)) ∀ g ∈ G

Suppose v1 is real. Then so is g−1w1 ∀ g and WK = 2. If we could prove RSC(K/k, S) in

this case then (24) would give a transcendental formula for ±εS ∈ K×. This would thus

lead to a solution of Hilbert’s 12th Problem – the construction of abelian extensions of k

– using special values of derivatives of partial-zeta functions that are intrinsic to k. (Or,

indeed, using ray-class L-functions, via (7) and (10).) If we only assume RSC(K/k, S), one

can still use (24) and the other facts about εS to identify it precisely on a computer. This

leads to a method for the algorithmic construction of certain ray-class fields which has been

implemented in PARI/GP [PARI].


David Solomon 143

5.3. The Case r ≥ 2. — For any r, Conjecture SC(K/k, S, r) can be proven under the

following hypothesis extending that of Remark 4.5:

(25) χ ∈ G, rS(χ) = r ⇒ ord(χ)|2

Indeed, every character of order 1 or 2 satisfies Conjecture I.5.1 of [Ta] for the sameK/k and S.

(This follows from properties of the latter under inflation, induction and addition of characters

and the caseK = k. See for example [Ta, I.7.1 and II.1.1]). As noted in Remark 4.10 it follows

that SC(K/k, S, r) holds whenever (25) does, and in particular whenever G is 2-elementary

(see above). If r ≥ 2, the only proven cases of SC(K/k, S, r) without (25) come by using

induction of characters to ‘raise the base field’ from known cases with r = 1 (see e.g. [Po]).

This applies, for instance, in some cases where when K is abelian over Q and [k : Q] = r.

There exist two integral refinements of SC(K/k, S, r) for r ≥ 2, both generalising

RSC(K/k, S). These are ‘Conjecture B′’ of Rubin mentioned in Remark 4.10 and a

version due to Popescu in [Po] which dispenses with Rubin’s auxiliary sets T . We shall not

give full statements here but note that both imply the following, rather crude generalisation

of condition (i) of RSC(K/k, S): ηS is of the form 1mWK

⊗ η for some η ∈ ∧rZ[G]US(K) and

some m ∈ N whose prime factors all divide |G|. (Cases are known in which m 6= 1 is forced.)

Rubin’s and/or Popescu’s refinements were established for |G| = 2 in [Ru], by Sands in

some other cases where G is 2-elementary (see his article in [BPSS]) and by Popescu [Po],

Cooper [Co] and Burns (see below) in other cases by base-raising.

Two other types of evidence support SC(K/k, S, r) and its refinements. Firstly, both Rubin’s

and Popescu’s Conjectures are shown in [Bu] to follow from a particular case of the very

general Equivariant Tamagawa Number Conjecture of Burns and Flach. This was proven

for k = Q and K/Q abelian in [BG] (see also Flach’s article in [BPSS] for the case p = 2).

Since it behaves well under raising the base field, one can even establish Rubin’s and Popescu’s

conjectures for such K, any k ⊂ K and any admissible S and r. Secondly, there is considerable

computational evidence in support of SC(K/k, S, r). We refer to Dummitt’s article in [BPSS]

for a survey concentrating on the case r = 1 (with v1 finite or infinite). Numerical confirmation

of some cases with r = 2, k real quadratic and v1, v2 real is given in [RS1] (along with an

analogous p-adic conjecture) and in [RS2].

6. The Brumer-Stark Conjecture and the Annihilation of Class Groups

For the rest of this article we shall assume that k is totally real, K is of CM-type and

G = Gal(K/k) is abelian with unique complex conjugation denoted c and maximal real

subfield K+ = K〈c〉. This forces |Smin| > 1. To simplify, we shall take S = Smin until further

notice and write Θ for ΘS. Theorem 5.1 implies that WKΘ(0) lies in (1− c)Z[G]. With these

assumptions, and temporarily using a multiplicative notation for Z[G]-actions, we can state

the



Conjecture BSC(K/k). — Brumer-Stark Conjecture (with S = Smin)

For every fractional ideal a ∈ I(K) there exists γa ∈ K× such that

(i) aWKΘ(0) equals the principal ideal (γa),

(ii) ||γa||w = 1 for all w ∈ S∞(K) and

(iii) the extension K(γ1/WKa )/k is abelian.

Note that conditions (i) and (ii) determine γa up to an element of µ(K). Also, property (i)

implies that ||γa||w = 1 for any place w of K above v ∈ Sram, since |Smin| > 1 implies

NDvΘ(0) = 0 (by (19) and (20) with r = 0). In our set-up, BSC(K/k) is therefore equivalent

to the case of Conjecture IV.6.2 of [Ta] with ‘T ’ equal to Smin and this implies all other cases

by Cor. 6.6 ibid..

The explanation of the name ‘Brumer-Stark’ is as follows. Firstly, one can show that

BSC(K/k) holds if and only if it holds with a = P for any prime ideal P above any prime

p of k that splits in K. (This follows from [Ta, Prop. IV.6.4] and the fact that the classes

of such P generate Cl(K).) But Remark 5.2 and the relation Θ′Smin∪p(0) = log(Np)Θ(0)

coming from (11) show that conditions (i)–(iii) with a = P are in fact equivalent to the

statement that 1WK

⊗ γP is the (canonical) solution of RSC(K/k, Smin ∪ p) with w1 = P.

Hence BSC(K/k) is equivalent to these cases of the Refined Stark Conjecture for r = 1.

Secondly, BSC(K/k) clearly implies that WKΘ(0) annihilates Cl(K) as a Z[G]-module, which

had previously been conjectured by Armand Brumer. We can go further. Given any a ∈ I(K),

it follows easily from condition (iii) of BSC(K/k) that for every x ∈ AnnZ[G](µ(K)) there

exists ya,x ∈ K× with γxa = yWK

a,x . It then follows from condition (i) that axWKΘ(0) = (ya,x)WK

and since xΘ(0) ∈ Z[G] by Thm. 5.1, we must have axΘ(0) = (ya,x). In particular, BSC(K/k)

implies the

Conjecture BC(K/k). — Brumer Conjecture (with S = Smin)

In the above situation Cl(K) is annihilated by the ideal Stick(K/k) ⊂ (1 − c)Z[G] defined

in (22).

Example 6.1. — Brumer and Brumer-Stark Conjectures for k = Q.

In this case, K is an imaginary abelian field and BC(K/Q) is simply Stickelberger’s Theo-

rem. The traditional proof of the latter (see e.g. [Wa]) uses the factorisation of (norms of)

cyclotomic Gauss sums attached to prime ideals P of OK . In fact, if P divides a rational

prime p ∈ Q split in K and the latter is the full cyclotomic field Q(ζf ) (notations as above, so

f |(p−1)) then exactly the same factorisations establish that WQ(ζf )th power of the Gauss sum

is (essentially) the element γP appearing in BSC(Q(ζf )/Q). As explained above, this means

that γP also gives rise to the solution of RSC(Q(ζf )/Q, Sf ∪ p) with v1 = p. In this way

one also establishes BSC(K/Q) for arbitrary imaginary abelian K and also RSC(K/Q, S)

for such K and arbitrary S ⊃ Smin containing a finite split place v1.

For arbitrary S′ ⊃ Smin one can make an ‘S′-Brumer Conjecture’ by replacing ΘSmin(0) with

ΘS′(0) in (22). This is the viewpoint of the excellent article by Greither in [BPSS]. (Of


David Solomon 145

course, the S′-version of BC(K/k) is weaker: it follows from our version because of (11).)

One or other version is now known in many cases, thanks to work of Greither and Wiles (see

below) and many others. For a survey, for higher analogues involving Θ(−n) with n ≥ 1 (the

Coates-Sinnott Conjecture) and for precise connections between BSC(K/k) and Rubin’s and

Stark’s Conjectures, we refer to ibid. Our focus now will be on strengthening the annihilation

statement of BC(K/k) in a different direction to that of BSC(K/k).

First, we localise: let p be a prime number and write Cl(K)p for the p-Sylow subgroup of Cl(K)

considered as a module for Rp := Zp[G]. If Stick(K/k)p denotes the Zp-span of Stick(K/k)

inside (1 − c)Rp then, clearly, BC(K/k) is equivalent to the following local statement for all

p, which we denote BC(K/k)p:

Cl(K)p is annihilated by Stick(K/k)p

We assume henceforth that p 6= 2 which implies that Rp is a product of rings R+p × R−

p =:

(1 + c)Rp × (1− c)Rp and, correspondingly, Cl(K)p = Cl(K)+p ⊕Cl(K)−p := (1 + c)Cl(K)p ⊕(1−c)Cl(K)p. Since Stick(K/k)p is an ideal of R−

p , it automatically annihilates Cl(K)+p which

is isomorphic to Cl(K+)p (since p 6= 2). This means firstly that BC(K/k)p tells us nothing

interesting about the class groups of totally real fields (at least, not directly). Secondly,

(26) BC(K/k)p ⇐⇒ Stick(K/k)p ⊂ AnnR−

p(Cl(K)−p )

7. Exact Annihilators and Fitting Ideals

We shall examine first the obvious question of whether one should actually expect equality on

the R.H.S. of (26). There are at least two reasons why this cannot hold in general, the first

being that AnnR−

pcontains |Cl(K)−p | so is of finite index in R−

p but Stick(K/k)p may not be,

essentially because of ‘trivial zeroes’ of Smin-truncated L-functions at s = 0. Indeed, Θ(0) and

hence Stick(K/k)p are killed by (1 − c)NDv for any v ∈ Smin (see above) which is a non-zero

element of R−p whenever c 6∈ Dv. To get around this, we need to enlarge Stick(K/k)p. One

way to do this is to define

(27) Stick(K/k)p :=∑

K⊃K ′⊃k

coresKK ′(Stick(K ′/k)p)

Here, K ′ runs through intermediate (CM) sub-extensions ofK/k and coresKK ′ : Zp[Gal(K ′/k)] →

Zp[G] = Rp is the Zp-linear map sending g′ ∈ Gal(K ′/k) to the sum of its pre-images under

πK/K ′. It is not hard to show that Stick(K/k)p is an ideal of finite index in R−p con-

taining Stick(K/k)p and, moreover, that if BC(K ′/k)p held for every K ′ then one would

have Stick(K/k)p ⊂ AnnR−

p(Cl(K)−p ). Other ‘enlargements’ of Stick(K/k)p appear in the

literature, mostly variants of (27) which agree with Stick(K/k)p when p ∤ |G|.However, one still cannot always have equality for another reason which becomes particularly

clear for [K : k] = 2, when all enlargements coincide with the basic Stick(K/k)p. We explain

briefly, leaving details to the reader. In this case, we can identify R−p = Zp(1 − c) with Zp



and evaluate Θ(0) precisely using the Analytic Class Number Formula for ζK(s) and ζk(s).

We get:

(28) If [K : k] = 2 then Stick(K/k)p = Stick(K/k)p = (hK/hk)Zp = |Cl(K)−p |Zp

This proves that Stick(K/k)p ⊂ AnnR−

p(Cl(K)−p ) (and hence BC(K/k)p) in this case but

equality clearly holds if and only if Cl(K)−p is a cyclic abelian group. This fails, for example,

when p = 3 and K/k = Q(√−974)/Q so that Cl(K)−3 = Cl(K)3 ∼= (Z/3Z)2.

Instead of comparing Stick(K/k)p with AnnR−

p(Cl(K)−p ) – which is generated by the exponent

of Cl(K)−p as an abelian group when [K : k] = 2 – equation (28) suggests that one should in

general compare it with an ideal of R−p which, in some sense, measures the ‘size’ of Cl(K)−p as

an R−p -module. The most obvious candidate is the (initial) Fitting ideal FittR−

p(Cl(K)−p ). We

briefly recall its definition and make some general remarks, referring to [No], the appendix

of [MW] or Greither’s article in [BPSS] for more details. For any finitely generated module

A over a commutative (noetherian) ring R one can choose a presentation

Rs M−→Rt −→ A→ 0

where s, t ∈ N and M is a t×s matrix with coefficients in R. One then defines FittR(A) to be

the ideal of R generated by all t× t minor-determinants of M (which is zero if s < t). It turns

out that this is independent of the choice of s, t and M and that AnnR(A)t ⊂ FittR(A) ⊂AnnR(A) for any possible t. If A is cyclic over R we can take t = 1 so FittR(A) = AnnR(A).

Now suppose A is finite. If R = Z or Zp one can take s = t and the theory of elementary

divisors gives FittR(A) = |A|R. If R = Zp[H] for a finite abelian group H, then s ≥ t

is forced and FittR(A) contains |A|t so is an ideal of finite index in R. Moreover, one can

show that s = t is possible if and only if A is a cohomologically trivial H-module. In this

case, FittR(A) is clearly principal, generated by det(M). (For a converse, see Prop. 2.2.2 of

Greither’s article in [BPSS]). If p ∤ |H| then A is always cohomologically trivial and Zp[H]

is a product of unramified extensions of Zp corresponding to Qp-valued characters of H. One

can decompose everything using such characters so that the Fitting ideal behaves much like

the case R = Zp. If, however, p||H| then the Fitting ideal can be non-principal and is, in

general, a far more subtle invariant.

We are interested in the case H = G where all the above remarks remain true with R = R−p ,

a direct factor of Zp[G]. In particular, FittR−

p(Cl(K)−p ) is an ideal of finite index in R−

P ,

contained in AnnR−

p(Cl(K)−p ) and one can ask:

(29) is Stick(K/k)p equal to FittR−

p(Cl(K)−p )?

Positive answers to this question were obtained by various authors under different hypotheses.

For example, when p ∤ |G|, characters are used to treat the case k = Q in [MW] and

k 6= Q (totally real) in [Wi] (under an additional hypothesis on characters). Without the

assumption p ∤ |G|, three other large classes with k = Q are treated in [Ku1] (using a

different enlargement of Stick(K/k)p) and [Gr1] treats the case in which K/k is ‘nice’ (a


David Solomon 147

condition which implies Stick(K/k)p = Stick(K/k)p among other things). In each case we can

deduce BC(K/k)p using (26). One should also mention certain ‘higher Stickelberger ideals’

defined by M. Kurihara in the case p ∤ |G| and k totally real. Using characters and Euler

Systems, he shows in [Ku2] how these can give more information on the R−p -structure of

Cl(K)−p , essentially determining the latter under certain hypotheses.

Nevertheless, in 2006 Greither showed that the Equivariant Tamagawa Number Conjecture

(ETNC) mentioned in Section 5 predicts a result that runs somewhat counter to (29): If A is a

finite Rp-module let us write A for the Pontrjagin dual HomZ(A,Q/Z) = HomZp(A,Qp/Zp) of

A made into an Rp-module by defining (gf)(a) to be f(ga) for any f ∈ A, g ∈ G. (This makes

sense because G is abelian and clearly A− ∼= A− as R−p -modules.) Theorem 8.8 of [Gr2] can

now be stated as

Theorem 7.1. — Suppose K/k and p satisfy our current assumptions, that G = Gal(K/k)

and in addition that

(i) µp∞(K) is G-cohomologically trivial and

(ii) the ETNC holds for the pair (K/k, h0(K))

Then SKu(K/k)p = FittR−

p(Cl(K)−p ).

Here, SKu(K/k)p is a variant of Stick(K/k)p satisfying

Stick(K/k)p ⊂ SKu(K/k)p ⊂ Stick(K/k)p

If p ∤ |G| then on the one hand SKu(K/k)p = Stick(K/k)p and on the other, any finite Rp-

module A is isomorphic to A as an Rp-module. So Thm. 7.1 predicts a positive answer to

question (29) in this case, agreeing with Wiles’ results. However, if p||G| things get decidedly

more complicated. On the one hand one can have SKu(K/k)p 6= Stick(K/k)p. On the

other, one can easily construct finite R−p -modules A with A 6∼= A. In this case, one still

has FittR−

p(A) ⊂ AnnR−

p(A) = AnnR−

p(A) so, in particular, Thm. 7.1 supports BC(K/k)p.

However, FittR−

p(A) and FittR−

p(A) will differ unless a special condition holds, e.g. A is

cohomologically trivial or G is p-cyclic. (The sufficiency of the latter follows from [MW,

Appendix, Prop. 1].)

An explicit counter-example to (29) was finally given by Greither and Kurihara. Taking

p = 3 they found an extension K/k with k = Q(√

29), µp∞(K) = 1 and G ∼= (Z/2Z) ×(Z/3Z)2 for which Stick(K/k)p – and a fortiori any enlargement of it – is not contained in

FittR−

p(Cl(K)−p ). (See [GK, §3.2] for more details, noting that our K is their K1.) In the

function-field case, counter-examples had previously been given by Popescu. We also mention

the recent, unconditional results of [GP] concerning Fitting ideals of duals in the function-field

case. Analogues for number fields may be forthcoming.

