stark-heegner points and the cohomology of ...people.ucalgary.ca/~mgreenbe/g2009.pdfh is that of a...

STARK-HEEGNER POINTS AND THECOHOMOLOGY OF QUATERNIONIC SHIMURAVARIETIES

MATTHEW GREENBERG

AbstractLet F be a totally real field of narrow class number one, and let E/F be a modular,semistable elliptic curve of conductor N �= (1). Let K/F be a non-CM quadraticextension with (Disc K, N) = 1 such that the sign in the functional equation ofL(E/K, s) is −1. Suppose further that there is a prime p|N that is inert in K . Wedescribe a p-adic construction of points on E which we conjecture to be rationalover ring class fields of K/F and satisfy a Shimura reciprocity law. These points areexpected to behave like classical Heegner points and can be viewed as new instancesof the Stark-Heegner point construction of [5]. The key idea in our construction is areinterpretation of Darmon’s theory of modular symbols and mixed period integralsin terms of group cohomology.

1. Introduction

1.1. Heegner points and signs in functional equationsLet F be a totally real field of narrow class number one, and let E/F be an ellipticcurve of conductor N ⊂ OF such that ordp N = 1 for some prime ideal p of OF .Assume, moreover, that E is modular, that is, that there exists a Hilbert modular formf ∈ S2(N) such that

L(E/F, s) = L(f, s).

By the Jacquet-Langlands correspondence, there is a Shimura curve X/F and amodular parametrization

� : Jac X −→ E

defined over F , where Jac X is the Jacobian variety of X. The curve X is a modulispace for algebro-geometric objects related to principally polarized abelian varieties

DUKE MATHEMATICAL JOURNALVol. 147, No. 3, c© 2009 DOI 10.1215/00127094-2009-017Received 7 January 2008. Revision received 20 October 2008.2000 Mathematics Subject Classification. Primary 14G05; Secondary 14G35.

541

542 MATTHEW GREENBERG

with quaternionic multiplication and level structure. When F = Q, the Shimura curveX is a moduli space for such varieties themselves. When F �= Q, the identificationof X as a moduli space is indirect and is actually defined relative to the choice of aquadratic extension of F . The resulting curve descends to F itself, however. For someuseful remarks regarding the nature of this moduli problem, see [9, §1.1]; for a preciseformulation, see [20, §1.1.2].

Let K/F be a CM extension with [K : F ] = 2, and let O ⊂ K be an OF -ordersuch that

(DiscO/OF , N) = 1.

There is a notion of a point of X(C) having CM by O (for the definition, see [20,§2.1]). It follows from the theory of complex multiplication that

�(Div0 CMO) ⊂ E(HO),

where HO is the ring class field of K associated to order O. Points of E(HO) arisingin this manner are called Heegner points. These points are of significant interest. Inparticular, the proof of instances of the conjecture of Birch and Swinnerton-Dyer forelliptic curves over F of analytic rank at most one, due to Gross-Zagier, Kolyvagin,and Zhang, depends essentially on their properties.

The existence of the Heegner point construction is compatible with the conjectureof Birch and Swinnerton-Dyer. If the set CMO is nonempty, then the sign (E/K) inthe functional equation of the completed L-function �(E/K, s) is −1:

�(E/K, s) = −�(E/K, 2 − s).

By the product formula,

L(E/HO, s) =∏

χ :PicO→C×L(E/K, χ, s).

Each of the completed twisted L-functions �(E/K, χ, s) satisfies a functional equa-tion in its own right of the form

�(E/K, χ, s) = sign(E/K, χ)�(E/K, χ, 2 − s).

Since E is defined over F and since χ , viewed as a character of A×K , is trivial on A×

F ,we have

L(E/K, χ, s) = L(E/K, χ, s).

STARK-HEEGNER POINTS 543

In addition, the intervening signs are all independent of χ :

sign(E/K, χ) = sign(E/K) = −1.

Therefore, each of the twisted L-functions L(E/K, χ, s) must vanish at s = 1,implying

ords=1

L(E/HO, s) ≥ [HO : K].

Thus, the conjecture of Birch and Swinnerton-Dyer leads one to believe that

rank E(HO) ≥ [HO : K]

or, in other words, that E possesses “many” HO-rational points. Thus, a systematicconstruction of global points on E rational over ring class fields of K is compatiblewith the conjecture of Birch and Swinnerton-Dyer.

1.2. Stark-Heegner pointsThe above logic suggests using the Birch and Swinnerton-Dyer conjecture and signsin functional equations to identify other scenarios for which it would be reasonable toexpect such systematic constructions of global points on E. Let K be any quadraticextension of F with (Disc K/F, N) = 1, and set

S(N, K) = {l|N prime : l is inert in K/F }. (1)

It can be shown that

sign(E/K) = (−1)#S(N,K)+r1(K)+r2(K), (2)

where r1(K) and r2(K) are the numbers of real and complex places of K , respectively.In our situation, r1(K) is always even, so sign(E/K) = −1 if and only if

#S(N, K) + r2(K) is odd. (3)

Therefore, by the discussion of §1.1, the existence of a Heegner point–like constructionis compatible with the conjecture of Birch and Swinnerton-Dyer whenever (3) issatisfied, not just in the case when r2(K) = [F : Q]; that is, K a is CM extension ofF .

The main result of this article is a conjectural realization of this heuristic fornon-CM K/F as above under the following additional condition.

Assumption 1S(N, K) is nonempty.


Following Darmon [5], we call the points we construct Stark-Heegner points becausethese points should play a role in the arithmetic theory of elliptic curves analogous tothat played by Stark units in the theory of unit groups in rings of algebraic integers.

1.3. Motivation: Darmon’s constructionLet E/Q be an elliptic curve of squarefree conductor N . Let f ∈ S2(N) be thenormalized cusp form satisfying L(f, s) = L(E/Q, s), and let ωf = 2πif (z) dz

be the associated holomorphic 1-form on the upper half-plane H. Classically, oneconstructs Heegner points on E as follows. One integrates ωf over an oriented pathwhose divisor is supported on imaginary quadratic irrationalities in H. One thenapplies the Weierstrass uniformization to the resulting quantity, thereby obtaining apoint on E. A fundamental obstruction to the existence of a naive analogue of thisprocess with imaginary quadratic irrationalities replaced by real quadratic ones is thatthere are no real quadratic irrationalities lying in H. The p-adic upper half-planeHp = Cp − Qp, however, contains many such. If K is a real quadratic field in whichp is inert, then K ∩ Hp is nonempty. Assume, in addition, that p divides N . Thenthere is a Tate uniformization � : C×

p → E(Cp) available to play the role of theWeierstrass uniformization. Darmon’s idea in [5] is to construct points in E(Kab) byreplacing the archimedean analysis present in the classical Heegner point constructionwith p-adic analysis. To properly motivate the constructions of this article, we mustreview those of [5] in some detail, albeit in a simplified form and paraphrased in orderto emphasize the analogy with our techniques.

The key step in the point construction of [5] is the identification of a suitablep-adic version of ωf .∗ The p-adic analogue of the notion of holomorphic 1-form onH is that of a rigid-analytic 1-form on Hp, the space of which we denote by �1

Hp.

Although there seems to be no natural method for viewing ωf as an element of �1Hp

, itis shown in [5] that one may naturally associate to f an �1

Hp-valued modular symbol

(defined below). In order to make this association precise, we need to identify thespace �1

Hpwith an appropriate space of measures.

