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Non-archimedean construction ofelliptic curves and abelian surfaces

ICERM WORKSHOPModular Forms and Curves of Low Genus:

Computational Aspects

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Universitat de Barcelona

2University of Warwick

3University of Sheffield

September 28th, 2015

Marc Masdeu Non-archimedean constructions September 28th , 2015 0 / 34

Modular Forms and Curves of Low Genus:Computational Aspects

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Universitat de Barcelona

2University of Warwick

3University of Sheffield

September 28th, 2015


Quaternionic automorphic forms of level N

F a number field of signature pr, sq, and fix N Ă OF .Choose factorization N “ Dn, with D square free.Fix embeddings v1, . . . , vr : F ãÑ R, w1, . . . , ws : F ãÑ C.Let BF be a quaternion algebra such that

RampBq “ tq : q | Du Y tvn`1, . . . , vru, pn ď rq.

Fix isomorphisms

B bFvi –M2pRq, i “ 1, . . . , n; B bFwj –M2pCq, j “ 1, . . . , s.

These yield BˆFˆ ãÑ PGL2pRqn ˆ PGL2pCqs ýHn ˆHs3.

R>0

C

H3

R>0

H

R

PGL2(R)PGL2(C)


Quaternionic automorphic forms of level N (II)

Fix RD0 pnq Ă B Eichler order of level n.

ΓD0 pnq “ RD

0 pnqˆOˆF acts discretely on Hn ˆHs

3.Obtain an orbifold of (real) dimension 2n` 3s:

Y D0 pnq “ ΓD

0 pnqz pHn ˆHs3q .

The cohomology of Y D0 pnq can be computed via

H˚pY D0 pnq,Cq – H˚pΓD

0 pnq,Cq.

Hecke algebra TD “ ZrTq : q - Ds acts on H˚pΓD0 pnq,Zq.

Hn`spΓD0 pnq,Cq “

à

χ

Hn`spΓN0 pnq,Cqχ, χ : TD Ñ C.

Each χ cuts out a field Kχ, s.t. rKχ : Qs “ dimHn`spΓD0 pnq,Cqχ.


Abelian varieties from cohomology classes

Definitionf P Hn`spΓD

0 pnq,Cqχ eigen for TD is rational if appfq P Z,@p P TD.

If r “ 0, then assume N is not square-full: Dp ‖ N.

Conjecture (Taylor, ICM 1994)1 f P Hn`spΓD

0 pnq,Zq a new, rational eigenclass.Then DEfF of conductor N “ Dn attached to f . i.e. such that

#Ef pOF pq “ 1` |p| ´ appfq @p - N.

2 More generally, if χ : TD Ñ C is nontrivial, cutting out a field K, thenD abelian variety Aχ, with dimAχ “ rK : F s and multiplication by K.

Assumption above avoids “fake abelian varieties”, and it is needed inour construction anyway.


Goals of this talk

In this talk we will:

1 Review known explicit forms of this conjecture.§ Cremona’s algorithm for F “ Q.§ Generalizations to totally real fields.

2 Propose a new, non-archimedean, conjectural construction.§ (joint work with X. Guitart and H. Sengun)

3 Explain some computational details.

4 Illustrate with examples.


F “ Q: Cremona’s algorithm for elliptic curves

Eichler–Shimura construction

X0pNq // JacpX0pNqq

ş

–H0pX0pNq,Ω1q

_

H1pX0pNq,ZqHecke// // CΛf – Ef pCq.

1 Compute H1pX0pNq,Zq (modular symbols).2 Find the period lattice Λf by explicitly integrating

Λf “

C

ż

γ2πi

ÿ

ně1

anpfqe2πinz : γ P H1

´

X0pNq,Z¯

G

.

3 Compute c4pΛf q, c6pΛf q P C by evaluating Eistenstein series.4 Recognize c4pΛf q, c6pΛf q as integers ; Ef : Y 2 “ X3 ´ c4

48X ´c6

864 .