In view of Thm. 7.1, it seems reasonable to ask whether one can use ΘS to construct an

ideal related to FittR−

p(Cl(K)−p ) rather than FittR−

p(Cl(K)−p ). A result of this type is given

in [KM] for k = Q but doesn’t seem to generalise. We shall return to this question in §9.1.



8. ΘS at s = 1

8.1. Plus- and Minus-Parts. — We maintain the above notations and assumptions on

K/k but once more allow S to contain Smin properly. A character χ ∈ G is called (totally)

even or odd according as χ(c) = 1 or −1. If R is a commutative ring in which 2 is invertible,

we write e+ and e− respectively for the idempotents 12(1 + c) and 1

2(1 − c) of R[G] and

A± for e±A where A is any R[G] module, so A = A+ ⊕ A−. The meromorphic functions

Θ+S (s) := e+ΘS(s) and Θ−

S (s) := e−ΘS(s) take values in C[G]+ and C[G]− respectively and

ΘS(s) = Θ+S (s) + Θ−

S (s) =∑

χ∈G

χ even

LS(s, χ−1)eχ +∑

χ∈G

χodd

LS(s, χ−1)eχ

by (10). Now, if χ is even, then Dv ⊂ ker(χ−1) for all v ∈ S∞ so that ords=0(LS(s, χ−1)) =

rS(χ−1) ≥ 1 by Prop. 4.4 (for χ = χ0, use |S| ≥ 2). Thus Θ+S (0) = 0 so that the value

ΘS(0) studied in Sections 6 and 7 is naturally equal to Θ−S (0). In contrast, equations (3)

and (2) show that ords=1(LS(s, χ−1)) = −δχ,χ0for both odd and even χ. Thus, even if we

subtracted the pole due to the trivial character, Θ+S would still make a non-zero contribution

to the value of ΘS at s = 1 and one, moreover, which we can expect to be of a very different

nature from that of Θ−S . Indeed, the functional equation (4) for L-functions and the case

r ≥ 1 of SC(K/k, S, r) mean that the former should – in a vague sense – ‘contain non-trivial,

transcendental regulators’. On the other hand, since τ(χ) is always algebraic, one can use the

case r = 0 of SC(K/k, S, r) – i.e. Θ−S (0) ∈ Q[G]− – and the functional equation to show that

Θ−S (1) ∈ πnQ[G]−, where we recall that n = [k : Q].

In the rest of this article we shall therefore consider only the minus-part Θ−S (1), follow-

ing [So1], [So2] and [RS2]. More precisely, we shall study

b−S = b−S,K/k := (i/π)nΘ−S (1)∗ = (i/π)n lim

s→1e−ΘS(s)∗ ∈ Q[G]−

where ∗ : C[G] → C[G] is the C-linear involution sending g ∈ G to g−1. The non-vanishing of

the L-functions at s = 1 shows that b−S lies in (Q[G]−)× and the limit (with s ∈ R>0) shows

that it has coefficients in inR. We also have have the following more precise algebraicity

result.

Proposition 8.1. — If fK/k ∩ Z = fZ with f ∈ Z>0 then the coefficients of√dkb

−S lie in

both Q(ζf ) and the normal closure of K over Q.

The proof is essentially that of [RS2, Prop. 2]. The latter also contains an integrality result

for the coefficients and assumes that n = 2 and that S is a particular set of places. However,

for the properties we want, the proof adapts immediately to our more general situation since√dkb

−S equals

√dka

−,∗K/k,S = dke

− ∏

q∈S\Smin(1−Nq−1σq)ΦK/k(0)

∗, where undefined notations

are as in ibid. and [So2, eq. (9)].

8.2. A p-adic Logarithmic Map. — Let p 6= 2 be a prime number as before, and hence-

forth take S = Sp(k)∪Smin where Sp(F ) denotes the set of places dividing p in a number field

F . We sometimes drop S from the notation. Fixing an embedding j : Q → Qp, we may apply


David Solomon 149

it to the coefficients of b− to get j(b−) ∈ Qp[G]−. We shall use this to construct a p-adic map

sK/k,p on (an exterior power of) the p-semilocal units of K.

For each P ∈ Sp(K), letKP denote the completion ofK at P (containing K). Let OKPdenote

its ring of valuation integers and let Kp denote the product∏

P∈Sp(K)KP with the product

topology. The diagonal map K → Kp extends Qp-linearly to a ring isomorphism K⊗Qp∼= Kp

which we regard as an identification. The natural action of G on K therefore gives rise to an

action (by continuous ring automorphisms) on Kp which ‘mixes up’ the factors KP. We write

U1(Kp) for∏

P∈Sp(K) U1(KP) ⊂ K×

p where U1(KP) denotes the group of principal units of

KP, i.e. 1 + POKP. The multiplicative Z-action on U1(Kp) extends by continuity to Zp so

we may regard U1(Kp) as a module for Rp = Zp[G]. Now, every τ ∈ Gal(Q/Q) gives rise to a

ring homomorphism (j τ) ⊗ 1 : Kp → Qp. We write simply jτ for its restriction to U1(Kp).

This map factors through the projection onto U1(KP) (where P is determined by j τ) and

takes values in the disc x ∈ Qp : |x − 1|p < 1 on which the p-adic logarithm logp, defined

by the usual power-series, converges. We may therefore define a ‘p-semilocal, G-equivariant

logarithm’, Logτ,p, by

Logτ,p : U1(Kp) −→ Qp[G]

u 7−→ ∑

g∈G logp(jτ(gu))g−1

(compare with Logw of Section 4). It is continuous and Z[G]-linear, so Rp-linear. Now fix a

choice τ1, . . . , τn of left-coset representatives of Gal(Q/k) in Gal(Q/Q). This gives a unique

Rp-linear p-semilocal regulator

Regp = Regτ1,...,τnp :

∧nRpU1(Kp) −→ Qp[G]

sending u1 ∧ . . . ∧ un to det(

Logτi,p(uj))n

i,j=1. Finally, we make the

Definition 8.2. — Let sp = sτ1,...,τn

K/k,p be the Rp-linear map

sp :∧n

RpU1(Kp) −→ Qp[G]−

θ 7−→ j(b−)Regp(θ)

and let Sp = SK/k,p be its image in Qp[G]−.

Of course, sp factors through the projection e− onto∧n

RpU1(Kp)

−. In particular, it van-

ishes on∧n

RpU1(Kp)

+. Also, changing the choice and ordering of τ1, . . . , τn only multiplies

Regτ1,...,τnp , and hence sK/k,p, by ±g for some g ∈ G, so has no effect on SK/k,p. The following

is proved in [So2, Props. 2.16, 2.17] using results of [So1].

Proposition 8.3. — With notations as above,

(i) sp is independent of j and takes values in Qp[G]−.

(ii) Sp is a fractional ideal of Qp[G]−, that is, a finitely generated Zp-submodule spanning

Qp[G]− over Qp.

(iii) ker(sp) ∩∧n

RpU1(Kp)

− is precisely the Zp-torsion submodule of∧n

RpU1(Kp)

−.



Example 8.4. — The Case K/k = Q(ζpt)/Q for t ≥ 1. In this case S = ∞, p and

Sp(K) = P where P := (1− ζpt) is totally ramified over p. Thus j induces an isomorphism

Kp = KP → Qp(j(ζpt)) which we treat as an identification, allowing us also to identify

G = Gal(KP/Qp) with Gal(Qp(j(ζpt))/Qp). From now on, we suppress j and write ζ for ζpt,

identified with j(ζpt). Computing Θ−(1) directly (see e.g. [So2, Lemma 7.1]) we find

b− = e−1

pt

∑

g∈G

g

(

ζ

1 − ζ

)

g

Now take τ1 to be 1 ∈ Gal(Q/Q) so that Logτ1,p(u) becomes simply∑

h∈G logp(hu)h−1 for

any u ∈ U1(Kp) = U1(Qp(ζ)). Since n = 1, this coincides with Regp(u). Therefore, assuming

w.l.o.g. that u ∈ U1(Kp)− and multiplying out, we get

(30) sp(u) = b−Regp(u) =1

pt

(

∑

h′∈G

h′( ζ

1 − ζ

)

h′)(

∑

h∈G

logp(hu)h−1

)

=∑

g∈G

a(g(u))g−1

where, for any v ∈ U1(Qp(ζ)) we have set

a(v) :=1

ptTrQp(ζ)/Qp

(

ζ

1 − ζlogp(v)

)

To take this example further, we observe that the coefficient a(v) appears in the explicit reci-

procity law of Artin and Hasse (see [AH]). More precisely, the reciprocity map of local class

field theory sends any α ∈ K×P = Qp(ζ)

× to an element sα = sα,KP/Qp, say, of Gal(Kab

P /KP)

where KabP denotes a given abelian closure. If β also lies in K×

P then any ptth root β1/pt

lies

in KabP because K, hence KP, contains the ptth roots of unity. For our purposes the Hilbert

symbol (α, β)KP ,pt can therefore be defined as sα(β1/pt

)/β1/pt

, a ptth root of unity in K de-

pending only on α and β. Now if v lies in U1(Qp(ζ)) and a(v) is as above, the Artin-Hasse

law states firstly that a(v) ∈ Zp and secondly that

(31) (1 − ζ, v)KP ,pt = ζ−a(v)

The first fact implies sp(u) ∈ Zp[G] for all u ∈ U1(Kp)−, in other words

(32) SQ(ζpt )/Q,p ⊂ R−p

Finally, we note that in this example, results of [Iw] allow one to calculate Sp exactly. Indeed,

it is shown in [So3] that

SQ(ζpt )/Q,p = Stick(Q(ζpt)/Q)p

where Stick(Q(ζpt)/Q)p is as in Section 6. One can also show Stick(Q(ζpt)/Q)p =

FittR−

p(Cl(Q(ζpt))−p ), using [Ku1, Thm. 0.5] for example. Thus we get

(33) SQ(ζpt )/Q,p = FittR−

p(Cl(Q(ζpt))−p )

We now consider some possible generalisations of equations (31), (32) and (33).


David Solomon 151

9. Conjectures at s = 1

We maintain the notations and assumptions on K/k, p and S introduced in §8.2.

9.1. The Ideal Sp: Integrality and Fitting Ideals. — The following generalisation of

(32) was conjectured in [So1], [So2]

Conjecture IC(K/k, p). — Integrality Conjecture (with S = Sp(k) ∪ Smin)

In the above situation, SK/k,p is contained in R−p .

We summarise the current evidence for this conjecture. First, it is proven in [So1] whenever

p is unramified in K and also whenever p splits completely in k. The latter case is consider-

ably harder than the former and requires a minor auxiliary condition. Next, the conjecture

is established in [So2] whenever p ∤ |G|. Results of A. Jones in [Jo] imply that a somewhat

stronger statement than IC(K/k, p) follows from a certain case of the ETNC (see below).

Since the latter is known whenever K is abelian over Q, the conjecture is then proven uncon-

ditionally. (For a direct proof in this case, not using the ETNC but imposing a mild condition

if k 6= Q, see [So2, Sec. 8].) Finally, as a by-product of the computations in [RS2] (see below)

one gets actual proofs of IC(K/k, p) in a dozen cases not otherwise covered, all with k real

quadratic and p = 3 or 5.

We now assume that IC(K/k, p) holds and ask what might replace the finer statement (33) in

the general case. The results of Jones mentioned above show that if the ETNC holds (so, in

particular, if K is absolutely abelian) then S(K/k)p is contained in FittR−

p(Clm(K)−p ) where

Clm(K) is a certain ray-class group. For more details and an unconditional result when p ∤ |G|,we refer to [So2], §4.2, §4.3 and Remark 6.2. The latter hints that the appearance of Clm(K)

– rather than its quotient Cl(K) – is explained by the imprimitivity of the L-functions making

up Θ−S (1). This does not lead to trivial zeroes as it does at s = 0 but still suggests enlarging

S(K/k)p to S(K/k)p, say, using intermediate fields as in (27). The above-mentioned results

then prompt the

Question. — Does one have S(K/k)p = FittR−

p(Cl(K)−p ) whenever µp∞(K) = µp∞(Kp)

−?

Here, µp∞(Kp) denotes the R−p -module

∏

Pµp∞(KP). (The given condition therefore fails,

for instance, whenever µp ⊂ K and |Sp(K+)| > 1 and one would like to relax it.) An obvious

first test is the extension K/Q(√

29) mentioned in Section 7, for which Greither and Kurihara

showed Stick(K/k)p 6⊂ FittR−

p(Cl(K)−p ) when p = 3. Unpublished computations of X. Roblot

and the author show that indeed S(K/k)p = S(K/k)p = FittR−

p(Cl(K)−p ) in this case (so

Stick(K/k)p 6⊂ S(K/k)p). However the above question is still open even for general abelian K

and k = Q. It is equally possible that some other enlargement of S(K/k)p, perhaps analogous

to SKu(K/k)p, should replace S(K/k)p.



9.2. The Map sp: a Conjectural Explicit Reciprocity Law for ηS. — Continuing to

assume that IC(K/k, p) holds, we now discuss a refinement in a different direction that makes

a connection between sK/k,p in the minus-part and the conjectural solution of SC(K+/k, S, n)

in the plus-part which was discussed in §4 and §5. As motivation, we first takeK/k = Q(ζpt)/Q

and reformulate equations (30) and (31) using the notations and conventions of that example.

For any ξ ∈ µpt we write Indpt(ξ) for the unique element x ∈ Z/ptZ such that ζxpt = ξ. Thus

the Hilbert symbol gives rise to a bilinear pairing

(34)

[·, ·]pt : US(K+) × U1(Kp) −→ (Z/ptZ)[G]

(α, β) 7−→∑

g∈G

Indpt

(

(α, gβ)KP ,pt

)

g−1

This ‘extends’ naturally so that the first (global) variable may lie in Z(p) ⊗ US(K+) where

Z(p) denotes the local subring a/b : a, b ∈ Z, p ∤ b of Q. Since US(K+) has no p-torsion,

Z(p) ⊗ US(K+) injects into Q ⊗ US(K+). Recall from Example 4.2 and Subsection 5.2 that

the canonical solution of SC(K+/Q, S, 1) in Q ⊗ US(K+) is

ηS,K+/Q =1

2⊗ ((1 − ζpt)(1 − ζ−1

pt ))−1

where w1 is given by the inclusion K+ → R. (The reader can check that this holds even

if pt = 3 when K+ = Q.) Since p 6= 2 we can therefore regard ηS,K+/Q as an element of

Z(p) ⊗US(K+) and one checks easily that equations (30) and (31) amount to the congruence

(35) sp(u) = [ηS,K+/Q, u]pt in (Z/ptZ)[G], for all u ∈ U1(Kp)

(Use the fact that (α, β)KP ,pt = 1 if α, β both lie in U1(Kp)− or in U1(Kp)

+ so, in particular,

both sides of (35) vanish if u ∈ U1(Kp)+.)

The Congruence Conjecture of [So2] generalises (35) for K/k, p and S as considered in this

section with the additional assumption that µpt ⊂ K for some given t ≥ 1. Thus if α = (αP)Pand β = (βP)P lie in K×

p we can regard (αP, βP)KP ,pt ∈ µpt(KP) as an element of µpt for

each P and define

(α, β)Kp,pt :=∏

P∈Sp(K)

(αP, βP)KP ,pt =∏

P∈Sp(K)

(βP, αP)−1KP ,pt ∈ µpt

Remark 9.1. — The second equality comes from the skew-symmetry of the Hilbert symbol.

It shows that if α lies in K× (regarded as a subgroup of K×p by the diagonal embedding) then

(α, β)Kp,pt = (ψβ(α1/pt

)/α1/pt

)−1 where α1/pt ∈ Kab and ψβ ∈ Gal(Kab/K) is the image of

β under the composition of the natural embedding K×p → Id(K) with the global reciprocity

map Id(K) → Gal(Kab/K).