There is a natural Poisson integral map from the space Meas(P1(Qp), Cp) ofCp-valued measures on P1(Qp) to �1

Hpdefined by

µ �→ ϕ(z) dz, ϕ(z) =∫

P1(Qp)

dµ(t)

t − z.

∗Of course, ωf has an obvious p-adic incarnation coming from algebraic geometry: ωf may be viewed as a1-form on the algebraic curve X0(N)/Q and thus, by base change, as a 1-form on X0(N)/Qp

. The key word in theabove sentence, however, is suitable, that is, suitable for facilitating a point construction. A basic attribute of theconstruction both of [5] and of this article is that it uses almost no algebraic geometry.


By a theorem of Teitelbaum [17, Corollary 11], this map is an isomorphism. It turns outto be more convenient to work with measures and measure-valued modular symbolsthan with differentials and differential-valued modular symbols.

Let V be an abelian group. A V -valued modular symbol is simply a grouphomomorphism from Div0 P1(Q) to V . If G ⊂ SL2(Q) and V is a G-module,we define the notion of G-equivariant modular symbol in the obvious way. WriteN = pM , let R ⊂ M2(Q) be the standard Eichler Z[1/p]-order of level M ,

R ={(

a b

c d

)∈ M2

(Z

[ 1

p

]): M divides c

},

and let

� = {γ ∈ R× : det γ = 1}.

There is a real period �+ such that for every D ∈ Div0 P1(Q),

1

�+

∫D

eωf ∈ Q. (4)

The period �+ is determined by (4) up to multiplication by elements of Q×. Notethat the integral (4) depends only on the image of D in the group of coinvariants(Div0 P1(Q))�0(N), a finitely generated abelian group. Therefore, the denominatorsof the integrals (4) are bounded. Let Meas0(P1(Qp), Q) be the group of Q-valuedmeasures on P1(Qp) with total measure zero.

PROPOSITION 1There exists a unique modular symbol

′E ∈ Hom�

(Div0 P1(Q), Meas(P1(Qp), Q)

)such that for all D ∈ Div0 P1(Q),

′E(D)(Zp) = 1

�+

∫D

eωf .

We now require a p-adic analogue of the operation of integrating ωf over divisorssupported on imaginary quadratic irrationalities. Formally, there is a natural integrationpairing∫

:(

Div0 P1(Qp)⊗Div0 Hp

)�×Hom�

(Div0 P1(Q), Meas0(P1(Qp), Q)

) −→ Cp


defined as follows. For D ∈ Div0 Hp, let gD denote a rational function on P1(Cp)with divisor D. Let qE be the Tate period of E, and set∫

〈D1 ⊗ D2, ϕ〉 =∫

P1(Qp)logqE

gD2 (t) dϕ(D1)(t).

Assume the following Heegner hypothesis: All primes divisors of N/p are splitin K . Let τ ∈ K ∩ Hp. Then the stabilizer �τ is an abelian group of rank one; let γτ

be a generator. Fix a cusp x ∈ P1(Q), and let

ϒ ′τ = ({x} − {γτx}) ⊗ {τ } ∈ Div0 P1(Q) ⊗ DivHp.

The short exact sequence

0 −→ Div0 Hp −→ DivHp −→ Z −→ 0

induces the following exact sequence in �-homology:

H1

(�, Div0 P1(Q)

) ∂−→ (Div0 P1(Q) ⊗ Div0 Hp

)�

−→ (Div0 P1(Q) ⊗ DivHp

)�

−→ (Div0 P1(Q)

)�.

One can show that (Div0 P1(Q))� is finite; let e be its exponent. Therefore, eϒ ′τ

lifts to an element �′τ of (Div0 P1(Q) ⊗ Div0 Hp)� , well defined up to elements of

∂H1(�, Div0 P1(Q)). Define

J ′τ =

∫〈�′

τ , ′E〉 ∈ Cp.

That J ′τ does not depend on our choice of lift �′

τ follows from the following theoremof Darmon.

THEOREM 2 ([5, Theorem 4])For all η ∈ H1(�, Div0 P1(Q)), we have 〈∂η, ′

E〉 = 0.

To the point τ ∈ K ∩ Hp we may attach an order Oτ ⊂ K , its ring of multipliers,which has an associated narrow ring class field H+

Oτ. The prime ideal pOK splits

completely in H+O . Therefore, choosing a prime of H+

O above p, we may view H+O as

a subfield of Kp. Let qE be the Tate period of E, and write logqEfor the unique branch

of the logarithm K×p → Kp such that logqE

qE = 0.

CONJECTURE 1 ([5, Conjecture 7])There is a point Pτ ∈ E(H+

O ) and a rational number t such that

J ′τ = t logqE

Pτ .


1.4. This articleIn order to generalize Darmon’s construction to elliptic curves over totally real fieldsF (and to elliptic curves over Q and real quadratic fields K with sign(E/K) = −1and #S(N, K) ≥ 2), we reinterpret the above theory in terms of group cohomology.Consider the exact sequence

0 −→ Div0 P1(Q) −→ Div P1(Q) −→ Z −→ 0.

Applying the functor Hom(−, Meas0(P1(Qp), Q)) (resp., Div0 Hp ⊗ −) and takingthe corresponding long exact sequence in �-cohomology (resp., �-homology), weobtain the coboundary (resp., boundary) map

δ : Hom�

(Div0 P1(Q), Meas0(P1(Qp), Q)

) −→ H 1(�, Meas0(P1(Qp), Q)

),

(resp., ∂ : H1(�, Div0 Hp) −→ (Div0 P1(Q) ⊗ Div0 Hp)�).

We construct the following objects:(1) a cohomology class

E ∈ H 1(�, Meas0(P1(Q), Q)

)such that δ(′

E) = E (see §8),(2) a homology class

�τ ∈ H1(�, Div0 Hp)

such that ∂�τ = �′τ (see §7),

(3) an integration pairing∫: H1(�, Div0 Hp) × H 1

(�, Meas0(P1(Qp), Q

) −→ Cp

such that

Jτ :=∫

〈�τ, E〉 =∫

〈�′τ ,

′E〉 = J ′

τ

(see §9).Key to our generalizations is the fact that these objects can be constructed purely

cohomologically, that is, independently of the modular symbol–based manipulationsof §1.3. This cohomological rephrasing of the above theory allows us to constructStark-Heegner points in the general situation described in §1.2 by giving us the flexi-bility to replace the groups � and �0(N) by unit groups of various orders in quaternionalgebras over F and replace H 1 and H1 by higher cohomology and homology groupsHn and Hn.


2. Quaternion algebras, orders, and unit groupsFor the remainder of the article, we let F be a totally real field of degree g over Q andlet E/F be an elliptic curve of squarefree conductor N ⊂ OF . Let K/F be a non-CMquadratic extension with (Disc K/F, N) = 1 such that sign(E/K) = −1. Assumefurther that Assumption 1 is satisfied. Relabeling if necessary, assume that the infiniteplaces v1, . . . , vn are split in K while vn+1, . . . , vg are inert. Note that n ≥ 1 as K isnot CM. The quantity n = n(K) plays a central role in this article, so we emphasizethat

n := number of infinite places of F which split in K = r1(K)/2.

Fixing a prime p ∈ S(N, K) and setting

S(p)(N, K) = S(N, K) − {p},

it follows from (3) that the set

S(p)(N, K) ∪ {vn+1, . . . , vg}

has even cardinality. Therefore, there is a unique quaternion F -algebra B ramifiedprecisely at the primes in this set. Write d for the discriminant of B,

d =∏

l∈S(p)(N,K)

l,

and define n by the equation

N = dpn.