F ‰ Q: constructions for elliptic curves

F totally real. rF : Qs “ n, fix σ : F ãÑ R.

S2pΓ0pNqq Q f ; ωf P HnpΓ0pNq,Cq; Λf Ď C.

Conjecture (Oda, Darmon, Gartner)CΛf is isogenous to Ef ˆF Fσ.

Known to hold (when F real quadratic) for base-change of EQ.Exploited in very restricted cases (Dembele, Stein+7).Explicitly computing Λf is hard.

§ No quaternionic computations (except for Voight–Willis?).

F not totally real: no known algorithms. . .TheoremIf F is imaginary quadratic, the lattice Λf is contained in R.

IdeaAllow for non-archimedean constructions.


Non-archimedean construction

From now on: fix p ‖ N.Denote by Fp “ alg. closure of the p-completion of F .

Theorem (Tate uniformization)There exists a rigid-analytic, Galois-equivariant isomorphism

η : Fˆp xqEy Ñ EpFpq,

with qE P Fˆp satisfying jpEq “ q´1E ` 744` 196884qE ` ¨ ¨ ¨ .

Choose a coprime factorization N “ pDm, with D “ discpBF q.Compute qE as a replacement for Λf .Starting data: f P Hn`spΓD

0 pmq,Zqp´new, pDm “ N.


Non-archimedean path integrals on Hp

Consider Hp “ P1pCpqr P1pFpq.It is a p-adic analogue to H:

§ It has a rigid-analytic structure.§ Action of PGL2pFpq by fractional linear transformations.§ Rigid-analytic 1-forms ω P Ω1

Hp.

§ Coleman integration ; make sense ofşτ2τ1ω P Cp.

Get a PGL2pFpq-equivariant pairingş

: Ω1HpˆDiv0 Hp Ñ Cp.

For each Γ Ă PGL2pFpq, get induced pairing (cap product)

H ipΓ,Ω1Hpq ˆHipΓ,Div0 Hpq

ş

// Cp

´

φ,ř

γ γ bDγ

¯

//ř

γ

ż

Dγ

φpγq.

Ω1Hp– space of Cp-valued boundary measures Meas0pP1pFpq,Cpq.


Measures and integrals

Bruhat-Tits tree of GL2pFpq, |p| “ 2.P1pFpq – EndspT q.Harmonic cocycles HCpAq “

tEpT q fÑ A |

ř

opeq“v fpeq “ 0u

Meas0pP1pFpq, Aq – HCpAq.So replace ω P Ω1

Hpwith

µω P Meas0pP1pFpq,Zq – HCpZq.

P1(Fp)

U ⊂ P1(Fp)µ(U)

v∗

v∗

e∗

T

Coleman integration: if τ1, τ2 P Hp, thenż τ2

τ1

ω “

ż

P1pFpq

logp

ˆ

t´ τ2

t´ τ1

˙

dµωptq “ limÝÑU

ÿ

UPUlogp

ˆ

tU ´ τ2

tU ´ τ1

˙

µωpUq.

Multiplicative refinement (assume µωpUq P Z, @U ):

ˆ

ż τ2

τ1

ω “ ˆ

ż

P1pFpq

ˆ

t´ τ2

t´ τ1

˙

dµωptq “ limÝÑU

ź

UPU

ˆ

tU ´ τ2

tU ´ τ1

˙µωpUq

.

Marc Masdeu Non-archimedean constructions September 28th, 2015 10 / 34

The tpu-arithmetic group Γ

Choose a factorization N “ pDm.BF “ quaternion algebra with RampBq “ tq | Du Y tvn`1, . . . , vru.

Recall also RD0 ppmq Ă RD

0 pmq Ă B.Fix ιp : RD

0 pmq ãÑM2pZpq.Define ΓD

0 ppmq “ RD0 ppmq

ÔˆF and ΓD0 pmq “ RD

0 pmqÔˆF .