We now take α ∈ US(K+) and β ∈ U1(Kp) and define [α, β]pt ∈ (Z/ptZ)[G] just as in (34)

but replacing (α, gβ)KP ,pt by (α, gβ)Kp ,pt for each g ∈ G. Let κ denote the homomorphism

Gal(Q/Q) → (Z/ptZ)× given by γ(ξ) = ξκ(γ) for all ξ ∈ µpt . The restriction of κ to Gal(Q/k)


David Solomon 153

clearly factors through a homomorphism G → (Z/ptZ)× also denoted κ. One checks that

[α, β]pt is (Z-)bilinear and also G-semi-bilinear in the sense that

(36) [gα, hβ]pt = κ(g)g−1h[α, β]pt for all α ∈ US(K+), β ∈ U1(Kp) and g, h ∈ G

Taking g = h = c, one deduces [α, e−β]pt = [α, β]pt . Writing G+ for Gal(K+/k) = G/〈c〉, it

also follows from (36) that there exists a unique pairing

HK/k,pt :∧n

Z[G+]US(K+) × ∧nRpU1(Kp) −→ (Z/ptZ)[G]−

satisfying HK/k,pt(ε1∧ . . .∧εn, u1∧ . . .∧un) = det(

[εi, ul]pt

)n

i,l=1for any ε1, . . . , εn ∈ US(K+)

and u1, . . . , un ∈ U1(Kp). Note also that HK/k,pt is Rp-linear in the second variable and

factors through the projection e− onto∧n

RpU1(Kp)

−, just like sK/k,p. As before, we extend

it naturally so that the first variable may lie in Z(p) ⊗∧n

Z[G+] US(K+).

For each i = 1, . . . , n the restriction of τi to K+ corresponds to real place wi lying above a

distinct place vi of k which splits in K+. Since S ⊃ Sp(k)∪S∞(k) = Sp(k)∪v1, . . . , vn, the

hypotheses of Conjecture SC(K+/k, S, n) are satisfied. We shall assume that it holds and has

canonical solution ηS ∈ Q⊗∧nZ[G+] US(K+) w.r.t. the choice of places w1, . . . , wn. We would

like to apply HK/k,pt to ηS but for n ≥ 2 we cannot simply treat the latter as an element of

Z(p)⊗∧n

Z[G+] US(K+). Indeed, if also p||G+| then the map νS : Z(p)⊗∧n

Z[G+] US(K+) → Q⊗∧n

Z[G+] US(K+) may not be injective. Furthermore, in these circumstances Rubin’s Conjecture

B′ does not imply ηS ∈ im(νS) (see §5.3). To get round this, we define in [So2, § 2.2] a

certain lattice Λ0,S = Λ0,S(K+/k) ⊂ Q⊗∧nQ[G+] US(K+) such that Z(p)Λ0,S contains im(νS),

and also a natural ‘extension’ HK/k,pt (there denoted ‘HK/k,S,n’) of the pairing HK/k,pt to

Z(p)Λ0,S × ∧nRpU1(Kp) such that for each θ ∈ ∧n

RpU1(Kp) the diagram

(37) Z(p) ⊗∧n

Z[G+] US(K+)

νS

HK/k,pt (·,θ)

,,XXXXXXXXXXXXXXXXXXXXXXXXXX

(Z/ptZ)[G]−

Z(p)Λ0,S

HK/k,pt (·,θ)

22fffffffffffffffffffffffffffffff

commutes. (This follows from [So2, eq. (20)]. Note that the vertical map is an isomorphism

whenever p ∤ |G|.) Finally, we show in [So2, Rem. 2.8] that the full version of Rubin’s

Conjecture B′ (for varying auxiliary sets T ) implies that ηS lies in Z(p)Λ0,S (in fact, in12Λ0,S). We can at last state the

Conjecture CC(K/k, pt). — Congruence Conjecture (with S = Sp(k) ∪ Smin)

Suppose that K/k, p and S are as in §8.2 and in addition that

(i) IC(K/k, p) holds,

(ii) µpt ⊂ K for some t ≥ 1,



(iii) SC(K+/k, S, n) holds and the canonical solution ηS w.r.t. the choice of places

w1, . . . , wn lies in Z(p)Λ0,S.

Then, for all θ ∈ ∧nRpU1(Kp), we have the congruence

(38) sp(θ) = κ(τ1 . . . τn)HK/k,pt(ηS , θ) in (Z/ptZ)[G]−

We recall that the choice of the coset representatives τ1, . . . , τn affects both the definition

of sp and of the places w1, . . . , wn, hence of ηS . However, one checks easily that the factor

κ(τ1 . . . τn) in (38) makes the Conjecture independent of this choice.

The motivation of both the Integrality and Congruence Conjectures came from the author’s

article in [BPSS] although neither is explicitly mentioned there. Their statements appear first

in [So1] but that of the latter conjecture is rather awkward. An improved formulation appears

in [So2] for any S containing Sp(k) ∪ Smin but it is shown in Prop. 5.4 loc. cit. that this is

implied by the special case S = Sp(k)∪Smin which is all we have given above. We summarise

the current evidence for CC(K/k, pt). Firstly, no connection with the ETNC is known but

in [So2] a generalisation of the Artin-Hasse law due to Coleman is shown to imply the CC in

the case k = Q, and also for K absolutely abelian (but with the same mild condition as for

the IC if k 6= Q). It is also shown that the congruence (38) holds as ‘0 = 0’ whenever p ∤ |G|and θ is a Zp-torsion element. M. Bovey considered the case k = K+, i.e. |G| = 2. The CC

is then trivial unless |Sp(k)| = 1. In this case conditions (i) and (iii) hold and both the map

sp and the element ηS can be written down in terms of certain S-class-numbers and S-units

of K and k etc. (See Rem. 4.5 for ηS .) Nevertheless, the congruence (38) seems to be new

and unknown. A weakening of it is proven in [Bo]. A variant with p = 2 is also stated and

one congruence or the other is fully numerically verified in over 100 cases. Finally, in the case

where k is real quadratic but K+/Q is not abelian, one cannot usually prove SC(K+/k, 2)

but high-precision computation allows one to identify the solution ηS with virtual certainty.

This was done in [RS2] allowing the verification of IC(K/k, p) and CC(K/k, pt) in nearly 50

such cases with varying p and t = 1 or 2.

References

[AH] E. Artin and H. Hasse, ‘Die beiden Ergänzungssätze zum Reziprozitätsgesetz der ln-ten Poten-zreste im Körper der ln-ten Einheitswurzeln’, Abh. Math. Sem. Univ. Hamburg, 6, (1928), p. 146-162.

[BG] D. Burns and C. Greither , ‘On the Equivariant Tamagawa Number Conjecture for Tate motives’,Inventiones Mathematicae, 153 (2003), p. 303-359.

[Bo] M. Bovey, ‘Explicit reciprocity for p-units and a special case of the Rubin-Stark Conjecture’,PhD. Thesis, King’s College London, 2009.

[BPSS] D. Burns, C. Popescu, D. Solomon, J. Sands eds.,‘Stark’s Conjectures: Recent Work and NewDirections’, Contemporary Mathematics 358, American Mathematical Society, 2004.

[Bu] D. Burns, ‘Congruences between Derivatives of Abelian L-functions at s = 0’, Inventiones Math-ematicae, 169 (2007), p. 451-499.

[C-N] Pi. Cassou-Noguès, ‘Valeurs aux Entiers Négatifs des Fonctions Zêta et Fonctions Zêta p-Adiques’, Inventiones Mathematicae, 51 (1979), p. 29-59.


David Solomon 155

[Co] A. Cooper, ‘Some Explicit Solutions of the Abelian Stark Conjecture’, PhD. Thesis, King’sCollege London, 2005.

[DR] P. Deligne and K. Ribet, ‘Values of Abelian L-functions at negative integers over totally realfields’, Inv. Math. 59, (1980), p. 227-286.

[EP] C. Emmons and C. Popescu, ‘Special values of Abelian L-functions at s = 0’, Journal of NumberTheory 129, (2009), p. 1350-1365.

[GK] C. Greither and M. Kurihara, ‘Stickelberger elements, Fitting ideals of class groups of CM-fieldsand dualisation’, Math. Zeitschrift 260, no. 4 (2008), p. 905-930.

[GP] C. Greither and C. Popescu, ‘Fitting ideals of ℓ-adic realizations of Picard 1-motives and classgroups of curves over a finite field’, preprint, 2009.

[Gr1] C. Greither, ‘Some cases of Brumer’s conjecture for abelian CM extensions of totally real fields’,Math. Zeitschrift 233, (2000), p. 515-534.

[Gr2] C. Greither, ‘Determining Fitting ideals of minus class groups via the Equivariant TamagawaNumber Conjecture’, Compositio Math., 143, no. 6 (2007), p. 1399-1426.

[Iw] K. Iwasawa, ‘On Some Modules in the Theory of Cyclotomic Fields’, J. Math. Soc. Japan, 20,(1964), p. 42-82.

[Jo] A. Jones, ‘Dirichlet L-functions at s = 1’, PhD. Thesis, King’s College London, 2007.[KM] M. Kurihara and T. Miura, ‘Stickelberger ideals and Fitting ideals of class groups for abeliannumber fields’, preprint, 2009.

[Ku1] M. Kurihara, ‘Iwasawa theory and Fittting ideals’, J. Reine Angew. Math. 561, (2003), p. 39-86.[Ku2] M. Kurihara, ‘On the structure of ideal class groups of CM fields’, Documenta MathematicaExtra Volume Kato (2003), p. 539-563

[La] S. Lang, ‘Algebraic Number Theory’, Graduate Texts in Math. 110, Springer-Verlag, New York,1986.

[Ma] J. Martinet, ‘Character Theory and Artin L-functions’, in ‘Algebraic Number Fields’, A. Fröhliched., Academic Press, New York, 1977.

[MW] B. Mazur and A. Wiles, ‘Class fields of abelian extensions of Q’, Inventiones Mathematicae,76 (1984), p. 179-330.

[No] D.G. Northcott, ‘Finite Free Resolutions’, Cambridge Tracts in Mathematics 71, CUP, 1976.[PARI] C. Batut, K. Belabas, D. Bernardi, H. Cohen and M. Olivier, PARI/GP System, available athttp://pari.math.u-bordeaux.fr

[Po] C. Popescu, ‘Base Change for Stark-Type Conjectures “Over Z” ’, J. Reine angew. Math., 542,(2002), p. 85-111.

[RS1] X.-F. Roblot and D. Solomon, ‘Verifying a p-adic Abelian Stark Conjecture at s = 1’, Journalof Number Theory 107, (2004), p. 168-206.

[RS2] X.-F. Roblot and D. Solomon, ‘Testing the Congruence Conjecture for Rubin-Stark Elements’,preprint, 2008.

[Ru] K. Rubin, ‘A Stark Conjecture “Over Z” for Abelian L-Functions with Multiple Zeros’, Annalesde l’Institut Fourier 46, No. 1, (1996), p. 33-62.

[Sh1] T. Shintani, ‘On Evaluation of Zeta Functions of Totally Real Algebraic Number Fields atNon-Positive Integers’, J. Fac. Sci. Univ. Tokyo, Sec. 1A, 23, no. 2 (1976), p. 393-417.

[Sh2] T. Shintani, ‘On certain ray class invariants of real quadratic fields’, J. Math. Soc. Japan, 30,no. 1 (1978), p. 139-167.

[Si] C. L. Siegel, ‘Über die Fourierschen Koeffizienten von Modulformen’, Nachr. Akad.Wiss Göttin-gen, 3 (1970), p. 15-56.

[So1] D. Solomon, ‘On Twisted Zeta-Functions at s = 0 and Partial Zeta-Functions at s = 1’, Journalof Number Theory 128, (2008), p. 105-143.



[So2] D. Solomon, ‘Abelian L-Functions at s = 1 and Explicit Reciprocity for Rubin-Stark Elements’,to appear in Acta Arith.

[So3] D. Solomon, ‘Some New Ideals in Classical Iwasawa Theory’, preprint, 2009.[St] H. Stark, ‘L-Functions at s = 1 I,II,III,IV’, Advances in Mathematics, 7, (1971), p. 301-343, 17,(1975), p. 60-92, 22, (1976), p. 64-84, 35, (1980), p. 197-235.

[Ta] J. T. Tate, ‘Les Conjectures de Stark sur les Fonctions L d’Artin en s = 0’, Birkhäuser, Boston,1984.

[Wa] L. Washington, ‘Introduction to Cyclotomic Fields’, 2nd Ed., Graduate Texts in Math. 83,Springer-Verlag, New York, 1996.

[Wi] A. Wiles, ‘On a Conjecture of Brumer’, Annals of Math., 131, No. 3, (1990), p. 555-565.

February 26, 2010

David Solomon, Department of Mathematics, King’s College, Strand, London WC2R 2LS, U.K.

E-mail : [email protected]


ON THE GALOIS MODULE STRUCTURE OF EXTENSIONS OF

LOCAL FIELDS

by

Lara Thomas

Abstract. — We present a survey of the theory of Galois module structure for extensions oflocal fields. Let L/K be a finite Galois extension of local fields, with Galois group G. We denoteby OK ⊂ OL the corresponding extension of valuation rings. The associated order of OL is thefull set, AL/K , of all elements of K[G] that induce endomorphisms on OL. It is an OK -orderof K[G] and the unique one over which OL might be free as a module. When the extension isat most tamely ramified, the equality AL/K = OK [G] holds, and OL is AL/K -free. But whenramification is permitted, the structure of OL as an AL/K -module is much more difficult todetermine. Recent progress has been made on this subject and motivates an exposition of thistheory.

Résumé. — Le sujet de cet article est la théorie des modules galoisiens pour les extensions decorps locaux. Précisément, soit L/K une extension galoisienne finie de corps locaux, de groupe G.Notons OK ⊂ OL les anneaux de valuation correspondants. L’ordre associé à l’anneau OL dansl’algèbre de groupe K[G] est l’ensemble, noté AL/K , des éléments λ ∈ K[G] tels que λOL estcontenu dans OL. Cet ensemble est le seul OK -ordre de K[G] sur lequel OL puisse être librecomme module. Lorsque l’extension est modérément ramifiée, on a l’égalité AL/K = OK [G] etOL est libre sur AL/K . Dans le cas contraire, la structure de OL comme AL/K -module est connueuniquement pour des extensions particulières et son étude donne lieu à de nombreuses questionsouvertes. Des progrès récents viennent d’être réalisés et sont exposés dans cet article.

2000 Mathematics Subject Classification. — 11R33, 11S15, 11S20, 20C05.Key words and phrases. — Galois module structure, Normal bases, Local fields, Number fields, Associatedorders, Representation theory of finite groups, Ramification.

The author is very grateful to Nigel Byott, Philippe Cassou-Noguès and Jacques Martinet for enrichingdiscussions and fruitful correspondence. She wishes to thank Erik Pickett for his suggestions in the Englishwriting of this paper, and she is indebted to Christian Maire for his constant support and encouragement.The author would also like to thank Christophe Delaunay, Christian Maire and Xavier Roblot for makingpossible the captivating conference “Fonctions L et Arithmétique” in June 2009. The author was supportedby a postdoctoral fellowship at the École Polytechnique Fédérale de Lausanne in Switzerland.

158 On the Galois module structure of extensions of local fields

Introduction

In its origin, the theory of Galois modules covered classical questions of algebraic numbertheory. For example, let L/K be a finite Galois extension of number fields, with Galois groupG, and let OK and OL denote the integer rings of K and L respectively. The ring OL isnaturally endowed with the structure of an OK [G]-module, and a deep question is concernedwith the freeness of this module. It is well known that a necessary condition for OL tobe free over OK [G] is for the extension to be at most tamely ramified, however this is notsufficient. In 1981, Taylor proved the conjecture of Fröhlich: when the extension L/K is tame,he established an explicit connection between the algebraic structure of OL as a Z[G]-modulewith some analytic invariants attached to certain characters of G [151].

Since this discovery, the subject has developed considerably into several directions, includingthe study of Galois modules over their associated order when ramification is permitted. Pre-cisely, when the extension L/K is wildly ramified, one natural question is to determine thestructure of the valuation ring OL as a module over its associated order in K[G], i.e., overthe full set AL/K of elements of K[G] that induce endomorphisms on OL:

AL/K = λ ∈ K[G] : λOL ⊂ OL.

This is a subring of K[G] which contains OK [G], with equality if and only if the extension isat most tamely ramified.

The most canonical way to attack the problem is via localization, i.e., by transition to localcompletions. Thus, we now suppose that L/K is a finite Galois extension of local fields, withGalois group G, and we denote by OK and OL the valuation rings of K and L. The previousconsiderations apply to this context as well. In this paper, we shall investigate the followingthree problems through the survey of previous articles, and outline the main contributions tothem since the works of Leopoldt and Fröhlich:

1. to give an explicit description of the associated order AL/K of OL in K[G];2. to describe the structure of OL as an AL/K -module, and in particular to determine

whether OL is free over, i.e., is isomorphic to, AK[G];3. if OL is AK[G]-free, to give an explicit generator of OL over its associated order.

It should be stressed that at present there is still no complete theory for associated orders,their structure being essentially known for prescribed extensions only. Also, there are stillpartial general criteria for determining whether a valuation ring is free over its associatedorder in some extension of local fields. However, several advances have recently been made,especially in positive characteristic, and it is our main goal to expose most of these results.