For each ideal a of OF with (a, d) = 1, choose an Eichler OF -order R0(a) inB of level a such that a1 | a2 implies R0(a2) ⊂ R0(a1). Thus, R0(1) = R0((1)) is amaximal order of B. For any subgroup T of B×, define subgroups T1 ⊂ T+ of T by

T1 = {α ∈ T : nrd α = 1},T+ = {

α ∈ T : vj (nrd α) > 0 for all j = 1, . . . , n}

= {α ∈ T : vj (nrd α) > 0 for all j = 1, . . . , g

},

where nrd : B× → F × is the reduced norm map. We define the arithmetic groups

�0(a) = R0(a)×1 /{±1}.

The natural map inclusion of R0(a) into B induces an embedding

�0(a) ↪→ B×/F ×.


LEMMA 3The inclusion R0(a)×1 ⊂ R0(a)×+ induces an isomorphism

�0(a) = R0(a)×1 /{±1} −→ R0(a)×+/O×F .

ProofNote that for u ∈ O×

F ⊂ R0(a), we have nrd u = u2. Thus, O×F is a subgroup

of R0(a)×+. It suffices to show that the induced map is surjective. To see this, letγ ∈ R0(a)×+. Since we assume that F has narrow class number 1, there is a unitu ∈ O×

F such that nrd γ = u2. Then γ u−1 ∈ R0(a)×1 and (γ u−1)u = γ . �

Let U+ be the group of totally positive units of F , and let

UB = {x ∈ O×

F : v(x) > 0 for infinite places v of F at which B is ramified}.

Since F has narrow class number one,

UB/U+ ∼= (Z/2Z)n.

By the norm theorem [19, Theorem III.4.1], the sequence

1 −→ R0(a)×1 −→ R0(a)×nrd−→ UB −→ 1

is exact. By Lemma 3,

�0(a) = R0(a)×1 /{±1} ∼= R0(a)×+/O×F = ker

(R0(a)×/O×

F

nrd−→ UB/U+).

Therefore, we have the exact sequence

1 −→ �0(a) −→ R0(a)×/O×F

nrd−→ UB/U+ −→ 1. (5)

For a set S of primes of OF , let OF,S be the set of S-integers of F . Set

R = R0(n) ⊗OFOF,{p} = R0(pn) ⊗OF

OF,{p} ⊂ B.

The ring R is an Eichler OF,{p}-order in B of level n. Define the {p}-arithmetic group� by

� = R×1 /{±1} ↪→ B×/F ×.

Let

R×ev = {α ∈ R× : ordp nrd α is even}.


Since p is principal, the natural map R×1 → R×

ev,+ induces an isomorphism

� ∼= R×ev,+/O×

F,{p}

and there is an exact sequence

1 −→ � −→ R×ev/O×

F,{p} −→ UB/U+ −→ 1. (6)

3. Hecke operatorsThe cohomology groups of �0(pn), �0(n), and � are endowed with the action ofcertain Hecke operators.

3.1. Involutions at infinityLet V be an (R0(a)×/O×

F )-module (resp., (R×ev/O×

F,{p})-module). By (5) (resp., (6)),there is a natural action of U ′

+/U+ on Hi(�0(a), V ) (resp., Hi(�, V )). For j =1, . . . , n, let εj ∈ U ′

+ be such that

vk(εj ) < 0 if k = j, vk(εj ) > 0 if k �= j .

The coset εjU+ gives rise to an involution of Hi(�0(a), V ) (resp., Hi(�, V )) whichwe denote by Wvj

.

3.2. An Atkin-Lehner involutionLet � = R×

+/O×F,{p}, and let N�(�0(pn)) be the normalizer of �0(pn) in �. Then there

is another exact sequence

1 −→ �0(pn) −→ N�

(�0(pn)

) ordp nrd−→ Z/2Z −→ 0.

Let wp be an element of R×+ which ordp nrd maps to the nontrivial element of Z/2Z.

If V is an N�(�0(pn))-module, the coset wpN�(�0(pn)) induces an Atkin-Lehnerinvolution of Hi(�0(pn), V ) which we denote Wp. The matrix wp also generates thequotient �/�. Therefore, we also obtain an involution Wp of Hi(�, V ).

Using the Wp-operator, we define a p-new subspace and quotient ofHn(�0(pn), V ) by

Hi(�0(pn), V

)p-new

= ker(Hi(�0(pn), V )

αi−→ Hi(�0(n), V )2),

H i(�0(pn), V

)p-new = coker(Hi(�0(n), V )2 βi

−→ Hi(�0(pn), V )),

where

αi(c) = (cor c, cor Wpc), βi(c1, c2) = res c1 − Wp res c2.


3.3. Double-coset operatorsLet V be a B×/F ×-module, and write G for �0(n) (resp., �0(pn) or �). Let l be aprime of OF not dividing d (resp., pd). Using the formalism of double cosets, onemay define Hecke operators

Tl : Hi(G, V ) −→ Hi(G, V ) if l � n,

Ul : Hi(G, V ) −→ Hi(G, V ) if l|n.

For a detailed description of this formalism, see [16, Chapter 8]. The operators Tl

and Ul generate a commutative subalgebra of Hi(G, V ). Moreover, these operatorscommute with the involutions at infinity Wvj

(resp., with the involutions at infinityWvj

and the Atkin-Lehner involution Wp).

3.4. Hecke algebrasRecall that N = dpn. Let [Tl], [Ul], and [Wvj

] be formal variables, and define thefollowing free polynomial rings:

T(N∞) = Z[{[Tl] : l � N}],

T(dp∞) = Z[{[Tl] : l � N}, {[Ul] : l | n}],

T(dp) = Z[{[Tl] : l � N}, {[Ul] : l | n}, {[Wvj

] : j = 1, . . . , n}].We refer to these rings as Hecke algebras. Since the Hecke operators defined abovecommute with one another, these Hecke algebras act on the various cohomologygroups of �0(pn), �0(n), and � in the way suggested by the notation.

Let T be any one of Hecke algebras defined above, let M be a T-module, andlet λ : T → Z be a ring homomorphism. We write Mλ for the λ-eigenspace for theaction of T on M , that is,

Mλ = {m ∈ M : tm = λ(t)m for all t ∈ T

}.

We say that T acts on M through λ if Mλ = M . For any commutative ring A, set

TA = T ⊗Z A.

Define the degree character deg : T(dp) → Z by

deg[Tl] = |l| + 1, deg[Ul] = |l|, deg[Wvj] = 1.

Fix signs at infinity σ1, . . . , σn ∈ {±1}. Define a character

λE : T(dp) → Z


corresponding to our elliptic curve E/F by

λE([Tl]) = al(E), λE([Ul]) = al(E), λE([Wvj]) = σj . (7)

The actions of the Hecke operators commute with most maps arising naturally inhomological algebra, for example, restriction maps, corestriction maps, and connectinghomomorphisms in long exact sequences corresponding to short exact sequences of(B×/F ×)-modules.

4. Cohomology of quaternionic Shimura varieties

4.1. Action on Hn

For i = 1, . . . , n, fix isomorphisms

ιvi: B ⊗F Fvi

−→ M2(R).

The isomorphisms ιv1, . . . , ιvninduce an embedding

ι∞ : B×/F × −→ PGL2(R)n

which identifies �0(pn) and �0(n) with subgroups of PSL2(R)n. To simplify matters,we make the following assumption.

Assumption 2The groups �0(pn) and �0(n) contain no nontrivial elliptic elements of finite order.

Thus, these groups act properly discontinuously on the product Hn of n copies of thecomplex upper half-plane H. The quotients

X0(pn) = �0(pn)\Hn and X0(n) = �0(n)\Hn

are n-dimensional complex varieties called quaternionic Shimura varieties. Thesevarieties are noncompact if and only if B = M2(F ), in which case X0(pn) and X0(n)are Hilbert modular varieties. If A is any abelian group, we have

Hi

(X0(pn), A

) = Hi

(�0(pn), A

)and Hi

(X0(n), A

) = Hi

(�0(n), A

)for all i and, similarly, for cohomology.

We write Hi! for internal cohomology (also called parabolic cohomology in our

setting). The group Hi! (X0(n), A) is by definition the image of the compactly supported

cohomology group Hic (X0(n), A) in Hi(X0(n), A); restriction and corestriction maps

send internal cohomology to internal cohomology. In addition, the internal subspaceis stable under the action of the Hecke operators.


4.2. Eisenstein cohomologyLet G be either �0(pn) or �0(n). When G is cocompact, that is, when G is not acongruence subgroup of the Hilbert modular group, we clearly have

Hi! (X, C) = Hi(X, C).

Suppose for the remainder of Section 4.2 that G is not cocompact. Let κ1, . . . , κt ∈P1(F ) be representatives for the cusps of G. Write Gκj

for the stabilizer in G of thecusp κj .

PROPOSITION 4 (Harder)(1) There is a Hecke-stable direct sum decomposition

Hi(G, C) = Hi! (G, C) ⊕ Hi

Eis(G, C).

(2) Consider the natural restriction map

res : Hi(G, C) −→t⊕

j=1

Hi(Gκj, C).

Then res maps HiEis(G, C) isomorphically onto the image of res.

(3) If 0 < i < n, then HiEis(G, C) = 0.

For an exposition of this result, see [10, Chapter III, §3].Suppose that κ1, . . . , κt are representatives for the cusps of �0(n). In this case,

it is well known that X0(pn) has 2t cusps—for each cusp κj of X0(n), there are twocusps κ ′

j and κ ′′j of X0(pn) lying over κj , and these two cusps are interchanged by the

Atkin-Lehner involution Wp. This remark has the following consequence.

LEMMA 5The map

βn : HnEis

(�0(n), C

)2 −→ HnEis

(�0(pn), C

)is injective.

The following is a standard property of Eisenstein series.

LEMMA 6The algebra T(N∞) acts on Hi

Eis(G, C) through the degree character.


4.3. Decomposition of the internal cohomologyWe continue to let G denote either �0(pn) or �0(n). For each nonempty subset a of{1, . . . , n}, define a differential 2|a|-form ωa on Hn by

ωa =∧j∈a

dzj ∧ dzj

y2j

= (−2i)#a∧j∈a

dxj ∧ dyj

y2j

. (8)

Each ωa is invariant under the full group GL+2 (R)n, and the classes of their pullbacks

represent cohomology classes [ωa] ∈ H 2|a|(G, C). Since these classes do not dependon the level structure, we call them universal cohomology classes. Define

Hiuniv(G, C) =

⎧⎪⎨⎪⎩⊕

|a|=i/2

C[ωa], i even,

0, i odd.

Since the forms ωa are invariant under the full group GL2(R)n, we obtain the following.

LEMMA 7The algebra T(dp) acts on Hi

univ(G, C) through the degree character.

PROPOSITION 8 (see [14])Suppose i �= n. Then

Hi! (G, C) = Hi

univ(G, C).

For an exposition of this result, see [10, Chapter III, §1].To describe the group Hn

! (G, C), we must take into account the contribution ofcusp forms. Let S2(G) denote the space of holomorphic cusp forms f : Hn → C oflevel G, equipped with its natural action of T(dp∞)

C . Each form f ∈ S2(G) may beviewed as a holomorphic (and hence harmonic) differential n-form on X = G\Hn,yielding a T(dp∞)

C -equivariant injection

S2(G) −→ Hn(G, C).

Let S denote the image of this map. One can show that S ⊂ Hn! (G, C). For any subset

a of {1, . . . , n}, we let Wa denote |a|-fold composition

Wa =∏j∈a

Wvj.


Define

Hncusp(G, C) =

⊕a⊂{1,...,n}

WaS.

PROPOSITION 9 (see [14])We have Hn

! (G, C) = Hnuniv(G, C) ⊕ Hn

cusp(G, C).

For an exposition of this result, see [10, Chapter III, §5].We define

Hncusp

(�0(pn), C

)p-new

= Hncusp

(�0(pn), C

) ∩ Hn(�0(pn), C

)p-new

,

Hncusp

(�0(pn), C

)p-new = im(Hn

cusp(�0(pn), C) −→ Hn(�0(pn), C)p-new).

The proof of the following result is easy.

LEMMA 10We have the following identifications:(1) Hn

cusp(�0(pn), C)p-new = ker αn|Hncusp(�0(pn),C);

(2) Hncusp(�0(pn), C) ∩ im βn = im βn|Hn

cusp(�0(n),C)2 .

THEOREM 11 (Ramanujan-Petersson conjecture)Let ap be an eigenvalue of Tp acting on S2(�0(n)). Then

|ap| ≤ 2|p|1/2.

Indication of proofUsing results of Carayol [3] on reductions of Shimura curves defined over numberfields, the statement of the theorem can be reduced to the Riemann hypothesis forcurves over finite fields, proved by Weil (for details, see [13]). �

LEMMA 12(1) The natural map

Hncusp

(�0(pn), C

)p-new

−→ Hncusp

(�0(pn), C

)p-new

is an isomorphism.(2) The sequence

Hnuniv

(�0(n), C

) c �→(c,0)−→ Hn(�0(n), C

)2 βn

−→ Hn(�0(pn), C

)is exact.


ProofConsider the diagram

Hncusp

(�0(n), C

)2βn

��

h

��Hncusp

(�0(pn), C

) αn

�� Hncusp

(�0(n), C

)2

defining the map h. An easy computation shows that h is given by the matrix(|p| + 1 Tp

Tp |p| + 1

).

Since Tp commutes with each involution Wa, the eigenvalues of Tp acting on S2(�0(n))are the same as those of Tp acting on Hn

cusp(�0(n), C). Therefore, it follows fromTheorem 11 that h is injective (cf. [8, Proposition 4.9]). Statement (1) is now easilydeduced. Statement (2) follows from (1) and Lemma 5. �

5. The Jacquet-Langlands correspondenceRecall that N = dpn. Let �0(N) ⊂ PSL2(OF ) be the congruence subgroup of level�0(N), and let S2(�0(N)) denote the corresponding space of holomorphic Hilbertmodular cusp forms f : Hg → C of parallel weight two, endowed with its naturalaction of T(dp∞)

C . We require a major result from the theory of automorphic forms, theJacquet-Langlands correspondence.

THEOREM 13 (Jacquet-Langlands correspondence)S2(�0(N))d-new and S2(�0(pn)) are isomorphic as T(dp∞)

C -modules.

COROLLARY 14The following hold:(1) dimC(S2(�0(pn))p-new)λE = 1;(2) dimQ(Hn(�0(pn), Q)p-new)λE = dimQ(Hn(�0(pn), Q)p-new)λE = 1.

ProofStatement (1) follows from Proposition 13 and the multiplicity-one theorem for Hilbertmodular forms. Statement (2) follows from (1) together with statement (1) of Propo-sition 4, Lemma 6, Lemma 7, Proposition 9, and the fact that, by Theorem 11,deg �= λE . �

6. The Bruhat-Tits tree of PGL2(Fp) and the cohomology of �

Set

G = PGL2(Fp), K = PGL2(OFp),


and

I ={(

a b

c d

)∈ K : c ∈ pOFp

}, G0 = ker(ordp det : G −→ Z/2Z).