The paper is organised as follows. In Section 1, we begin with a brief survey of the theory ofGalois modules for number fields, including associated orders of integer rings. This will givemotivation for the rest of the paper. We then restrict to the algebraic structure of valuationrings as modules over their associated order in extensions of local fields. Since this studydepends on the ramification of the extension, Section 2 comprises a short preliminary chapterof definitions and properties on the ramification theory for local fields. Sections 3 and 4 are


Lara Thomas 159

then concerned with the questions of describing the associated order of the top valuationring in certain extensions of local fields and determining whether the valuation ring is a freemodule over it, in both mixed (Section 3) and equal (Section 4) characteristic cases. Lastly,Section 5 exposes further comments towards these investigations.

1. Classical Galois module theory for number fields

To give context for what follows, we first recall the classical theory of Galois modules, i.e., forextensions of number fields. For more details about this section, we refer the reader, e.g., toChapter 1 of [92], as well as to the articles [34], [70], [122] and [126].

1.1. Normal integral bases in tame extensions. — Let L/K be a finite Galois extensionof number fields, with Galois group G. We denote by OK (resp. OL) the ring of integers ofK (resp. L).

The normal basis theorem asserts that the field L is free of rank 1 as a left module overthe group ring K[G] (see [18] for two recent short proofs). A more delicate problem is theanalogue question for the study of the Galois module structure of the ring OL. Precisely,the natural action of G on L induces on OL an OK [G]-module structure: understanding thisstructure and determining whether OL is a free module are deeper questions. Note that ifOL is free over OK [G], it is of rank one and the Galois conjugates of any generator form aK-basis of L that is called a normal integral basis. The existence of normal integral bases forthe extension L/K is thus equivalent to the freeness of OL over OK [G].

There are many obstructions to studying the OK [G]-module structure of OL. In particular,when K 6= Q, the ring OK might not be principal. Moreover, OL might not even be free overOK , and, even if it is free, OL might not be OK [G]-free. Some examples will be given below.In fact, the freeness of OL over OK [G] is closely related to the ramification of the extension.

A necessary condition for OL to be free over OK [G] is for it to be OK [G]-projective, i.e., tobe a direct summand of a free OK [G]-module. The following theorem characterizes OK [G]-projective modules in a more general context (see e.g. Theorem II.I of [126]). Recall thatthe extension L/K is said to be at most tamely ramified (“tame") if every prime ideal thatramifies has a ramification index prime to the characteristic of its residue field.

Theorem 1.1. — Let A be a Dedekind domain, with field of fractions K. Let L/K be a finiteGalois extension with Galois group G. We denote by B the integral closure of A in L, and byTrL/K =

∑

σ∈G σ the trace map of L/K. The following conditions are equivalent.

1. B is a projective A[G]-module ;2. TrL/K(B) = A ;3. L/K is at most tamely ramified.

Note that the equivalence 2 ⇔ 3 is a consequence of the characterisation of the differentDL/K of the extension L/K. Indeed, TrL/K(B) 6= A if and only if TrL/K(B) is contained ina prime ideal p of A, i.e., if and only if DL/K ⊂ pB. Thus, according to ([143], Chap. III,Prop. 13), this is equivalent to the existence of prime ideals of OL above p that are not



tamely ramified. As for 1 ⇒ 2, it is essentially the statement that, since B is an A-moduleof finite type which is torsion-free, if B is projective then it is cohomologically trivial, i.e.,H0

(G,B) = A/TrL/KB = 0.

When applied to extensions of number fields, Theorem 1.1 provides necessary conditions forOL to be free as an OK [G]-module, but they are not sufficient. On the other hand, a theoremof Swan [147] asserts that every projective OK [G]-module M of finite type is locally free: foreach prime ideal p of OK , the localization of M at p, that is Mp = M ⊗OK

OKp , is free overOKp [G], where Kp is the p-adic completion of K (for a complete proof, see also Theorem 32.11of [74]). In particular, if M is such a module, its rank is well defined: it is given by the rankof the free K[G]-module M ⊗OK

K. This rank is finite and also equals the rank of Mp as anOKp [G]-module, for every p.

Therefore, Theorem 1.1 implies the following criterion which is usually known as Noether’scriterion, part of which goes back to Speiser [146] (he proved the necessary condition), andwhich is presented as the basic starting point of Galois module structure theory:

Theorem 1.2 (Noether’s criterion). — Let L/K be a finite Galois extension of numberfields, with Galois group G. Let OK ⊂ OL be the corresponding integer rings. Then OL islocally free over OK [G] if and only if the extension is tamely ramified.

In particular, when the extension L/K is tame, the ring OL determines an element in the classgroup Cl(OK [G]) of locally free OK [G]-modules, and one is interested in understanding thisclass in terms of the arithmetic of the extension L/K. Recall that this group is defined as thekernel of the rank map from K0(OK [G]) to Z, where K0(OK [G]) is the Grothendieck groupof the category of locally free OK [G]-modules with addition given by direct sums. If M is alocally free OK [G]-module, we denote by M its class in K0(OK [G]) and by [M ] the elementM − mOK [G] of Cl(OK [G]), where m is the rank of M . Since every locally free OK [G]-module of rank ≥ 2 has the cancellation property ([96], result IV), we have M = N inK0(OK [G]) if and only if M ⊕ OK [G] ≃ N ⊕ OK [G], which implies M ≃ N whenever theranks are strictly greater than 1. Finally, Cl(OK [G]) is a finite abelian group whose neutralelement is formed by the classes of all stably free OK [G]-modules, and in fact by the classesof all locally free OK [G]-modules M of rank 1 such that M ⊕ OK [G] ≃ OK [G] ⊕ OK [G], asa consequence of ([96], results I and IV). Algorithms for explicit computations of the locallyfree class group Cl(OK [G]) were recently worked out in [20, 22].

1.1.1. Extensions over Q.— We first suppose K = Q, consider a tame extension L/Q withGalois group G, and address the question of determining whether OL is free over Z[G]. Whenthe extension L/Q is abelian, a result of Hilbert, as part of the well-known Hilbert-Speisertheorem, implies that OL is Z[G]-free (originally, this result was restricted to abelian exten-sions L/Q whose degree is relatively prime to the discriminant of L, and Leopoldt extended itto abelian tame extensions [117]). The proof is based on the Kronecker-Weber theorem: L is

a subfield of a cyclotomic field Q(e2iπn ) with n squarefree, and the trace of e

2iπn in L generates

a normal integral basis for L/Q. Now, this argument does not apply when G is not abelianor when K 6= Q.


Lara Thomas 161

The existence of normal integral bases in tame non-abelian extensions over Q was widely inves-tigated during the 1970’s by several authors, including Armitage, Cassou-Noguès, Cougnard,Fröhlich, Martinet, Queyrut and Taylor. In particular, Martinet first proved that the ring OL

is free over Z[G] when G is a dihedral group of order 2p, for some odd prime number p [126].But then, in 1971, he constructed tamely ramified extensions L/Q whose Galois group G isa quaternion group of order 8 and such that OL is not free over Z[G] [124]. This providedthe first known counter-example for the existence of normal integral bases, and motivated toa very large extent the conjecture of Fröhlich. Several contributions and computations led tothe proof, by Taylor, of this conjecture, in 1981 [151]. A precise account of them is given inChapter 1 of [92].

The conjecture of Fröhlich determines the class of OL in the locally free class group Cl(Z[G])

in terms of some analytic invariant. Taylor’s proof is based on the combination of severalingredients: a generalization to non-abelian characters of the classical Lagrange resolvent andGalois Gauss sums, the logarithm for local group rings which was first introduced by Taylorhimself, as well as the famous Fröhlich’s Hom-description of the class group Cl(Z[G]) allowingmuch of the work to be conducted at a local level (see e.g. Chapter II of [92], or [34]).

Precisely, for any character χ of the Galois group G, there is an extended Artin L-functionΛ(s, χ) attached to L/Q and which satisfies a functional equation Λ(1− s, χ) = W (χ)Λ(s, χ),where χ is the complex conjugate character and W (χ) is a constant called the Artin rootnumber attached to χ. Fröhlich’s conjecture is related to the equality [OL] = t(WL/Q) inCl(Z[G]), where t(WL/Q) is the so-called analytic root number class. This invariant wasfirst defined by Cassou-Noguès [50] solely in terms of the values of Artin root numbers ofsymplectic characters of G. As these values are ±1, t(WL/Q) is an element of order 1 or 2.The theorem of Taylor can thus be stated as follows.

Theorem 1.3 (Taylor, 1981). — Let L/Q be a finite Galois extension of number fields,with Galois group G. Denote by OK ⊂ OL the corresponding integer rings. If the extensionis at most tamely ramified, then:

1. [OL ⊕ OL] = 1 in Cl(Z[G]);2. the only obstructions to the vanishing of the class of OL are the signs of the Artin

root numbers of symplectic characters. In particular, if G has no irreducible symplecticcharacter, then [OL] = 1 in Cl(Z[G]).

Equivalently, assertion 1 states that the module OL⊕OL is stably free over Z[G], and thus freesince it is of rank 2. This means that we always have OL ⊕ OL ≃ Z[G] ⊕ Z[G]. If, moreover,t(WL/Q) = 1, then OL⊕Z[G] ≃ Z[G]⊕Z[G] and OL is stably free. This happens in particularwhen G has no irreducible symplectic characters (assertion 2): in this case, OL is in fact freebecause Z[G] satisfies Jacobinski’s cancellation theorem since no simple component of Q[G]

is a totally definite quaternion algebra (see e.g. [96], Par. 3).

More generally, Theorem 1.3 can be applied to determine the Z[G]-structure of OL in somerelative tame extension L/K with Galois group G: in this case, the module OL ⊕ OL is

isomorphic to Z[G]2[K:Q]. In particular, OL is Z[G]-free whenever [OL] = 1 in Cl(Z[G]) and

Z[G] has the cancellation property. Specifically, OL is Z[G]-free in the following supplementary



cases: when the order of G is odd [151] or not divisible by 4 [51, 70], when G is symmetric,or when K contains the m-th roots of unity if m is the exponent of G [93]. In particular,for K = Q, this provides new examples of tame extensions of Q with integral normal bases.Note also that, in 1978, Taylor already proved the analogue of the Hilbert-Speiser theorem inthis context: if L/K is tame, then OL is Z[G]-free. Now, Theorem 1.3 cannot be extendedto determine the relative Galois structure of OL in general, i.e., as an OK [G]-module whenK 6= Q (see next paragraph).

On the opposite side, the conjecture of Fröhlich has given rise to the construction of new tameextensions L/Q without integral normal basis, among them certain quaternion extensions. Forinstance, if L/Q has Galois group G = H32, Fröhlich had proved that OL is stably free overZ[G]. However, this doesn’t necessarily imply that OL is free. Indeed, Cougnard constructedsuch an extension without integral normal basis [69]. Note that on H8 and H16, every stablyfree module is free. In particular, in [124], the quaternion extensions L/Q with Galois groupH8 and without integral normal basis are such that OL is not stably free over Z[G].

To conclude this section, we should stress the fact that Theorem 1.3 does not lead to anydescription of generators when OL is free over OK [G]. However, explicit generators or al-gorithms to find them when K = Q and G = A4, D2p (with p odd prime), H8, H12, H32

or H8 × C2 are given in [65, 67, 73, 68, 126]. More recently, in 2008, Bley and Johnstonimplemented an algorithm which, amongst other things, determines such generators for othergroups G; in particular, abelian or dihedral D2n with “small" orders [21].

1.1.2. Relative extensions of number fields.— The case of relative extensions is much moredifficult, even for abelian extensions. In 1999, Greither et al. proved that the field Q is theonly base field over which all tame abelian extensions have a normal integral basis [104].

In fact, the question of the existence of normal integral bases for tame relative extensions issolved for prescribed extensions only, including e.g. certain cyclic extensions. For example,one key argument in [104] is that for any number field K 6= Q, there exists a prime number pand a tame cyclic extension L/K of degree p without normal integral basis. In 2001, Cougnardgave other examples of relative cyclic extensions without normal integral basis, generalizingresults of Brinkhuis [29, 30]. Furthermore, in 2009, Ichimura proved that when K/Q isunramified at some odd prime number p, any tame cyclic extension L/K of degree p has anormal integral basis if the extension L(ζp)/K(ζp) has a normal integral basis, where ζp is ap-th root of unity [108].

Kummer extensions of number fields have been investigated by several authors, such as Fröh-lich [99], Kawamoto [113, 114, 115], Okutsu, Gomez-Ayala [101], Ichimura [109], and veryrecently Corso and Rossi [75]. Gomez-Ayala gave an explicit criterion for the existence of nor-mal integral bases in tame Kummer extensions of prime degree, along with explicit generators.Del Corso and Rossi (2010) have just generalized this result to cyclic Kummer extensions ofarbitrary degree, precising [109]. Their result is based on an explicit formula for the rami-fication index of prime ideals in such extensions. As an application, Ichimura proved that,given an integer m ≥ 2 and a number field K, there exists a finite extension L/K depending


Lara Thomas 163

on m and K such that for any abelian extension M/K of exponent dividing m, the extensionLM/L has a normal integral basis.

Other relative extensions without normal integral basis have been investigated (see e.g. [115]).Among several contributions, one could cite Brinkhuis’ result for CM fields [29]: if L/K isan unramified abelian extension of number fields, each of which is either CM or totally real,and if the Galois group of L/K is not 2-elementary, then L/K has no normal integral basis.

We conclude this section with the notion of weak normal integral bases whose non existenceis a further obstruction to the existence of normal integral bases [30]: if L/K is a tamefinite abelian extension of number fields with Galois group G, we say that L/K has a weaknormal integral basis if the projective M-module M⊗OK [G] OL is in fact free, where M is theunique maximal OK -order of K[G]. The investigation of this notion has just led Greither andJohnston to establish a necessary and sufficient condition for the existence of normal integralbases ([102], Theorem 5.5]):

Proposition 1.4 (Greither & Johnston, 2009). — Let L/K be a tame finite extensionof number fields such that L/Q is abelian of odd degree. Suppose that either [L : K] is notdivisible by 3 or that for all primes q dividing [K : Q] the field L(ζ3∞) contains no q-th root ofunity. Then L/K has a normal integral basis if and only if the tower L/K/Q is arithmeticallysplit.

Here, a tower of number fields K ⊂ M ⊂ L is said to be arithmetically split if there exists anextension L′/K such that L = L′M and the extensions L′/K and M/K are arithmeticallydisjoint, i.e., they are linearly independant and no finite prime p ramifies both in L′/K andM/K.

1.2. Wildly ramified extensions. — When the extension L/K is wildly ramified, i.e., nottamely ramified, we are faced with a very different situation since the integer ring OL is notlocally free over OK [G]. We also note another failure concerned with the following result dueto Fröhlich (which was a key argument in Theorem 1.3): suppose L/Q is a tame extensionwith Galois group G and let M be a maximal order in Q[G] containing Z[G], then the locallyfree M-module OLM is stably free over M. This result was first conjectured by Martinet,and Cougnard proved that it does not hold in general for wild extensions [71]. There are anumber of approaches to circumventing these difficulties, and we present some of them here(see also Appendix C of [92]).

We can first investigate which results from the classical tame theory can be generalized tothe wild case, by adapting the framework. For example, Queyrut developed a K-theoreticapproach, replacing the locally free class group Cl(Z[G]) with the class group of anothercategory of Z[G]-modules (see e.g. [52], [139]). One should cite [53] as well, where the classof OL in Cl(Z[G]) is replaced with the class of a certain submodule of OL.

We can also consider the Ω-conjectures of Chinburg which extend and generalize Fröhlich’sconjecture [63, 145], and their relations with Equivariant Tamagawa Number conjectures inspecial cases (see e.g. [31]). Chinburg’s conjectures give equalities between new invariants inthe class group Cl(Z[G]) that involve the Galois structure of OL.



Another approach is concerned with the indecomposable Z[G]-modules or OK [G]-modulesthat can occur in OL as well as the question of the decomposability of OL as an OK [G]-module. In this direction, one should cite the contributions for certain p-extensions of Yokoi,Miyata (e.g. [132]), Bertrandias [14], Bondarko & Vostokov (e.g. [27]), Rzedowski Calderon,Villa Salvador & Madan [141], as well as Elder & Madan (e.g. [78, 79, 80, 81]). Note thatthe question depends very much on ramification invariants of the extension, and most of theresults are stated under some technical restrictions on ramification numbers.