LetV and E denote the sets of vertices and directed edges of the Bruhat-Tits tree Tof G, respectively, (see [6, Chapter 5]). Let v0 and v1 be the vertices of T representedby the lattices OFp

× OFpand OFp

× pOFp, respectively, and set e0 = (v0, v1). For

an edge e = (x, y) ∈ E , we write s(e) and t(e), respectively, for the source x andtarget y of e, and we write e for the edge (y, x) = (t(e), s(e)). Call a vertex even(resp., odd) if it belongs to the G0-orbit of v0 (resp., v1), and let V0 (resp., V1) be theset of all even (resp., odd) vertices of T . We call an edge e even (resp., odd) if s(e) iseven (resp., odd), and we write E0 (resp., E1) for the set of all even (resp., odd) edges.The assignments g �→ gv0 and g �→ ge0 induce identifications

V ∼= G/K, E ∼= G/I, V0∼= G0/K, and E0

∼= G0/I.

Fix an isomorphism

ιp : B ⊗F Fp −→ GL2(Fp)

such that the induced maps

ιp : R0(1) ⊗OFOFp

−→ M2(OFp), ιp : R0(p) ⊗OF

OFp−→ M0(pOFp

)

are isomorphisms, where

M0(pOFp) =

{(a b

c d

)∈ M2(OFp

) : c ∈ pOFp

}.

If a ⊂ OF is an ideal prime to pd, it follows that ιp induces an isomorphism ofR0(a) ⊗ OFp

(resp., R0(pa) ⊗OFOFp

) with M2(OFp) (resp., M0(pOFp

)) as well.The map ιp induces an embedding of B×/F × into PGL2(Fp). Thus, we may view

the groups �0(pn), �0(n), and � as subgroups of G. We have

�0(pn) ⊂ I, �0(n) ⊂ K, � ⊂ G0,

�0(n) ∩ I = �0(pn), � ∩ I = �0(pn) � ∩ K = �0(n).

Let wp be as defined in §3.2. Then wpv0 = ιp(wp)v0 = v1, resulting in identifi-cations

V1∼= G0/wpKw−1

p and E1∼= G0/wpIw−1

p = G0/I.


PROPOSITION 15The group � acts transitively on the groups E0, E1, V0, and V1.

ProofThis result can be deduced from the strong approximation theorem [19, TheoremIII.4.3] together with the fact that F has narrow class number one. �

For any sets X and Y , write F(X, Y ) for the set of functions from X into Y . For anabelian group A, let F0(E, A) ⊂ F(E, A) be the subset consisting of those functionsµ satisfying µ(e) + µ(e) = 0. If X is a left G-set, we define a left action of G onF(X, A) by the rule

(γf )(x) = f (γ −1x).

We obtain the following corollary to Proposition 15.

COROLLARY 16We have the following identifications of �-modules:

E0∼= �/�0(pn) ∼= E1, V0

∼= �/�0(n), V1∼= �/wp�0(n)w−1

p ,

F(V0, A) = Ind��0(n) A, F(V1, A) = Ind�

wp�0(n)w−1p

A, F0(E, A) = Ind��0(pn) .

Consider the short exact sequence

0 −→ A −→ F(V, A) = F(V0, A) ⊕ F(V1, A)δ−→ F0(E, A) −→ 0 (9)

of G-modules, where

δf (e) = f(t(e)

) − f(s(e)

).

Writing down the corresponding long exact sequence in �-cohomology and applyingShapiro’s lemma using the identifications of Corollary 16, we get the Mayer-Vietorissequence

· · · −→ Hi−1(�0(n), A

)2 βi−1

−→ Hi−1(�0(pn), A

) −→ Hi(�, A)

−→ Hi(�0(n), A

)2 βi

−→ Hi(�0(pn), A

) −→ · · · . (10)


PROPOSITION 17If n is odd, then Hn(�, C) = 0. If n is even, then the restriction map

res : Hn(�, C) −→ Hn(�0(n), C

)maps Hn(�, C) isomorphically onto Hn

univ(�0(n), C).

ProofWe prove the assertion for n odd, the argument for n even being similar. ThenHn

univ(�0(n), C) = 0, so ker βn = 0 by statement (2) of Lemma 12. The restrictionmap

Hn−1(�0(n), C

) = Hn−1univ

(�0(n), C

) −→ Hn−1univ

(�0(n), C

) = Hn−1(�0(pn), C

)is clearly surjective since the universal cohomology classes do not depend on the levelstructure, so coker βn−1 = 0. Therefore, by the exactness of (10), Hn(�, C) = 0. �

LEMMA 18We have dim Hn+1(�, C)λE = 1.

ProofFrom the long exact sequence (10), we extract the short left-exact sequence

0 −→ Hn(�0(pn), C

)p-new −→ Hn+1(�, C) −→ Hn+1(�0(n), C

)2.

Since taking eigenspaces is a left-exact operation and Hn+1(�0(n), C)λE = 0, weobtain the isomorphism

Hn+1(�, C)λE ∼= (Hn(�0(pn), C)p-new

)λE.

The result now follows from statement (2) of Corollary 14. �

7. Divisor-valued homology classes associated to embeddingsRecall that K is a quadratic extension of F , as in §2. Write E(K, B) for the set of F -algebra embeddings of K into B. Since all the primes at which B is ramified are inertin K , the set E(K, B) is nonempty. The group B× acts on E(K, B) by conjugation onthe target. Since F × clearly acts trivially, this action descends to B×/F ×. It is wellknown that the stabilizer of ψ ∈ E(K, B) in B× is the nonsplit torus ψ(K×).

Let O ⊂ OK be an order of conductor prime to N .

Definition 19An embedding ψ ∈ E(K, B) is (O, R0(n))-optimal if ψ−1(R0(n)) = O. Denote theset of all such by E(O, R0(n)).


The group R0(n)× acts by conjugation on E(O, R0(n)), and the natural inclusionof E(O, R0(n)) into E(K, B) is R0(n)×-equivariant. Let Pic+ O be the narrow classgroup of the order O. The following result is also well known (see [19, Chapter III,§5C] or [12]).

LEMMA 20The set R0(n)×+\E(O, R0(n)) admits a natural free action of the group Pic+ O. Thenumber of orbits under this action is the number of divisors of the ideal dn.

We note that by Lemma 3, we have

R0(n)×+\E(O, R0(n)

) = �0(n)\E(O, R0(n)

).

Let O×1 denote the set of units u of O such that NormK/F u = 1. By Dirichlet’s

unit theorem,

O×1 /torsion ∼= Zn.

We fix a basis u1, . . . , un ofO×1 /torsion. The orientation class of this basis corresponds

to a choice of generator � of Hn(O×1 /torsion, Z) ∼= Z.

Let ψ ∈ E(O, R0(n)). Then

stabR0(n)× ψ = ψ(K×) ∩ R0(n)× = ψ(O×).

Thus, the embedding ψ induces an identification

�0(n)ψ ∼= O×1 /torsion, (11)

where we write �0(n)ψ for stab�0(n) ψ . Let �ψ be the generator of Hn(�0(n)ψ, Z)which corresponds to � under the isomorphism induced by the identification (11).

Following [4], we describe a particular geometric model of the restriction mapHn(�0(n)ψ, Z) → Hn(�0(n), Z). For each j = 1 . . . , n, the group ιvj

(�0(n)) acts ona copy of the complex upper half-plane. (The notation ιvj

is defined in §4.1.) The torusιvj

(ψ(K×)) has two fixed points τ ′j and τ ′′

j on the boundary of H. The points τ ′j and τ ′′

j

actually belong to P1(K) and are interchanged by the nontrivial element of Gal K/F .Let ϒj be the geodesic path in H joining τ ′

j to τ ′′j , and let

ϒψ = ϒ1 × · · · × ϒn ⊂ Hn.

The group �0(n)ψ acts freely on ϒ via ι∞ with compact quotient homeomorphic toZn\Rn. It is essentially tautological that the map

Hn

(�0(n)ψ\ϒψ, Z

) −→ Hn

(�0(n)\Hn, Z

)


induced by the inclusion of spaces corresponds to the restriction map in group coho-mology under the canonical identifications Hn(�0(n)ψ, Z) = Hn(�0(n)ψ\ϒψ, Z) andHn(�0(n), Z) = Hn(�0(n)\Hn, Z).

LEMMA 21 (cf. [4, Lemma 3.1])Let [ω] ∈ Hn

univ(�0(n), C). Then 〈�ψ, [ω]〉 = 0.

(Here the pairing is the natural one between Hn(�0(n), Z) and Hn(�0(n), C).

ProofSince Hn

univ(�0(n), C) is zero for n odd, we may assume that n is even and ω = ωa

for a subset a ⊂ {1, . . . , n} of size n/2. It follows formally that

〈�ψ, [ωa]〉 = ±∫

ϒψ

ωa.

(The ±-indeterminacy in the above equation is a reflection of the fact that we havenot been careful about orienting ϒψ compatibly with our choice of generator �ψ ofHn(�0(n)ψ, Z).) The result now follows from the fact that, by its definition (see (8)),the restriction of ωa to the j th copy of H is a form of real dimension zero or two whileϒj has real dimension one. �

Define the p-adic upper half-plane Hp by

Hp = Kp − Fp.

There is a unique τψ ∈ Hp such that

ψ(α)

(τψ

1

)= α

(τψ

1

).

We define

HOp = {

τψ : ψ ∈ E(O, R0(n))}.

The stabilizer �0(n)τψof τψ is canonically identified with �0(n)ψ .

By Shapiro’s lemma,

Hn

(�0(n), DivHO

p

) =⊕

τ∈�0(n)\HOp

Hn

(�0(n)τ , Z

)=

⊕ν∈�0(n)\E(O,R0(n))

Hn

(�0(n)ν, Z

).


Therefore, there is a natural inclusion

j : Hn

(�0(n)ψ, Z

)↪→ Hn

(�0(n), DivHO

p

).

Consider the natural restriction map

res : Hn

(�0(n), DivHO

p

) −→ Hn

(�, DivHO

p

)as well as the map

deg : Hn(�, DivHOp ) −→ Hn(�, Z)

induced by the degree map deg : DivHp → Z. Let �ψ be the generator ofHn(�0(n)ψ, Z) constructed above.

LEMMA 22Let ψ ∈ E(O, R0(n)). Then the element deg res j (�ψ ) ∈ Hn(�, Z) is torsion.

ProofIf n is odd, then Hn(�, Z) is finite by Proposition 17. Therefore, we may assume thatn is even. By the commutativity of the diagram

Hn(�0(n), DivHOp )

res��

deg

��

Hn(�, DivHOp )

deg

��Hn(�0(n), Z)

res�� Hn(�, Z)

it suffices to show that res deg �ψ is torsion. Let 〈−, −〉 denote the natural C-valuedpairing between Hn(�, Z) and Hn(�, C). It is enough to show that

〈res deg j (�ψ ), c〉 = 0 for all c ∈ Hn(�, C).

But

〈res deg j (�ψ ), c〉 = 〈deg j (�ψ ), res c〉 = 〈res �ψ, res c〉.

But by Proposition 17, res c ∈ Hnuniv(�0(n), C). The result now follows from

Lemma 21. �

From the exact sequence

0 −→ Div0 HOp −→ DivHO

p −→ Z −→ 0,


we obtain the following long exact sequence in �-homology:

Hn+1(�, Z)∂−→ Hn(�, Div0 HO

p ) −→ Hn(�, DivHOp ) −→ Hn(�, Z). (12)

Since Hn(�, Z) is finitely generated, its torsion subgroup is finite, of exponent e, say.By Lemma 22, the class e res j (�ψ ) ∈ Hn(�, DivHO

p ) has a preimage

�ψ ∈ Hn(�, Div0 HOp ), (13)

well defined up to elements of ∂Hn+1(�, Z).

8. Harmonic cocycle-valued cohomology classes associated to E

Let A be an abelian group. Define two degeneracy maps

ϕs, ϕt : F(E, A) −→ F(V, A)

by the rules

ϕs(µ)(v) =∑

s(e)=v

µ(e) and ϕt (µ)(v) =∑

t(e)=v

µ(e).

Definition 23An A-valued harmonic cocycle is an element of

HC(A) := F0(E, A) ∩ ker ϕs.

LEMMA 24The function ϕs maps F0(E, A) surjectively onto F(V, A).

Sketch of proofLet V (m) be the set of vertices a distance at most m from v0. Thus, V (0) = {v0} and|V (m)| = |p|m + |p|m−1. Let E (m) be the subset of E consisting of edges whose sourceor target lie in V (m). For each m, the map ϕs induces a map

ϕ(m)s : F0(E (m), A) −→ F(V (m), A)

the obvious way.Let ν ∈ F(V, A), and write ν(0) ∈ F(V (0), A) for its restriction to V (0). Write

e0, . . . , e|p| for the |p| + 1 edges with source v0. There is a unique element µ0 ∈


F0(E (0), W ) such that

µ0(ei) ={

ν(v0) if i = 0,

0 otherwise.

Clearly, ϕ(0)s (µ0) = ν(0). It is also clear that

F0(E, A) = lim←−F0(E (m), A), F(V, A) = lim←−F(V (m), A),

and

ϕs = lim←− ϕ(m)s .

Therefore, to complete the proof it suffices to show that given µ(m) ∈ F0(E (m), A)and an extension ν(m+1) ∈ F(V (m+1), A) of ϕ(m)

s (µ(m)), there is an extension µ(m+1) ∈F0(E (m+1), A) of µ(m) such that ϕ(m+1)

s (µ(m+1)) = ν(m+1). Exploiting the fact that weare working on a tree, it is an easy exercise to verify that such an extension exists.

The proof is best understood graphically. Figure 1 illustrates one possible con-struction of a function µ(1) such that ϕ(1)

s = ν(1) for an arbitrary ν in the special case|p| = 2. �

By Lemma 24, we have the following exact sequence of �-modules:

0 −→ HC(A) −→ F0(E, A) −→ F(V, A) −→ 0. (14)

Using Shapiro’s lemma together with the identifications of Corollary 16, the longexact sequence in �-cohomology associated to (14) takes the form

· · · −→ Hn−1(�0(pn), A

) αn−1−→ Hn−1(�0(n), A

)2 −→ Hn(�, A)

−→ Hn(�0(pn), A

) −→ Hn(�0(n), A

)2 −→ · · · . (15)

We extract the following short exact sequence from (15):

0 −→ coker αn−1 −→ Hn(�, HC(A)

) ρ−→ Hn(�0(pn), A

)p-new

−→ 0. (16)

PROPOSITION 25The map ρ of (16) restricts to an isomorphism

ρ : Hn(�, HC(Q)

)λE −→ (Hn(�0(pn), Q)p-new

)λE.