In this paper, the approach we consider is due to Leopoldt [117]; it was initiated by Leopoldt,Fröhlich [99] and Jacobinski [111]. The idea is to replace the group ring OK [G] by a largersubring of K[G], namely the associated order of OK in K[G]:

AL/K = AL/K(OL) = λ ∈ K[G] : λOL ⊂ OL,with the idea that OL may have better properties as a module over AL/K than over OK [G].The Galois module structure of OL over its associated order, for extensions of global or localfields, is our main topic of interest in the rest of the paper.

1.3. The associated order of integer rings in extensions of number fields. —

1.3.1. General properties.— Let L/K be a finite Galois extension of number fields, withGalois group G. In this section, we give an account of the general properties of the associatedorder of OL in K[G]. They all hold in more general Galois extensions L/K, e.g. when K isthe field of fractions of some Dedekind domain OK and OL is the integral closure of OL in L.However, to simplify the exposition of the paper, we describe them in the number field case.For further details, we refer to [7], [8], [11] and [125].

The associated order AL/K of OL in L/K is an OK-order in K[G], i.e., it is a subring ofK[G] and a finitely generated module over OK which contains a K-basis of K[G]. It is alsoa free OK -module of rank [L : K] over OK , since it is a subring of the endomorphism ringEndOK

(OL) with OK a principal ideal domain.

The ring OL is a module over its associated order which is torsion free and finitely generated.One can thus define its rank as the dimension over K of OL ⊗AL/K

L, and see that it is equalto 1, by the normal basis theorem. Moreover, the associated order AL/K is the only OK -orderof K[G] over which OL can be free as a module (Par. 4 of [125], Prop. 12.5 of [60] or Par. 5.8of [154] ).

In [11], Bergé described some further general results about AL/K obtained by Jacobinski, inparticular when viewed as a subring of the ring EndOK

(OL) of OK -endomorphisms of OL.Note that, when the extension L/K is abelian, the associated order AL/K is isomorphic tothe ring EndOK [G](OL) of OK [G]-endomorphisms of OL.

Finally, the equality AL/K = OK [G] holds if and only if the extension L/K is at mosttamely ramified (see e.g. [11], Theorem 1), in which case OL is locally free as an OK [G]-module. However, the question of determining whether OL is locally free (or even free)over its associated order is much more delicate in the wildly ramified case than in the tame


Lara Thomas 165

case. First, OL might not be projective over its associated order [17]. Moreover, there existprojective AL/K -modules that are not locally free (see [122], Par. 3). Also, the algebraicstructure of AL/K is known for prescribed extensions of global and local fields only and stillyields open questions.

Remark 1.5. — One could address the question of whether Artin root numbers of real char-acters of G can be related to the structure of OL over its associated order, generalizingFröhlich’s conjecture in this setting. However, as far as we know, this idea fails. In [122](Par. 3.3), Martinet raised several obstructions to such a connection and gave a counter-example for quaternion extensions of degree 8, citing a result of Fröhlich [98].

1.3.2. Extensions over Q.— Again, when K = Q, some partial results are known. First,in 1959, Leopoldt proved that for any abelian extension L/Q the ring OL is free over itsassociated order [117], generalizing in this way the Hilbert-Speiser theorem. Once more, theproof is based on the Kronecker-Weber theorem but the arguments are much more difficult.In 1964, Jacobinski [111] gave an alternate proof to this theorem, extending the explicitdescription of the associated order in terms of the ramification structure to a larger class ofextensions (see Subsection 3.2). A simplified proof was also given by Lettl in [120]. Note thatthe theorem of Leopoldt is very explicit, in the sense that it determines AL/K and providesan explicit Galois generator in terms of the classical abelian Galois Gauss sums. See ([54],Chapter I) for further details.

In 1972, generalizing a theorem of Martinet, Bergé [10] proved that OL is free over its as-sociated order when L/Q is a dihedral extension of order 2p when p is an odd prime. Butdihedral extensions over Q of order 6= 2p give counter-examples of the projectivity of OL overits associated order. At the same time, Martinet proved that every quaternion extension ofdegree 8 over Q that is wildly ramified is such that OL is free over its associated order [123],which is not always true when the extension is tamely ramified [124].

1.3.3. Leopoldt extensions.— Let L/K be an extension of number fields where L/Q is abelian;it is said to be Leopoldt if the ring of integers OL is free over its asociated order. A field issaid to be Leopoldt if every finite extension L/K with L/Q abelian is such that OL is freeover its associated order. Results of Leopoldt [117], Cassou-Noguès & Taylor ([54], Chap. 1,Thm. 4.1), Chan & Lim [57], Bley [19], and Byott & Lettl [48] culminated in the proofthat the n-th cyclotomic field Q(ζn) is Leopoldt for every n. See also [100] for another typeof Leopoldt extensions. Johnston has generalized these results by giving more examples ofLeopoldt fields, along with explicit generators [112]. He has also obtained some freeness resultin intermediate finite layers of certain cyclotomic Zp-extensions ([112], Cor.8.4).

The result of Chan and Lim is the following [57]: let m and m′ be positive integers with m|m′,let K = Q(ζm) and L = Q(ζm′), where ζn denotes a primitive n-th root of unity. Then, thering of integers Z[ζm′ ] of L is free over its associated order in K[G] and the authors give explicitgenerators (Aiba investigated an analogue of this result for function fields, see Subsection 4.3).As noticed by Byott, this order is in fact the maximal order in K[G]. Later, for an extension



L/K with L/Q abelian, Byott and Lettl [48] gave an explicit description of the associatedorder of OL when K is a cyclotomic field, and proved that OL is free over it. Their papercontains some intermediate results about maximal orders in K[G] (see Subsection 5.1).

1.3.4. Other extensions.— One should also mention the reference ([54], Chaper XI) whereCassou-Noguès and Taylor determine the Galois structure of rings of integers of certain abelianextensions over quadratic imaginary number fields, by evaluating suitable elliptic functionsat singular values. See also [5] where Bayad considers the Galois module structure of rings ofintegers attached to elliptic curves without complex mutliplication and admitting a rationalpoint of finite order: this contains a freeness result over associated orders, with explicitgenerators.

1.3.5. Intermediate results.— The paper of Johnston [112] is interesting also because it gath-ers several properties of associated orders that might be very useful, some of them are originallyissued from [48] and [57]. For example, the next two propositions show how associated ordersin composite fields and subfields can be determined under certain additional assumptions,which sometimes permits the reduction of the problem to simpler extensions:

Proposition 1.6 ([48], Lemma 5). — If L/K and M/K are arithmetically disjoint exten-sions of number fields, then ALM/M = AL/K⊗OK

OM and ALM/K = AL/K⊗OKAM/K . More-

over, if OL = AL/K · α1 and OM = AM/K · α2, then OLM = ALM/K · (α1 ⊗ α2).

More generally, Greither & Johnston recently obtained an arithmetically disjoint capitulationresult for certain extensions of number fields ([103], Cor. 1.2 ), generalizing [109]. As noticedby the authors ([103], Remark 5), there is no “arithmetically disjoint capitulation" for finiteGalois extensions of p-adic fields ([119], Proposition1.b). Moreover, one can prove that ifL/K is a finite Galois extension of number fields such that OL is not locally free over itsassociated order AL/K , then there exists no extension M/K arithmetically disjoint from L/Ksuch that OLM = OL ⊗OK

OM is free over ALM/K = AL/K ⊗OKOM .

An interesting result for certain intermediate extensions is the following:

Proposition 1.7 ([48], Lem. 6 - [112], Cor. 2.5). — Let L/K and M/K be Galois ex-tensions of number fields with K ⊂ M ⊂ L and L/M at most tamely ramified. Put G =

Gal(L/K) and H = Gal(M/K). Let π : K[G] → K[H] denote the K-linear map induced bythe natural projection G → H. If OL = AL/K · α for some α ∈ OL, then AM/K = π(AL/K)

and OM = AM/K · TrL/M (α).

Another line of attack is to reduce the problem of the existence of elements α such thatOL = AL/K ·α to the computation of certain discriminants, based on explicit computation ofresolvents;

Proposition 1.8 ([112], Cor. 4.4). — Let L/K be a finite extension of number fields, with

Galois group G, and let G denote the group of characters of G. Suppose that OL is locallyfree over AL/K . Then, for any α ∈ OL, OL = AL/K · α if and only if

∏

χ∈G < α|χ > divides


Lara Thomas 167

∏

χ∈G < β|χ > for all β ∈ OL, where < α,χ >=∑

g∈G χ(g−1)g(α) is the resolvent attached

to α and χ.

This result has to be compared with Lemma 1 of [3] in the case of function fields, where Galoisgenerators over associated order when they exist are characterized by a minimality conditionon discriminants.

Finally, let us mention the article [6] in which Bergé investigated the genus of the ring of inte-gers of an extension of number fields. She analyzed the obstacles for projectivity encounteredat various stages of reduction, linking them to a bad functorial behavior of the associatedorder.

1.3.6. From local freeness to global freeness.— Other examples are derived from the local case.Mathematicians rapidly considered Galois module structure of rings of integers for extensionsof local fields, mainly motivated by Noether’s theorem and because the local context is easierto deal with. We should also mention the following proposition which allows us to reduce tothe local case (see Prop. 1 of [8], as well as Prop. 2 of [17]):

Proposition 1.9. — Let L/K be a finite Galois extension of number fields, with Galois groupG. Let P be a prime ideal of OL whose decomposition group coincides with G and writep = P ∩ OK . Then, the associated order of OLP

in Kp[G] is the p-adic completion of AL/K .Moreover, if OL is free (resp. projective) over AL/K , then OLP

is free (resp. projective) overUKp [G].

One can generalize this to all prime ideals p of OK , i.e., without the condition on the decom-position group, and prove that OL is projective over its associated order in K[G] if and onlyif it is locally projective (see [8], Chap. I, Par. 1 & 2, where p-adic completions correspond totensor products over OKp [D], D being the decomposition group of an ideal above p). How-ever, being locally free is not a sufficient condition for OL to be free over AL/K . Nevertheless,recent results of Johnston and Greither & Johnston show that local freeness is close to globalfreeness in the following sense;

Proposition 1.10 (Johnston, 2008 ([112], Proposition 3.1))If OL is locally free over AL/K , then, given any non-zero ideal a of OK , there exists β ∈ OL

such that a + [OL : AL/K · β] = OK , where [OL : AL/K · β] denote the OK-module index ofAL/K · β in OL.

Proposition 1.11 (Bley & Johnston, 2008 ([21], Prop. 2.1))Let L/K be a finite Galois extension of number fields with Galois group G. Let M be amaximal order in K[G] containing AL/K . Then, OL is AL/K-free if and only if:

1. OL is locally free over AL/K , and;2. there exists α ∈ OL such that M ⊗AL/K

OL = M · α.

When this is the case, then OL = AL/K · α.



In fact, Proposition 1.11 is stated in a more general context. Furthermore, when the Wed-derburn decomposition of K[G] is explicitly computable and under certain extra hypothesis,Bley and Johnston derive from this proposition an algorithm that either determines a AL/K -generator of OL or determines that no such element exists ([21], Par. 8).

In what follows, we then restrict to finite Galois extensions of local fields and consider thestructure of the top valuation rings over their associated order.

2. Local setup

From now on, we suppose that K is a local field, i.e., a complete field with respect to adiscrete valuation vK : K∗ → Z (with vK(0) = +∞). Let OK be its valuation ring, i.e.,OK = x ∈ K : vK(x) ≥ 0, and let pK denote the unique maximal ideal of OK . We thendefine the residue field of K as the quotient k := OK/pK , and we shall always suppose thatit is perfect.

Let p be a prime number. When k has characteristic p, this leads to the following cases;

- equal characteristic case (p, p): K has characteristic p, in which case it can be identifiedwith the field of formal power series k((T )) for some element T ∈ K with vK(T ) = 1;

- unequal characteristic case (0, p): K has characteristic 0, i.e., it is an extension of the fieldQp of p-adic numbers.

Next, we fix a separable closure Ksep of K and we consider a finite Galois extension L/Kwith Galois group G. Let OK ⊂ OL denote the corresponding valuation rings. In our setup,i.e., when K is a local field, the ring OL is always free as an OK -module, since it is of finitetype over a principal ring ([143], Chap. 2, Prop. 3).

2.1. Galois module theory for extensions of local fields.— The more general Hattori’sapproach to Swan’s theorem ([74], 32.A) enables us to derive again, from Theorem 1.1, aNoether’s criterion for extensions of local fields:

Proposition 2.1. — If L/K is a finite Galois extension of local fields, with Galois group G,then OL is OK [G]-free if and only if the extension is at most tamely ramified.

When the extension is ramified, we introduce the associated order AL/K of OL in K[G], givenby AL/K = λ ∈ K[G] : λOL ⊂ OL. The classical considerations described in the firstsection, and specifically in Subsection 1.3, apply to this local context as well.

In particular, it is a ring containing OK [G], with equality if and only if L/K is tame, andOL is an AL/K-module. Moreover, since K[G] acts faithfully on L, AL/K is an OK -orderin K[G] and we address the question of whether OL is free over AL/K . If it is, and if wecan find an explicit generator, then we can say that we have determined the structure of theOK [G]-module OL. If, on the other hand, OL is not free over AL/K , then we have at leastobtained one information about the Galois structure of OL: its structure is too complicated tobe rendered free by enlarging OK [G]. In both cases, the question is difficult, not least becauseit is difficult to describe AL/K as an OK-module since it requires a detailed understanding of


Lara Thomas 169

the action of G on OL. Many answers are in fact given without an explicit determination ofthis order.

This question is solved for prescribed extensions of local fields only, and our goal is to exposemost of the known answers in what follows.

2.2. Ramification of local fields.— The description of the associated order of the valua-tion ring in any extension of local fields involves higher ramification invariants. We thus recallsome facts about the ramification theory of local fields with perfect residue field, and pre-cisely the notion of ramification groups and jumps. For further details, see for example ([143],Chap. IV), and for a complete investigation of the possible values of ramification jumps inp-extensions of local fields, we refer the reader to [116] and [90], as well as the works ofMarshall, Maus, Miki and Wyman.

Let L/K be a finite Galois extension, with group G. The ramification groups Gi, for i ∈ Z≥−1,of L/K are defined by:

Gi := σ ∈ G : σ(x) − x ∈ pi+1

L .In particular, the ramification groups form a decreasing filtration of normal subgroups of G:

G = G−1 ⊇ G0 ⊇ G1 ⊇ · · · ⊇ Gm 6= Gm+1 = 1,

for some integer m ≥ −1. Note that, for i = 0, the ramification group G0 is the inertia groupof L/K, and the ramification index of the extension is defined as eL/K = card(G0).

The extension L/K is said to be unramified if G0 = 1; tamely ramified if G1 = 1, equivalentlyif its ramification index is prime to the residue characteristic of K; and totally ramified ifG0 = G. Moreover, when the residue field k of K has characteristic 0, the group G1 is trivialand G0 is cyclic; when char(k) = p, G1 is a p-group and the quotient group G0/G1 is cyclicof order prime to p. In particular, if char(k) = 0, then the extension L/K is at most tamelyramified. This is the reason why we exclude this case and only consider local fields withpositive residue characteristic p, since we are interested in wild extensions. In this context, ifL/K is a p-extension, then G0 = G1: in particular, tamely ramified implies unramified.

A notion that arises naturally is that of ramification jumps, which are defined as the integersb ≥ −1 such that Gb 6= Gb+1. They form an increasing sequence: b1 < · · · < br (with br = mwith regards to the previous notation). When the residue characteristic of K is some primenumber p and L/K is a totally ramified p-extension, i.e., b1 ≥ 1, jumps are all congruentmodulo p ([143], IV.2, Proposition 11) :

∀i, j, bi ≡ bj mod p.

Furthermore, when the extension is abelian, the Hasse-Arf theorem induces more advancedcongruences (see e.g. [153], Proposition 5).

If L/K is a totally ramified p-extension, we have the following. When char(K) = p, i.e.,in the context of Artin-Schreier theory, one can prove that all ramification jumps of L/Kare relatively prime to p whenever the residue field of K is perfect. Moreover, in this case,ramification jumps are not bounded.



When char(K) = 0, things are rather different: jumps are bounded and might be divisibleby p. Precisely, suppose L/K be of degree pn and let b1 ≤ b2... ≤ bm denote its ramification

jumps. Let eK = vK(p) be the absolute ramification index of K. Then, bm ≤ eK [L:K]

p−1, and

so bm − ⌊ bm

p ⌋ ≤ pn−1eK . Moreover, if bm < pneK

p−1then p 6 |bi for all i, and if bm =

pneK

p−1, then

all jumps are divisible by p and L/K is a Kummer cyclic extension (see [143], Chap. IV,Exercise 3).