Both of these eigenspaces are one dimensional.


a

bc

d

e

fg

h

i j

a

0 0–b

0–c

0

0 –a–d

e+b

f0g+c0

h

0

i

0 j+a+d

0

ϕs

Figure 1. Surjectivity of ϕs . Even vertices are shaded. The top tree representsa function ν(1) in which each vertex is labeled with its image. The bottom tree

depicts a (choice of ) preimage µ(1) of ν(1) under ϕ(1)s , each positively oriented edge

being labeled with its image. (A function µ(1) ∈ F0(E (1), A) is completely determinedby its values on positively oriented edges.)


ProofLet T = T(dp)

Q , and set m = ker λE . Observe that MλE = HomT(T/m, M) forany T-module M . Applying the left-exact functor HomT(T/m, −) to the short exactsequence (16), we obtain the exact sequence

0 −→ (coker αn−1)λE −→ Hn(�, HC(Q)

)λE

−→ (Hn(�0(pn), Q)p-new

)λE −→ Ext1T(T/m, coker αn−1)

As coker αn−1 is a quotient of Hn−1(�0(n), Q)2 and Hn−1(�0(n), C) consists en-tirely of universal cohomology, T acts on coker αn−1 through the degree character byLemma 7. Since T acts on T/m through λE and λE �= deg, it follows that

(coker αn−1)λE = Ext1T(T/m, coker αn−1) = 0.

Therefore,

Hn(�, HC(Q)

)λE ∼= (Hn(�0(pn), Q)p-new

)λE.

The result now follows from Proposition 14. �

Applying the functor − ⊗ Q to the exact sequence (15) with A = Z, it follows fromthe five lemma that

Hn(�, HC(Q)

) ∼= Hn(�, HC(Z)

) ⊗ Q.

Therefore, we may choose a nontorsion cohomology class

E ∈ Hn(�, HC(Z)

)λE. (17)

9. The p-adic integration pairingLet A be an abelian group, and let X = P1(Fp).

Definition 26An A-valued measure on X is a finitely additive A-valued function µ on the set ofcompact-open subsets of X.

Let Meas(X, A) denote the set of such measures. The group G = PGL2(Fp) acts onthe set Meas(X, A) from the left by the rule

(γµ)(U ) = µ(γ −1U ).


Let

Meas0(X, A) = {µ ∈ Meas(X, A) : µ(X) = 0

}be the G-stable subspace of measures with total measure zero. The collection B ofcompact-open balls in X is a basis for its algebra of compact-open subsets. Choosinga coordinate on X, we have the fact that the stabilizer of the ball OFp

is precisely I,yielding an identification

B ∼= G/I.

Thus, the map

ge0 �→ gOFp

from edge set E of T to B is a bijection.

LEMMA 27To give an A-valued measure on X with total measure zero is equivalent to giv-ing an A-valued harmonic cocycle on T ; that is, there is a canonical isomorphismMeas0(X, A) → HC(A).

In the remainder of the article, we feel free to identify measures and harmonic cocyclesin this manner. In particular, we allow ourselves to integrate against harmonic cocycles.

Let C(X, Kp) denote the ring of continuous Kp-valued functions on X. Forf ∈ C(X, Kp)× and µ ∈ Meas(X, Z), we consider the following limit of Riemannproducts:

lim‖U‖→0

∏U∈U

f (tU )µ(U ).

Here we are taking the limit over increasingly fine covers U of X by finitely manycompact open subsets. The argument tU denotes an arbitrary point of U . One can showthat this limit converges to a limit in K×

p which we call the multiplicative integral off against µ and denote

×∫

〈f, µ〉

exists. The group G acts on C(X, Kp) from the right by the rule

(γf )(x) = f (γ −1x).


One may verify the identity

×∫

〈γf, γµ〉 = ×∫

〈f, µ〉.

In other words, the pairing

×∫

〈−, −〉 : C(X, Kp)× × Meas(X, Z) −→ K×p

is G-invariant. We immediately obtain an induced pairing

×∫

〈−, −〉 : C(X, Kp)×/K×p × Meas0(X, Z) −→ K×

p .

There is a natural G-equivariant embedding Div0 Hp → C(X, Kp)×/K×p sending a

divisor D to a rational function over Kp with divisor D. (Such a function is welldefined up to multiplication by elements of K×

p .) This gives us yet another inducedpairing

×∫

〈−, −〉 : Div0 Hp × Meas0(X, Z) −→ K×p . (18)

We wish to give a combinatorial description of the pairing ordp ×∫

〈−, −〉. Define

∑〈−, −〉 : Div0 V × F0(E, Kp) −→ Kp (19)

defined by ∑〈{v2} − {v1}, µ〉 =

∑e:v1→v2

µ(e).

(The sum on the right-hand side denotes the sum over the directed edges e in a path onT from v1 to v2.) We wish to describe the relationship between the pairings ×∫ 〈−, −〉and

∑〈−, −〉. Consider the reduction map

r : Hp −→ V

and the natural inclusion map ρ : Meas0(X, Kp) ↪→ F0(E, Kp).


LEMMA 28The following diagram commutes:

Div0 Hp × Meas0(X, Z)×∫ 〈−,−〉

��

r×ρ

��

K×p

ordp

��

Div0 V × F0(E, Kp) ∑〈−,−〉�� Kp

ProofSee [2, Lemma 2.5] or [8, Lemma 4.2]. �

10. Abel-Jacobi maps and Stark-Heegner pointsThe G-equivariant pairing (18) induces a pairing

×∫

〈−, −〉 : Hn(�, Div0 HOp ) × Hn

(�, Meas0(X, Z)

) −→ K×p .

Let E ∈ Hn(�, HC(Z)) be the cohomology class defined in (17). Define

AJE : Hn(�, Div0 HOp ) → K×

p

by the rule

AJE(ξ ) = ×∫

〈ξ, E〉,

and set

L = AJE(∂Hn+1

(�, Z)

) ⊂ K×p .

Let

AJE : Hn(�, Div0 HOp ) −→ K×

p /L (20)

be the “Abel-Jacobi map” induced by AJE .Recall the homology class �ψ associated to ψ defined in (13). This class is

determined by ψ up to elements of ∂Hn+1(�, Z). It is now evident that

Qψ = AJE(�ψ ) = ×∫

〈�ψ, E〉 ∈ K×p /L


is completely determined by ψ . Thus, (20) induces a map

�0(n)\E(O, R0(n)

) −→ K×p /L, [ψ] �→ Qψ.

Definition 29A lattice � in K×

p is an infinite, finitely generated, discrete subgroup K×p . Two lattices

�1 and �2 in K×p are homothetic if �1 ∩ �2 has finite index in both �1 and �2.

PROPOSITION 30L is a lattice in K×

p .

We prove Proposition 30 in the next section.

CONJECTURE 2L and the Tate lattice 〈qE〉 of E/Kp are homothetic.

Let � be a uniformizer of OKp, and let log� : K×

p → Kp be the branch of thelogarithm satisfying log� � = 0 (cf. §11). Let λ be a nontorsion element of L. Thehomothety class of the lattice L is completely characterized by its L-invariant

L� (L) := logp λ

ordp λ.

It is not difficult to verify that Conjecture 2 is equivalent to the statement that

L� (L) = L� (〈qE〉) =: L� (E).