3. Local Galois module structure in mixed characteristic

Let K be a local field of characteristic (0, p), for some fixed prime number p. We seekto determine AL/K and the module structure of OL as an AL/K-module, where the action isinduced by that of K[G] acting on L. We will often suppose that K is in fact a finite extensionof Qp (and say it is a p-adic field), equivalently, that its residue field k is finite, since most ofthe known results are stated in this setting even if they can be generalized to local fields withperfect residue field.

3.1. On the p-adic version of Leopoldt’s theorem. — The archetypal result is that ofLeopoldt. For extensions of p-adic fields, it says that OL is AL/K -free whenever K = Qp andG is abelian. However, it is proved that the field Qp is the only base field which satisfies thisproperty (see e.g. Subsections 3.2 and3.3).

In 1998, Lettl strengthened the local version of Leopoldt’s theorem as follows [119]. Heproved that if L/Qp is abelian, then OL is again free over AL/K for any intermediate field Kof the extension L/Qp. Moreover, writing G0 for the inertia group of L/K, and M0 for themaximal order in K[G0], the author shows that if p 6= 2, then AL/K = OK [G] ⊗OK [G0] M0.Unfortunately, his argument does not give an explicit generator of OL over its associated orderin general. As a corollary, Lettl deduced a global result: if L/K is an extension of numberfields with L/Q abelian, then OL is locally free over AL/K .

In fact, the property of Lettl characterises Qp among its finite extensions, as a local analogueof [104]. If F 6= Qp, then there exist fields F ⊂ K ⊂ L with L/F abelian but OL not free overAL/K . An example is given by Lubin-Tate extensions in ([42] Theorem 5.1): let K be a finite

extension of Qp and write K(n) for the n-th division field of K with respect to a Lubin-Tate

formal group, then K(n) is an abelian extension of K, but OKm+r fails to be free over itsassociated order in K(m+r)/K(r) whenever m > r ≥ 1 and K 6= Qp.

Another extension of the p-adic Leopoldt’s theorem is due to Byott. If L/K is an abelianextension of p-adic fields, then OL is AL/K -free whenever L/K is at most weakly ramified,i.e., its second ramification group is trivial ([41], Cor. 4.3).

3.2. Extensions with cyclic inertia group (Bergé). — According to the local version ofLeopoldt’s theorem, if L/Qp is a finite abelian extension with Galois group G and ramificationgroups Gi, the associated order AL/K of OL is the subring of Qp[G] obtained by adjoining to


Lara Thomas 171

Z[G] the idempotents

ei =1

card(Gi)

∑

σ∈Gi

σ,

and OL is free over this subring. In [7], Bergé investigated the analogue of this propertyfor extensions over any absolutely unramified p-adic field with cyclic inertia group, extendingresults of [111]. In this section, we shall describe most of her results.

Let K be a p-adic field which is absolutely unramified, i.e., eK = 1. Let L/K be a finiteGalois extension, with Galois group G and cyclic inertia group G0. Bergé described explicitlythe associated order AL/K of such an extension, and investigated criteria for the top valuationring to be free over its associated order. However, in her more general setting, Bergé did notconsider the problem of giving explicit generators for those cases in which OL is free over itsassociated order.

As another consequence of her investigation, she constructed extensions for which the valua-tion ring is not free. This fact was already surprising since the conditions imposed on K andon G by Bergé are merely the abstraction of conditions satisfied by all abelian extensions ofQp, and these extensions satisfy Leopoldt’s result.

Using a result of Jacobinski [111], Bergé first reduced the problem to totally ramified exten-sions. So, let L/K be a totally ramified cyclic extension of order rpn, with p 6 |r and eK = 1.The cyclic group G/G1 has order r. Let C be the multiplicative group of characters of G/G1

of degree 1. For each χ ∈ C, we write eχ for the idempotent

eχ =1

r

∑

σ∈G/G1

χ(σ−1)σ

of the group algebra K[G/G1]. For each ramification group Gi of L/K, we also write

ei =1

card(Gi)

∑

σ∈Gi

σ

which is an idempotent of K[G]. According to Leopoldt’s result, if K = Qp and L/K isabelian, all ei belong to K[G]. This does not hold in general, and Bergé provided explicitcounter examples in her setting. Her main result is the following ([7], Thm. 1);

Proposition 3.1 (Bergé, 1978 ). — Let K be a p-adic field such that K/Qp is unramified.Let L/K be a totally ramified cyclic extension of degree rpn with p 6 |r. Let σ be a generatorof its highest non trivial ramification group, and write f = σ − 1. Then AL/K is the subringof K[G] generated by OK [G], the elements eif for 1 ≤ i ≤ n, and the idempotents eχei for allχ ∈ C and all i such that eχei ∈ AL/K .

Bergé then investigated the existence of a criterion for OL to be free over its associated order.This yields the following criterion ([7], Cor. of Thm. 3);



Proposition 3.2 (Bergé, 1978). — Under the assumptions of Proposition 3.1, and if t1denotes the first ramification jump of L/K, the ring OL is free as a AL/K-module if and only

if rpp−1

− t1 < pn

pn−1−1, with pn

pn−1−1= +∞ if n = 1.

In particular, if the property “OL is free over its associated order" is true for L/K, then it istrue for subextensions L′/K.

Bergé then derived from these investigations the structure of AL/K , as well as certain criteriafor freeness, for non totally ramified extensions (see e.g. [7], Cor. of Thm 2). For example,if L/K has cyclic inertia group and if eK = 1, she proved that AL/K is included in the OK -order of K[G] generated by OK [G] and all the idempotents ei attached to the ramificationgroups Gi. Moreover, the equality holds if and only if rp

p−1− 1 ≤ t1, where t1 is the first

ramification jump of L/K ([7], Cor. of Prop. 3 and Cor. 3 of Thm. 1). When this is the case,we say that the extension is almost maximally ramified. We shall come back on this notionin Subsection 5.1.

Finally, note that most of Bergé’s results have been extended by Burns in [33].

3.3. Cyclic p-extensions. — We now consider a general p-adic field K, i.e., without anyassumption on eK . On such a field, there are several results for cyclic p-extensions, but onlythe case of extensions of degree p is completely solved.

3.3.1. Cyclic extensions of degree p.— Let L/K be a totally ramified cyclic extension ofdegree p. Contributions of Bergé, Bertrandias (F. and J.-P.) and Ferton in the 1970’s cul-minated in a complete answer for such an extension: they determined an explicit descriptionof the associated order AL/K , obtained full criteria for OL to be free over it, and describedgenerators.

First, Bergé [12] and Bertrandias & Ferton [17] obtained independently and by differentmethods an explicit description of AL/K when L/K is a totally ramified cyclic extension ofdegree p;

Theorem 3.3 (Bergé - Bertrandias & Ferton, 1972). — Let K be a p-adic field, withuniformizing element πK . Let L/K be a totally ramified extension of degree p. Let t be itsunique ramification jump, and let σ be a generator of its Galois group. Write f = σ−1. Then,the associated order of OL in K[G] is the OK-submodule of K[G] generated by the elementsf i

πni

K

for i = 0, ..., p − 1, where the integers ni are given by:

ni = ⌊ it + ρi

p⌋,

with rj the least non-negative residue of −jt modulo p, and ρi = infi≤j≤p−1rj .

Moreover, Bertrandias (F. and J.-P.) and Ferton ([16, 17]) determined explicitly when OL isfree over its associated order;


Lara Thomas 173

Theorem 3.4 (F. Bertrandias - J.P. Bertrandias - M. J. Ferton, 1972)Let K be a finite extension of Qp. Let L/K be a totally ramified extension of degree p withramification jump t.

1. If p|t, then OL is free over AL/K .2. If p 6 |t, write t = pk + a with 1 ≤ a ≤ p − 1, we have:

(a) if 1 ≤ t < peK

p−1− 1, then OL is AL/K-free if and only if a|p − 1 ;

(b) if t ≥ peK

p−1− 1, then OL is AL/K-free if and only if N ≤ 4, where N denotes the

length of the continued fraction expansion:

t

p= a0 +

1

a1 + ...1

... +1

aN

,

with aN ≥ 2.

Note that cases 2.(a) and 2.(b) are treated differently. Moreover, case 2.(b) is preciselywhen the extension is said to be almost maximally ramified (see Subsection 5.1 for furtherdetails). For an extension of degree p, this is equivalent to the condition where the idempotent1

p

∑

σ∈G σ belongs to AL/K .

When eK = 1, then t = 1 = a if p 6= 2, and t = a = 1 or p|t if p = 2. In particular, case2.(a) never happens. Therefore, if K is an absolutely unramified p-adic field, extensions ofdegree p over K are almost maximally ramified whenever p 6= 2, and they are such that OL

is AL/K -free, according to cases 1 and 2.(b). In particular, we recover Leopoldt’s theorem forcyclic extensions of degree p over Qp.

Moreover, the authors determined explicitly Galois generators for OL, when it is free overUL/K , in terms of t, p and a generator of G. They also deduced conditions of projectivity forinteger rings in cyclic extensions of degree p of number fields ([17], Cor. 1).

3.3.2. Cyclic extensions of degree pn.— Following these results, Bergé, Bertrandias (F.) andFerton attacked the problem for cyclic extensions of degree pn with n ≥ 2, but this situationis more difficult. For p = 3, Bergé described the OK-generators for the associated order in aparticular extension of degree 9 with prescribed ramification jumps [12]. In parallel, Fertonobtained partial results for cyclic extensions of degree p2 ([86], Par. 3).

In 1978 and 1979, Bertrandias (F.) generalized case 2.(b) of Theorem 3.4 to cyclic extensionsof degree pn when p 6 |eK [15, 13];

Proposition 3.5 (Bertrandias, 1979 ([13], Thm. 4)). — Let K be a p-adic local field,with p 6 |eK . Let L/K be a totally ramified cyclic extension of degree pn, with p ≥ 1. Letti denote its ramification jumps, for i = 1, ..., n. We suppose that the ramification is almost

maximally ramified, i.e., that ti ≥ pieK

p−1− 1, for all i. Then OL is free over AL/K if and only

if N ≤ 4, where N is the length of the continued fraction expansion of t1/p.Moreover, the structure of AL/K , and Galois generators, are determined explicitly.



More than twenty years later, case 2.(a) of Theorem 3.4 was partially generalized to certainKummer extensions by Miyata [133], with improvements by Byott in 2008 [36]. For such anextension L/K of degree pn, the ramification jumps all lie in the same residue class modulo pn.Write b for the least non negative residue of these jumps modulo pn. In [36], Byott introduceda set

S(pn) ⊂ c : 1 ≤ c ≤ pn − 1, p 6 |c.

The precise definition of this set is a little elaborate. However, a more easily defined set closelyrelated to S(pn

) is S0(pn), with

S0(pn) =

⋃

m=1,...,n

c : c divides pm − 1.

In particular, S0(pn) ⊂ S(pn

), with equality if n ≤ 2, and in most cases when n ≥ 3. Thecriterion of Miyata, reformulated by Byott, is then that OL is free over AL/K if and only ifb ∈ S(pn

).

Note also that Nigel Byott [38] had dealt with cyclic extensions of degree p2 when char(K) = 0

in 2002, in the language of Hopf algebras.

3.4. Lubin-Tate extensions. — Another extension of local fields for which the structureof the valuation ring over its associated order has been investigated are Lubin-Tate extensions.For background on Lubin-Tate theory, see for example [142].

Let K be a finite extension of Qp, and let q be the cardinality of its residue field. Let π bea uniformizing element of K. Let f(X) be a Lubin-Tate series for K, corresponding to theparameter π, and let F (X,Y ) be the formal group admitting f(X) as an endomorphism. Letm be the maximal ideal of the valuation ring of a fixed algebraic closure of K, and, for alln ≥ 0, set

G(n)= λ ∈ m : f (n)

(λ) = 0,where f (0)

(X) = X and f (n)(X) = f(f (n−1)

(X)) for n ≥ 1. Then, the division fields K(n)

are defined by K(n)= K(G(n)). For every n ≥ 1, the extension K(n)/K is totally ramified

and abelian, of degree qn−1(q − 1), and every element of G(n)\G(n−1) generates the maximal

ideal of OK(n) . Furthermore, K(1)/K is cyclic of order q − 1.

Just as the cyclotomic theory allows an explicit constructive treatment of class field theoryfor Qp, so the extensions Kn provide a constructive treatment of class field theory of totallyramified extensions for K [142]. This is probably the main motivation to consider the fields

K(n) as good candidates for an investigation of integral Galois module structure, in the lightof the theorem of Leopoldt. Interest in these questions arose from the work of Taylor [150],and its subsequent applications to CM fields.

Example 3.6. — Take K = Qp and π = p. Consider f(X) = (1+X)p−1. Then F (X,Y ) =

X + Y + XY = (1 + X)(1 + Y )− 1, so the group operation is the usual multiplication with a

change of variable to shift the identity from 1 to 0. Thus F (n)= Qp(ζpn) for some primitive

pn-th root of unity ζpn . In this sense, Lubin-Tate extensions can be presented as generalizedcyclotomic extensions.


Lara Thomas 175

The integral Galois module structure of extensions of the form K(m+r)/K(r) was consideredin some detail ([40, 41, 42, 55, 56, 149, 150]), and there is now a complete theory for theGalois structure of the top valuation ring over its associated order in such an extension:

Theorem 3.7 (Lubin-Tate extensions). — Let m, r ≥ 1 be two integers. Let Om,r denote

the valuation ring of K(m+r) and write Gm,r := Gal(K(m+r)/K(r)) for m, r ≥ 1. We have:

1. if m ≤ r, then Om,r is free over AK(r)[Gm,r] [150] ;2. if m > r and K = Qp, then OL is free over AK(r)[Gm,r] [55, 56] ;3. if m > r and K 6= Qp, then OL is not free over AK(r)[Gm,r] [42].

Case 1 corresponds to the Kummer case. In cases 1 and 2, an explicit Galois generator is given,as well as the determination of the associated order (in case 2, AK(r)[Gm,r] is determined bya ”transport of structure“ from the cyclotomic case [56]). Note also that if one take π = p incase 2, we have a relative extension of cyclotomic fields, and the result was already provedin [57].

The proof of case 3 uses a study of the ramification jumps of the extension, and it doesn’tprovide any explicit determination of the associated order. However, in [41], Byott gives anexplicit description of AK(r)[Gm,r] when r = 1 and m = 2, under the additional assumptionthat the field K has absolute ramification index eK > q2. Similarly for the works of Bergé,Bertrandias and Ferton for certain cyclic p-extensions, this thus provides an infinite familyof totally ramified extensions over local fields in which the valuation ring is not free over itsassociated order, but for which this order is known explicitly. This is worth noting, sinceorders and freeness are usually established simultaneously.

The investigation of the extensions K(n)/K, with K itself as base field, probably startedwith the work of Byott [39]. In particular, for n = 2, he proved that OL is not free overAK(2)/K , whenever K/Qp is ramified and the residue field of K has cardinality at least 3.

Byott also considered the integral Galois module structure of intermediate fields of K(2)/K.He determined explicitly the associated orders in all cases, and when freeness holds he gave agenerator. Today, we do not know whether this result has been generalized to other extensionsK(n)/K with n ≥ 2.

We close this section with the following remark. As noticed by Byott, there is a strikingsimilarity to R. Miller’s work [129], who considered the corresponding problem for functionfields in characteristic p, where Lubin-Tate formal groups are replaced with Carlitz modules.We shall come back on this setting in subsection 4.3.

3.5. Elementary abelian extensions. — In characteristic 0, few results are known forelementary abelian extensions. In 2007, Miyata gave conditions for the valuation ring not tobe free over its associated order when L/K is a totally ramified abelian Kummer extension ofthe type L = K(α, β), where α and β are suitably normalized elements with αp, βp ∈ K andsuch that K(α)/K and K(β)/K have ramification numbers t in the range 2p < t < peK/(p−1)

([130], Theorem 5).



3.6. Dihedral extensions. — We suppose now that L/Qp is a dihedral extension of degreer2p, with p 6= 2 and p 6 |r. We suppose that the inertia group is cyclic, which happens wheneverr ≥ 3. In [9], Bergé proved that OL is free over its associated order if and only if r < p.In [86], Ferton investigated the case of dihedral extensions of degree 2p. This case is alsoderived from works of Bergé (see e.g [7] and [10], ).

4. Local Galois module structure in equal characteristic

When the local field K is of characteristic p and the extension L/K is of order a power of p,the group algebra K[G] is a local ring whose maximal ideal is its augmentation ideal, i.e.,the left ideal generated by all σ − 1 when σ runs through G (Thm. 19.1 of [121]). Moreover,it has a unique minimal left ideal, which is generated by the trace element

∑

σ∈G σ (see e.g.[153], Chap. 3).

In this setting, the associated order AL/K is a local ring as well, and there exists a canonicalisomorphism between AL/K/mAL/K and the residue field k of K, if m denotes the uniquemaximal ideal of AL/K ([154], Prop. 5.10 and Cor. 5.2).

4.1. Cyclic extensions of formal power series fields. — Over a local field K of char-acteristic p, the Galois module structure of the top valuation ring has been entirely solved forcyclic extensions of degree p. Precisely, let L/K be a totally ramified extension of degree p.We denote by t its unique ramification jump: it is prime to p and we write t = pk + a with1 ≤ a ≤ p−1. Recall that, by Artin-Schreier theory, one can find A ∈ K such that L = K(α)

with αp − α = A and vK(A) = −t.

In 2003, Aiba established the following criterion [1], which was precised by Lettl [118]:

Proposition 4.1 (Aiba, 2003 - Lettl, 2005). — The valuation ring OL is AL/K-free ifand only if a divides p − 1.

This criterion is the same as the one given by Bertrandias (F.) and Ferton [17] for thecorresponding problem in characteristic 0. Note also that it is based on the following propertyderived from ([3], Lemma 1) and ([1], Lemma 2), and which characterises p-extensions incharacteristic p:

Lemma 4.2 (Aiba, 2003). — Suppose L/K is an abelian p-extension with Galois group G.If OL is AL/K-free, then OL = AL/K · α if and only if TrL/K(α) divides TrL/K(β) for anyβ ∈ OL.

Then, in 2005, Proposition 4.1 was made more explicit and reinterpreted in algebraic terms bythe author in her Ph.D. [154]. Precisely, let edim(AL/K) := dimkm/m2 denote the embeddingdimension of AL/K . She proved that OL is AL/K -free if and only if edim(AL/K) ≤ 3 ([154],Thm. 5.2, Prop. 5.23).

Finally, de Smit and the author [144] generalized these criteria by computing efficiently theminimal number of AL/K-module generators of OL from p and t with a continued fraction


Lara Thomas 177

expansion. In particular, this provides an algorithm that given p and a computes d in polyno-mial time, i.e., in time bounded by a polynomial in log(p). The main result is the following.Note that, in this case, the continued fraction expansion of −a/p, instead of +a/p (comparingwith Theorem 3.4), codes the Galois structure of the ring OL; moreover, the criterion is basedon the values of the coefficients of this expansion, instead of its length.

Theorem 4.3 (de Smit & Thomas, 2007). — Let K be a local field of characteristic p,and let L/K be a totally ramified cyclic extension of degree p. Let t be the unique ramificationjump of L/K, and write t = pk + a with 1 ≤ a ≤ p − 1. Let d be the minimal number ofAL/K-generators of OL. Then d = 1 if and only if OL is AL/K-free, and we have;

1. if a = p − 1, then d = 1 and edim(AL/K) = 2;2. if a < p− 1, then edim(AL/K) = 2d + 1 and d =

∑

i odd,i<n ai, where the coefficients ai’sare the unique integers given by

−a

p= a0 +

1

a1 +1

. . .. ..

an−1 +1

an

with a1, ..., an ≥ 1 and an ≥ 2. In particular, OL is AL/K-free if and only if a|(p − 1).

Moreover, in all cases, a set of OK-generators for AL/K and a minimal set of AL/K-generatorsfor OL are given explicitly.

The proof has two basic ingredients: graded rings and balanced sequences. Precisely, theassociated order AL/K is given the structure of a graded ring and OL the structure of agraded module over it. The proof is then based on an explicit combinatorial description ofthe gradings on AL/K and OL in terms of the balanced sequence associated to the fraction a

p .

It is worth noting that this result is probably the first one where the minimal number ofAL/K -generators of OL is given in the case of non freeness, whereas previous works in thisdirection had concentrated only on determining when OL is AL/K -free. Moreover, the explicit

use of combinatorics, through subtle properties of the sequence ⌈ iap ⌉i≥1, is very intriguing

and constitutes another novel approach of the authors.

These contributions also prove that there is no Leopoldt type result over Fp((T )) since onecan derive infinitely many cyclic extensions over Fp((T )) for which the valuation ring is notfree over its associated order. Indeed, for any prime number p, if t > 0 is a positive integersuch that p 6 |t and a 6 |(p − 1) (a is the least non negative residue of t modulo p), then theextension L/Fp((T )) given by L = Fp((T ))(α) with αp −α = T−t is cyclic of order p and suchthat OL is not free over its associated order.

The consideration of cyclic p-extensions of higher degree in positive characteristic is still inprogress.

4.2. Elementary abelian extensions. — In parallel, Byott and Elder have obtainedresults for a family of elementary abelian extensions [44], and obtained a criterion which



agrees with the condition found by Miyata for certain Kummer extensions in characteris-tic 0 [36, 133]).

For these extensions, it is the existence of a particularly well-behaved “Galois scaffold" thatallows the structure of the top valuation ring over its associated order to be determined. Suchstructure was introduced by Elder [77], it corresponds to some variant of a normal basis thatallows for an easy determination of valuation and thus has implications for the questionsof the Galois module structure. Byott and Elder develop the idea to use it to determine anecessary and sufficient condition for OL to be free over its associated order for larger classesof extensions in mixed and equal characteristic.

4.3. Note on the function Field case. — Since function fields can be viewed as theglobalisation of local fields of positive characteristic, it is natural to consider analogue Galoismodule structure questions for extensions of such fields.

Let p be a prime number, and let q be a power of p. Let K = k(T ) be a global function fieldover the finite field k = Fq of characteristic p. One can think of K as being the set of functionsdefined over k of a certain projective nonsingular curve C defined over k. In general, thereis no canonical way to define a ring of integers OK for K. To study integral Galois modulestructure, we fix a finite non-empty set S of places of K, and we let OK = OK,S be the setof all x ∈ K having no pole outside S. If L is a finite extension of K, then we let OL be theintegral closure of OK in L. Let L/K be a finite Galois extension with Galois group G; wecan consider OL as an OK [G]-module and investigate its structure. We can also be interestedin the existence of analogue results between the number field case and the function field case,when Fq(T ) plays the same role as Q.

Recall that we derived Noether’s criterion from Theorem 1.1 and Swan’s Theorem. Now,Swan’s theorem was originally stated for modules over group algebras A[G], when e.g. G isabelian or the ring A has characteristic 0. According to Martinet, in a private communication,this also holds in positive characteristic once the order of the group G is prime to char(A).However, for general extensions of function fields of characteristic p, we can derive a Noether’scriterion from the local case. Indeed, the ring of integers OL is locally free over its associatedorder AL/K if and only if each completion OL,p is free over its associated order in the cor-responding local extension. Then, using the characterisation of tameness of the trace beingsurjective at integral level, and since taking the trace commutes with completion, we deducethat OL is locally free over OK [G] if and only if the extension is at most tamely ramified.

First, if G has order prime to p, tameness is automatic and Chapman gave a version ofthe “Hom-description" of Fröhlich for the class group of locally free OK [G]-modules [58].Furthermore, if G is cyclic, Chapman used class field theory and Kummer theory to calculatethe isomorphism classes explicitly.

Then, Ichimura proved the converse of Noether’s criterion, in the particular case where G isan abelian p-group. Precisely, if L/K is a finite abelian p-extension, then Ichimura proves thatOL is free over OK [G] if and only if L/K is unramified outside S. The method of the proof isquite explicit. Since the problem reduces to the case where G is cyclic, one can suppose L/K


Lara Thomas 179

to be cyclic, in which case it can be described explicitly in terms of Witt vectors. This allowsa free generator of OL as an OK [G]-module to be written down.

In the particular case where OK = Fq[T ], i.e., S = ∞T , Chapman gave a constructive proofof an analogue of the Hilbert-Speiser theorem ([59],Theorem 1). This result is based on ananalogue of the Kronecker-Weber theorem for function fields due to Carlitz and Hayes [107]:all abelian extensions of Fq(T ) can be obtained by adjoining roots of unity, division points ofthe Carlitz module for Fq[T ], and division points of the Carlitz module for Fq[

1

T ]. The resultof Chapman is the following;

Theorem 4.4 (Chapman, 1991). — If L/Fq(T ) is a finite abelian extension which is wildlyramified at no prime of OK = Fq[T ], then OL is a free module of rank 1 over the group ringOK [G]. Moreover, a generator can be constructed explicitly.

Note that normal integral bases can be afforded by Thakur’s analogue of Gauss sums, usinga Carlitz module.

When raising the bottom field K to a finite extension, the situation is more difficult. In [4],Anglès investigated the existence of integral normal bases for intermediate extensions of a tamecyclotomic extension over Fq[T ]. Precisely, let K ⊂ M ⊂ N ⊂ L be a tower of extensionsover the function field K = Fq(T ). Suppose that the field L is obtained by adjoining to K theP -division points of the Carlitz module, for some irreducible polynomial P ∈ Fq[T ] (we saythat L is a cyclotomic function field). Anglès gave several sufficient conditions for N/M tobe without normal integral basis. In particular, if p 6= 2 and if M is the quadratic subfield ofL/K, then N/M has a normal integral basis if and only if the polynomial P ∈ Fq[T ] definingL/K has degree at most 2. This provides some analogue of results of Brinkhuis and Cougnardfor cyclotomic extensions of number fields.

Finally, for wild extensions of function fields, the analogue of Leopoldt’s theorem on Fq(T )

is no longer true for function fields since it is not true for wild extensions of the local fieldFp((T )) (see Subsection 4.1). Aiba obtained another counter example which is more elaborate.Let L/K be a finite Galois extension of function fields with Galois group G, and let OL bethe integral closure of OK in L. One can define the associated order of OL in K[G] asUL/K = λ ∈ K[G] : λOL ⊂ OL, and study the structure of OL as a module over it. IfK = Fq(T ) and OK = Fq[T ], Aiba constructed examples of extensions L/Fq(T ) for which OL

is not free over its associated order using Hayes modules [3]. Moreover, for certain extensionsof cyclotomic function fields L/K, Aiba also investigated an analogue of a result of Chanand Lim [57] on cyclotomic number fields. In particular, in [2], he found the existence ofconditions for OL not to be free over its associated order, contrary to the characteristic 0case.

5. Further comments

5.1. On the maximality of associated orders. — Let K be a local field of residuecharacteristic p and L/K be a finite abelian p-extension over K, with Galois group G. In thissection, we consider the question of whether the associated order of OL in K[G] is a maximal



order, which might help in the investigation of the Galois module structure of OL. Most ofthe required information about maximal orders is contained in [140].

If K has characteristic p, the algebra K[G] has no maximal order. This is due to the factthat the K-agebra is not separable (see e.g. [154], Prop. 5.9).

When K has characteristic 0, and since G is abelian, the algebra K[G] contains a unique max-imal OK -order M; it is the integral closure of OK in K[G]([140], remark after Theorem 8.6).The Wedderburn decomposition of K[G] into simple K-algebras is K[G] = ⊕K[G]eχ, wherethe eχ are primitive idempotents indexed by a set of representatives for the classes of charac-ters of G which are conjugate under the action of the absolute Galois group of K. This yieldsthe decomposition M = ⊕Meχ and each Meχ is the maximal order of K[G]eχ. Therefore, M

is the OK -module generated by the group ring OK [G] and idempotents of K[G].

Moreover, each summand K[G]eχ is isomorphic to a cyclotomic field Kχ, with Kχ = K(ζm)

if m is the order of χ and ζm a primitive mth root of unity. Therefore, M ≃ ∏

χ∈ΓKMχ,

where Mχ is the valuation ring of Kχ. In particular, the components Mχ are principal idealdomains, so that, if the associated order AL/K equals M, then OL is free over it. The equalityAL/K = M thus provides a condition of freeness.

5.1.1. Criteria for AL/K = M.— Since L/K is abelian, the associated order AL/K can onlyequal the maximal order M if the extension is cyclic. As noticed by Byott, this can be shownusing Frohlich’s notion of “factorisability" [91]. See also Corollary 1.8 of [33]. One can alsoprove it in a more restrictive context, using some other criteria to determine whether OL isOK [G]-indecomposable.

Indeed, one necessary condition for AL/K to coincide with M is that it must contain somenontrivial idempotents. On the other hand, the ring OL is indecomposable as an OK [G]-module if it cannot be written as a direct sum of two non-zero OK [G]-submodules. Thisamounts to the fact that the ring of OK [G]-endomorphisms of OL contains no nontrivialidempotents. But since G is supposed to be abelian, this ring is precisely the associated orderAL/K . Hence, if AL/K = M, then OL is OK [G]-decomposable. Vostokov and Miyata haveinvestigated criteria for OL to be OK [G]-indecomposable [131, 159].

For example, if L/K is an abelian p-extension, and if the order of the first ramification groupG1 does not divide the different, then OL is OK [G]-indecomposable, and so AL/K 6= M. Thiscomes from the fact that if the order of G1 divides the different, then the associated ordercontains the central idempotent attached to the trace element for G1. In particular, if L/Khas ramification index pn, and if its biggest ramification jump tm satisfies tm−⌊ tm

p ⌋ ≤ pn−1eK ,

then OL is indecomposable as an OK [G]-module ([160], Theorem 4). For p ≥ 3, Byott provedthat if this condition does not hold, then L/K is cyclic ([42], Prop. 3.7).

5.1.2. Link with almost maximal ramification.— Bertrandias investigated the OK [G]-decomposability of OL when L/K is a cyclic extension of degree p [13]. In particular, sheproved that OL is OK [G]-decomposable if and only if the idempotent 1

pTrL/K belongs to


Lara Thomas 181

AL/K , and, when this is true, she described the decomposition of OL into indecomposableOK [G]-submodules in terms of the value of the ramification jump t modulo p ([13], Thm. 2and Thm. 3). Moreover, she proved that the condition is actually equivalent to the doubleinequality p

p−1eK − 1 ≤ t ≤ p

p−1eK : we say that the extension is almost maximally ramified.

The notion of almost maximal ramification is due to Jacobinski [111]. An extension L/K withGalois group G is said to be almost maximally ramified if all idempotents eH =

1

|H|∑

σ∈H σ

belong to the associated order UL/K , when H run over all subgroups of G included betweentwo consecutive ramification groups of the extension.

When L/K is a totally ramified cyclic extension of degree pn, this is equivalent to the followingconditions (see e.g. [7], Cor. of Prop. 3, and [13], Prop. 1). For each integer i, 0 ≤ i ≤ n,write Hi for the subgroup of G of order pi, and put ei = eHi

. Clearly, the groups H ′is are the

ramification groups of the extension. Moreover, each ei is an idempotent of K[G], and piei

coincides with the trace of the extension L/LHn−i . We also write t1 < t2 < ... < tn for the nramification jumps of L/K.

Proposition 5.1. — The extension L/K is almost maximally ramified if and only if it sat-isfies one of the following equivalent conditions:

1. eH ∈ AL/K for all subgroups H ⊂ G included between two consecutive ramificationgroups;

2. ei ∈ AL/K for all i ∈ 1, ..., n;

3.pieK

p − 1− 1 ≤ ti ≤

pieK

p − 1for all i = 1, 2, ..., n;

4. ti =pie − a

p − 1for all i, where a is the least non-negative residue modulo p of t1.

Note that, in this context, if we set e′0

= en and e′i = en−i−en−i+1, then the e′i are orthogonalidempotents whose sum is 1, and K[G] = ⊕0≤i≤nK[G]e′i.

Suppose now that K is absolutely unramified (eK = 1), and that the extension L/K iscyclic and totally ramified, of order rpn with p 6 |r. In 1978, Bergé obtained an explicitdescription of the maximal order M of K[G]: this is the OK -module generated by OK [G] andthe idempotents ei ([7], Proposition 5). Moreover, the equality AL/K = M holds if and onlyif the extension is almost maximally ramified ([7], Corollary 3 of Theorem 1).

When K is not absolutely unramified, almost maximal ramification is not sufficient for AL/K

to equal the maximal order M. If L/K is cyclic of order pn, this is due to the fact that, in thissetting, the idempotents ei defined above are not sufficient to generate the maximal order M

of K[G]. As a consequence of Theorem 3.4, Bertrandias (F. and J.-P.) and Ferton obtainedthe following criterion for a cyclic extension of degree p ([16], Theorem 2);

Proposition 5.2 (Bertrandias & Bertrandias & Ferton, 1972)Let K be a local field of mixed characteristic (0, p). Let L/K be a totally ramified extensionof degree p, with Galois group G. Let t be its ramification jump; let a be its least non-negativeresidue modulo p. Then AL/K coincides with the maximal order M in K[G] if and only if the



extension is almost maximally ramified and a satisfies one of the following conditions :

a = 0 ou a|(p − 1) ou a|p − 2 ou a|2p − 1.