For the remainder of this paragraph, assume that E is defined over F = Q. Thenthere is a natural choice for � , namely, p. In the case where all primes � �= p

dividing the conductor of E are split in K , Conjecture 2 is a theorem of Darmon (cf.Theorem 2). The main ingredient in the proof of this result is the Greenberg-Stevenstheorem [11, Theorem 0.3], formerly the exceptional zero conjecture of Mazur, Tate,and Teitelbaum [15, Conjecture 1]. In [11], it is shown how the L-invariant Lp(E) canbe extracted from the p-adic Galois representation associated to E/Qp. On the otherhand, the quantity Lp(L), as defined above, is an invariant of p-adic automorphicdata associated to E. Thus, Conjecture 2 may be viewed as a connection betweenautomorphic forms and Galois representations. It therefore seems natural to attempt toattack this conjecture in general by relating it to the p-adic Langlands correspondence.The author hopes to pursue these ideas in future work.

Granting Conjecture 2, we may find an isogeny

β : K×p /L −→ K×

p /〈qE〉.


Let

� : K×p /〈qE〉 −→ E(Kp)

be Tate’s uniformization, and define

Pψ = �(β(Qψ )

) ∈ E(Kp).

Let H+O /K be the narrow ring class field extension associated to the OF -order O,

and let

rec : Pic+ O −→ Gal(H+O /K)

be the isomorphism given by class field theory. (Pic+ O was defined in §7.) Recallfrom Lemma 20 that E(O, R0(n))/�0(n) admits an action of Pic+ O.

CONJECTURE 3(1) Pψ belongs to E(H+

O ) (rationality).(2) For each α ∈ Pic+ O, we have Pψα = P rec α

ψ (Shimura reciprocity).

11. Proof of Proposition 30The arguments of this section are inspired by those of [8]. Let � : K×

p → Kp be agroup homomorphism. The function � is called a branch of the logarithm if(1) �(1 + x) = ∑

n≥1 (−1)n/nxn for all x ∈ pOKp, and

(2) ker � is not torsion.For all q ∈ pOKp

, there is a unique branch logq of the logarithm such that logq q = 0.Let � be a uniformizer of Kp, and let � be a subgroup of K×

p .

LEMMA 31� is a lattice if and only if(1) ordp � �= {0},(2) there is a constant L� ∈ Kp such that

log� λ = L� ordp λ

for all λ ∈ �.

ProofEasy. �

Let ∂ be the boundary map of (12), and define functions

κordp: Hn+1(�, Z) −→ Kp, κlog�

: Hn+1(�, Z) −→ Kp


by the rules

κordp(ξ ) = ordp ×

∫〈∂ξ, E〉, κlog�

(ξ ) = log� ×∫

〈∂ξ, E〉.

The universal coefficients theorem for cohomology implies that the natural map

Hn+1(�, Kp) −→ Hom(Hn+1(�, Z), Kp

)is an isomorphism. Therefore, we may freely view κordp

and κlog�as elements of

Hn+1(�, Kp).By Lemma 31, to prove Proposition 30 it suffices to establish the following two

lemmas.

LEMMA 32The function κordp

is not identically zero.

LEMMA 33The functions κordp

and κlog�are Kp-linearly dependent.

For the proof of Lemma 32, we require a result from basic homological algebra.Let G be a group, and let

0 −→ A −→ B −→ Z −→ 0

be a short exact sequence of G-modules. From the corresponding long exact sequencein G-homology, we extract the boundary map

∂ : Hn+1(G, Z) −→ Hn(G, A).

Let M be a trivial G-module. From the long exact sequence in G-cohomology asso-ciated to the sequence

0 −→ M −→ Hom(B, M) −→ Hom(A, M) −→ 0,

we extract the coboundary map

δ : Hn(G, Hom(A, M)

) −→ Hn+1(G, M).

There are natural pairings

〈−, −〉n : Hn(G, A) × Hn(G, Hom(A, M)

) −→ M,

〈−, −〉n+1 : Hn+1(G, Z) × Hn+1(G, M) −→ M.


Let η ∈ Hn+1(G, Z), and let c ∈ Hn(G, Hom(A, M)). Then a standard computationshows that

〈∂η, c〉n = 〈η, δc〉n+1. (21)

Proof of Lemma 32Let

δ : Hn(�,F0(E, Kp)

) = Hn(�0(pn)

) −→ Hn+1(�, Kp)

be the coboundary map arising in the (Mayer-Vietoris) long exact sequence (10)with A = Kp. The map δ factors through the natural projection Hn(�0(pn), Kp) →Hn(�0(pn), Kp)p-new, and the induced map Hn(�0(pn), Kp)p-new → Hn+1(�, Kp)is injective. As ρ(E) is a nonzero element of Hn(�0(pn), Kp)p-new, it follows thatδ(ρ(E)) is nonzero as well.

It remains to verify that

δ(ρ(E)

) = κordp. (22)

Setting

G = �, A = (Div E)/〈{e} + {e} : e ∈ E〉, B = DivV, M = Kp

and noting that there are natural identifications

Hom(A, M) ∼= F0(E, Kp), Hom(B, M) ∼= F(V, Kp),

it follows from (21) that ⟨ξ, δ(ρ(E))

⟩n+1

= 〈∂ξ, ρ(E)〉n.

But by Lemma 28,

〈∂ξ, ρ(E)〉n =∑

〈∂ξ, ρ(E)〉 = ordp ×∫

〈∂ξ, E〉 = 〈ξ, κordp〉n+1,

where the pairing∑〈−, −〉 is as in (19). As⟨

ξ, δ(ρ(E))⟩n+1

= 〈ξ, κordp〉n+1

for all ξ ∈ Hn+1(�, Z), identity (22) follows from the universal coefficient theoremfor cohomology. �


Proof of Lemma 33By the Hecke-equivariance of the universal coefficient theorem, the classes κordp

and κlog�both lie in the eigenspace Hn+1(�, Kp)λE . But by Lemma 18, we have

dim Hn+1(�, Kp)λE = 1, completing the proof. �

12. Concluding remarksModular symbols are intimately related to special values of L-functions. This relation-ship is exploited in [5, proof of Theorem 2] and is the key ingredient in the proof of theBertolini-Darmon theorem on the rationality of Stark-Heegner points over the genusfield of a real quadratic field [1]. This theorem is the main theoretical evidence forthe conjectures of [5]. Due to the lack of a theory of modular symbols in general, theconnection between the Stark-Heegner points constructed in this article and specialvalues of L-functions is less clear. Clarifying this relationship, with an eye to provinganalogues of the Bertolini-Darmon theorem, is an important open problem.

Modular symbols have also proved essential in the gathering of numerical ev-idence for the conjectures of [5] (see [7]). It would be highly desirable to havesufficiently developed algorithms for computing in cohomology groups of arithmeticgroups which would allow for the accumulation of evidence supporting the conjecturesof this article.

Acknowledgments. The author thanks Pierre Charollois and Samit Dasgupta for theirnumerous helpful comments regarding this article. He is especially grateful to HugoChapdelaine for his extremely dedicated proofreadings of early drafts of this work aswell as for many valuable discussions related to its content. Finally, the author wishesto express his appreciation to the anonymous referees whose thorough reviews andinsightful comments contributed greatly to the final form of this article.

References

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[2] M. BERTOLINI, H. DARMON, and P. GREEN, “Periods and points attached to quadraticalgebras” in Heegner Points and Rankin L-series (Berkeley, Calif., 2001), Math.Sci. Res. Inst. Publ. 49, Cambridge Univ. Press, Cambridge, 2004, 323 – 367.MR 2083218 569

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Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, CanadaT2N 1N4; [email protected]

















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