5.1.3. Number field case.— These considerations still hold for extensions of number fields.As an illustration, let us mention the following criterion due to Byott and Lettl, where theassumption on linear disjointness is crucial;

Proposition 5.3 (Byott-Lettl, 1996 [48]). — Let L/K be a cyclic and totally ramifiedextension of number fields. Suppose it is linearly disjoint to Q(ζm)/Q, where m denotes theconductor of K. Then AL/K is the maximal order of K[G].

Is the associated order a local ring ? The previous consideration also lead us to thequestion whether the associated order is a local ring. For an abelian p-extension L/K, thisalways holds when char(K) = p (see e.g. [154], Prop. 5.10). In zero characteristic, this isrelated to the existence of nontrivial idempotents as well, and one can prove that if OL isOK [G]-indecomposable, then AL/K is a local ring. According to ([42], Prop. 3.7),the conditionthat AL/K is a local ring is thus very weak.

5.2. Hopf structures in Galois module theory. — The use of Hopf theory is anotherone of the most innovative approaches to the wild situation in recent years. This idea, initiatedby Fröhlich, was developed by Taylor and Childs in the mid 1980’s to solve Galois modulequestions for extensions of local fields of unequal characteristic. Hopf orders had first beenconsidered by people studying group schemes (Tate, Oort, Raynaud, Larson). Most of thecontributions towards the connection between Hopf orders and Galois module structure aredue to Byott, Childs, Greither, Pareigis and Taylor. For more details about this theory, werefer the reader to [60]. Among other investigations of the relation between Hopf orders andGalois module structure, one should cite, e.g., the recent contributions of Agboola, Bley &

Boltje, Miyata and Truman.

If R is a commutative ring, a Hopf R-algebra is an R-bialgebra with antipode. It is said tobe finite if it is finitely generated and projective as an R-module. If L/K is a finite Galoisextension of number fields or of local fields of mixed characteristic (0, p), the group ring K[G]

provides the easiest example of a Hopf algebra. We then call a Hopf order any sub Hopfalgebra of K[G] which is also an OK -order in K[G].

In 1985, Taylor considered local extensions constructed using division points of Lubin-Tateformal groups [150]: using the formal group structure, he gave an explicit description ofthe associated order, and showed that the top valuation ring was free over it. Then, in1987, he generalised and reinterpreted this in terms of Kummer theory with respect to theformal group [149]. In particular, he made it explicit that the construction works becausethe associated order is a Hopf order.

More generally, Childs and Moss proved the following criterion [61, 62];


Lara Thomas 183

Theorem 5.4 (Childs - Moss, 1994). — Let L/K be a finite Galois extension of p-adicfields or of number fields, with Galois group G. If the associated order AL/K of OL is a Hopforder in K[G], then OL is AL/K-free of rank one.

The converse is false. Indeed, there are many wildly ramified Galois extensions L/K whosevaluation rings are free over their associated order AL/K but AL/K is not a Hopf order (seeTheorem 5.1 of [61] and its corollaries).

In parallel, Greither and Pareigis [106] proved that L is also an H-Hopf Galois extension ofK for various K-Hopf algebras H (the terminology means that L is an H-module algebra);one of them is the group algebra K[G]. If the field extension is one of p-adic fields, one candefine the associated order in each Hopf-Galois structure, prove that Theorem 5.4 still holds,and compare freeness results between them. For example, Childs did this for cyclic extensionsof degree p2; Byott then did the same for elementary abelian extensions of degree p2 [38].It can happen that the valuation ring is free over its associated order with respect to somenon-classical Hopf-Galois structure, whereas it is not free in the classical case.

In 1992, Greither essentially classified most of the Hopf orders in the group algebra K[Z/p2Z]

for a p-adic field K, and found which of them occur as associated orders of valuation rings. In-dependently to this, Byott found almost all the Hopf orders in both K[Z/p2

Z] and K[Z/pZ×Z/pZ], including some excluded by Greither’s hypothesis. Furthermore, in 2004, Byott de-termined all Hopf-Galois structures on Galois extensions of fields of degree pq, where p, q aredistinct primes such that q ≡ 1(mod p) [37].

Moreover, for an abelian p-extension of p-adic fields, Bondarko proved that if the top valuationring is free over its associated order, then the associated order must be a Hopf order andthe extension can be produced from a one-dimensional formal group [25]. In [42], Byottinvestigates the ramification numbers of abelian p-extensions L/K for which the associatedorder of OL is a Hopf order in K[G].

Finally, Hopf structures have been investigated in certain p-extensions of degree pn by Miy-ata [131] and Byott (see e.g. [36, 38, 42]). One recent result is the following;

Theorem 5.5 (Byott, 2008). — If K is a p-adic field and L/K a Kummer extension ofdegree pn of the form L = K(α) with αpn ∈ K and vK(α − 1) > 0, vK(α − 1) coprime to p.Then AL/K is a Hopf order if and only if a = pn − 1, where a denotes the least non-negativeresidue of the first ramification jump modulo pn.Moreover, if pn/2 < a < pn − 1, then OL is not free over its associated order.

5.3. Valuation criteria for normal basis generators. — Investigating the algebraicstructure of the top valuation ring over its associated order in abelian elementary p-extensions,Byott and Elder [45] raised the question of the existence of a valuation criterion for normalbasis generators of some extension L/K of local fields, i.e., of the existence of an integer vsuch that every element x ∈ L with valuation v generates a normal basis for L/K.

If char(K) = p, Elder and the author proved that every totally ramified p-extension of Ksatisfies such a valuation criterion, for a prescribed value of v;



Theorem 5.6 (Thomas 2008, Elder 2010). — Let K be a local field of characteristic pand let L/K be a totally ramified p-extension. Write d for the valuation of the different of theextension. Then each element x ∈ L with valuation congruent to −d − 1 modulo [L : K] is anormal basis generator for L/K.

Nigel Byott has reinterpreted this result in terms of Hopf-Galois structures [35].

Florence, de Smit and the author have just solved the question entirely in all characteris-tics [89]. Let L/K be a finite Galois extension of local fields. To simplify, say that V C(L/K)

holds if L/K satisfies a valuation criterion for normal basis generators. They first proved thatV C(L/K) holds if and only if the tamely ramified part of the extension L/K is trivial andevery non-zero K[G]-submodule of L contains a unit. Moreover, the integer v can take onevalue modulo [L : K] only, namely −dL/K − 1, where dL/K is the valuation of the different ofL/K. When K has positive characteristic, they recover the result of Elder and the author.When char(K) = 0, they identify all abelian extensions L/K for which V C(L/K) is true,using algebraic arguments. These extensions are determined by the behaviour of their cyclicKummer subextensions.

Therefore, one can then address the question of the existence of a valuation criterion forGalois generators of valuation rings over their associated orders, when they are free. Severalresults in this direction have now been obtained, and the existence of such a criterion canalso provide arguments to determine non-freeness results. Note also that the UL/K -generatorsof OL founded for cyclic extensions L/K of degree p in both characteristic cases 0 and p(according to the results of Subsection 3.3 and 4.1) satisfy the valuation criterion of Florence,de Smit and the author.

5.4. Galois module structure of ambiguous ideals. — Let L/K be a finite Galoisextension of number fields or local fields, with Galois group G. Instead of investigatingthe Galois module structure of the integer ring OL, one can consider ambiguous ideals, i.e.,fractional ideals of L that are stable under the action of G. In particular, if a is such an ideal,one may define its associated order in K[G] by:

AL/K(a) = α ∈ K[G] : αa ⊂ a.Similarly to the ring OL, a is a module over AL/K(a) and one can address the question ofwhether it is free. In what follows, we give a brief account of the investigation that has beendone on this subject when K is a p-adic field. In this case, every fractional ideal of L isambiguous.

If the extension L/K is tame, then AL/K(a) = OK [G] ([11], Thm. 1). Now, whereas therelation AL/K(OL) = OK [G] characterizes tamely ramified extensions (see Subsection 1.3),this is false if we replace OL with another ambiguous ideal. Indeed, in ([11], Par. I.3), Bergégives the following counter-example. If L = Q2(i) with i2 = −1, and if a = (1 + i)OL, thenthe extension L/Q2 is wildly ramified whereas AL/K(a) = Z2[G].

Moreover, when K is a local field, Ullom proved that if he extension L/K is tame, then everyambiguous ideal of L is a free OK [G]-module [155]. An explicit set of generators for eachideal can be derived from the construction of normal integral bases by Kawamoto [113].


Lara Thomas 185

If L/K is wild, the situation is very different, and only special cases are known. First, Ullomshowed that the freeness of any ambiguous ideal a of L over OK [G] is a strong restriction onboth the ramification of L/K and the L-valuation of a ([156], Theorem 2.1). He also provedthat if an ambiguous ideal in L is free over OK [G], then L/K must be weakly ramified, i.e.,its second ramification group is trivial [156].

The Galois module structure of ambiguous ideals over their associated orders has been inves-tigated for cyclic extensions. Suppose first that L/K is an extension of degree p. Write t forthe unique ramification jump of L/K, and PL for the maximal ideal of OL. In [87], Fertoncharacterized the ideals Pr

L of OL which are free over their associated order, in terms of thevalues of t and r. Her results generalized Theorem 3.4. In particular, answering a question ofJacobinski, she proved that every ideal Pr

L such that r ≡ t mod(p) is free over its associated

order. Note that two ideals PrL and Pr′

L have the same associated order if r ≡ r′(mod p).Later, in [33], and under the assumption that eK = 1, Burns proved that if L/K is a cyclicextension of order pnr with p 6 |n, then OL is free over its associated order if and only if thereexists a fractional ideal of L which is free over its associated order, and this happens if andonly if n = 1, or n = 2 and r < p2, or n > 2 and r < p(p − 1) (Thm. 3 of [33]). See alsoLemma 1.1 of [32].

Finally, in [32], Burns gave an almost complete answer to the question when L/K is a finitetotally ramified abelian extension of p-adic fields, for an odd prime p, extending [7], [155]and [33]. A key tool is the notion of factorisability introduced by Fröhlich [91], as well asthe factorisable quotient function, introduced by Burns in 1991 [33] and which allows thequestion of whether a is free over its associated order to be answered by computing moduleindices. Note that an appendix by W. Bley describes an algorithm to determine whether anambiguous ideal in the ring of algebraic integers in a number field is locally or globally freeover its associated order.

In particular, denoting by G the Galois group of L/K, Burns investigated the structure offractional ideals of L, a, over their associated order in Qp[G], i.e., over AQp[G](a) := λ ∈Qp[G] : λa ⊂ a. When K is ramified over Qp, there are only two types of extensions forwhich there is an ideal free over its associated order in Qp[G]: the weakly ramified extensions,and the cyclic extensions that are almost maximally ramified. When K/Qp is unramified, theresult is much more complicated, and Burns investigated necessary conditions on the existenceof ideals free over their associated orders in Qp[G] in terms of the ramification jumps.

We now develop two examples of the Galois module structure of ambiguous ideals.

5.4.1. Galois module structure of the inverse different.— Let L/K be a totally ramifiedabelian extension of degree pn with Galois group G. We denote by b1 ≤ b2 ≤ ... ≤ bn, withb1 ≥ 1 and possibly bi = bi+1 for some i, the ramification jumps of L/K, and let eK = vK(p)

be the absolute ramification index of K. Let D−1

L/K be the inverse different of L/K, defined

by:

D−1

L/K = x ∈ L : TrL/K(xOL) ⊂ OL.In ([42], Theorem 3.10), Byott proved the following:



Theorem 5.7 (Byott,1997). — If bn − ⌈ bn

p ⌉ 6= pn−1eK , and if bi 6≡ −1 (mod pn) for some

i, then D−1

L/K is not free over its associated order.

5.4.2. Square root of the inverse different.— In the global case, the study of the Z[G]-structureof other G-stable ideals of OL began in a special case in [85], where Erez studied the squareroot of the inverse difference of some extensions of number fields answering a question ofConner and Perlis. Let L/K denote an odd degree Galois extension of number fields. ByHilbert’s formula for the valuation of the different DL/K of L/K, there exists a fractionalideal AL/K of the ring of integers OK of K such that:

A2

L/K = D−1

L/K.

This ideal is known as the square root of the inverse different. It is an ambiguous ideal. Asan analogue of Noether’s criterion, Erez showed AL/K to be locally free if and only if L/K isweakly ramified, i.e., if the second ramification group of any prime ideal p of OL is trivial [83].Moreover, in [82], Erez and Taylor proved that when L/K is at most tamely ramified, thenAL/K is always free over Z[G]. For a precise account on the Galois module structure of thesquare root of the inverse different until 1991, see [84].

The question of whether AL/K is free as a Z[G]-module when L/K is wildly but weaklyramified is still open. Pickett and Vinatier [138] have recently proved that AL/K is a freeZ[G]-module when L/K is an odd degree weakly ramified Galois extension of number fieldssuch that, for any wildly ramified prime p of OL, the decomposition group is abelian, theramification group is cyclic and the localised extension F℘/Qp is unramified, where ℘ =

p ∩ F and pZ = p ∩ Q. This result generalises Theorem 1.2 of [158], which is the naturalanalogue in the absolute case K = Q. The proof of this result uses Lubin-Tate theory and theexplicit descriptions by Pickett of self-dual normal basis generators for cyclic weakly ramifiedextensions of an unramified extension of Qp [137]. These generators are constructed with thehelp of Lubin-Tate theory and Dwork’s p-adic exponential power series.

It should be interesting to pursue this investigation and determine the Galois structure ofAL/K as a module over its associated order, when higher ramification is permitted. In thisdirection, one result is due to Burns whose proof is given in Appendix A of [84]:

Proposition 5.8 (Burns, 1991). — If L/Q is a finite abelian extension such that thesquare root of the inverse different AL/K exists, then AL/K is locally free over its associatedorder.

5.4.3. Existence of valuation criteria.— Finally, one can also consider the question of theexistence of valuation criterion for Galois generators of ambiguous ideals when they are freeover their associated order, and partial answers are already obtained. For example, if L/K isan abelian and weakly ramified extension of p-adic fields with Galois group G of odd order,the square root of the inverse different exists and is a free module over OK [G]. Generalizinga result of Byott, Vinatier proved that every element β ∈ L with valuation vL(β) = 1 − eK

generates AL/K over OK [G] ([157], Cor. 2.5).


Lara Thomas 187

5.5. On the sufficiency of the ramification invariants. — We close this paper by thefollowing remark. Let L/K be a finite extension of local fields of residue characteristic p.Results about the Galois module structure of ideals of L (e.g. [42], [32], [87]) indicate thepossibility of nice general patterns governing some relationship with the ramification invariantsof the extension, precisely its ramification jumps as well as the absolute ramification index ofK. Moreover, when L/K is of degree p, the single ramification break determines whether ornot OL is free over its associated order in both mixed and equal characteristic cases. It seemsthat the ramification invariants actually control the question of freeness to a considerableextent, and Byott and Elder have noticed that they are sufficient to determine the structureof ideals when their number is maximal [46].

Nevertheless, we do not expect that such invariants will give the required information for allextensions. For instance, in ([12], Chap. 4), Bergé constructed two wild extensions over afixed 3-adic field K with the same ramification jumps but such that the associated orders oftheir top valuation rings are different. More recently, this insufficiency was also observed forbiquadratic extensions of 2-adic fields with one ramification jump [47]. These observationshave led Byott and Elder to introduce a refined ramification filtration for some totally ramifiedelementary abelian p-extensions, i.e., one with more ramification jumps [44, 46], and theninvestigate whether they are sufficient. It will be interesting to address the question of whichinvariants determine the Galois module structure of ideals for more general extensions.

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May 27, 2010

Lara Thomas, Ecole Polytechnique Fédérale de Lausanne - Chaire de Structures Algébriques et Géométriques- FSB - IMB - Station 8, Bureau 594 CH - 1015 Lausanne • E-mail : [email protected]


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2010

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20

10M. J. Bertin Mesure de Mahler et série L d’une surface K 3 singulière

F. Brunault Régulateurs p-adiques explicites pour le K2 des courbes elliptiques

H. Cohen Some formulas of Ramanujan involving Bessel functions

C. Delaunay et C. Wuthrich Some remarks on self-points on elliptic curves

G. Gras Analysis of the classical cyclotomic approach to Fermat’s last Theorem

F. Hajir Asymptotically good families

D. Solomon Equivariant L-functions at s = 0 and s = 1

L. Thomas On the Galois module structure of extensions of local fields

Revue du Laboratoire de mathématiques de Besançon (CNRS UMR 6623)


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