Let Γ “ RD0 pmqr1ps

ÔF r1psˆ ιp

ãÑ PGL2pFpq.

ExampleF “ Q and D “ 1, so N “ pM .B “M2pQq.Γ0ppMq “

`

a bc d

˘

P GL2pZq : pM | c(

t˘1u.Γ “

`

a bc d

˘

P GL2pZr1psq : M | c(

t˘1u ãÑ PGL2pQq Ă PGL2pQpq.


The tpu-arithmetic group Γ

LemmaAssume that h`F “ 1. Then ιp induces bijections

ΓΓD0 pmq – V0pT q, ΓΓD

0 ppmq – E0pT q

V0 “ V0pT q (resp. E0 “ E0pT q) are the even vertices (resp. edges) of T .

Proof.1 Strong approximation ùñ Γ acts transitively on E0 and V0.2 Stabilizer of vertex v˚ (resp. edge e˚) is ΓD

0 pmq (resp. ΓD0 ppmq).

Corollary

MapspE0pT q,Zq – IndΓΓD

0 ppmqZ, MapspVpT q,Zq –

´

IndΓΓD

0 pmqZ¯2.


Cohomology

Γ “ RD0 pmqr1ps

ˆOF r1psˆ ιp

ãÑ PGL2pFpq.

MapspE0pT q,Zq – IndΓΓD

0 ppmqZ, MapspVpT q,Zq –

´

IndΓΓD

0 pmqZ¯2.

Want to define a cohomology class in Hn`spΓ,Ω1Hpq.

Consider the Γ-equivariant exact sequence

0 // HCpZq //MapspE0pT q,Zq β //MapspVpT q,Zq // 0

ϕ // rv ÞÑř

opeq“v ϕpeqs

So get:

0 Ñ HCpZq Ñ IndΓΓD

0 ppmqZ βÑ

´

IndΓΓD

0 pmqZ¯2Ñ 0


Cohomology (II)

0 Ñ HCpZq Ñ IndΓΓD

0 ppmqZ βÑ

´

IndΓΓD

0 pmqZ¯2Ñ 0

Taking Γ-cohomology, . . .

Hn`spΓ,HCpZqq Ñ Hn`spΓ, IndΓΓD

0 ppmq,Zq β

Ñ Hn`spΓ, IndΓΓD

0 pmq,Zq2 Ñ ¨ ¨ ¨

. . . and using Shapiro’s lemma:

Hn`spΓ,HCpZqq Ñ Hn`spΓD0 ppmq,Zq

βÑ Hn`spΓD

0 pmq,Zq2 Ñ ¨ ¨ ¨

f P Hn`spΓD0 ppmq,Zq being p-new ô f P Kerpβq.

Pulling back get

ωf P Hn`spΓ,HCpZqq – Hn`spΓ,Ω1

Hpq.


Holomogy

Consider the Γ-equivariant short exact sequence:

0 Ñ Div0 Hp Ñ DivHpdegÑ ZÑ 0.

Taking Γ-homology yields

Hn`s`1pΓ,ZqδÑ Hn`spΓ,Div0 Hpq Ñ Hn`spΓ,DivHpq Ñ Hn`spΓ,Zq

Λf “

#

ˆ

ż

δpcqωf : c P Hn`s`1pΓ,Zq

+

Ă Cˆp

Conjecture A (Greenberg, Guitart–M.–Sengun)

The multiplicative lattice Λf is homothetic to qZE .

F “ Q: Darmon, Dasgupta–Greenberg, Longo–Rotger–Vigni.F totally real, |p| “ 1, B “M2pF q: Spiess.Open in general.


Lattice: explicit construction

Start with f P Hn`spΓD0 ppm,Zqqp´new.

Duality yields f P Hn`spΓD0 ppmq,Zqqp´new.

Mayer–Vietoris exact sequence for Γ “ ΓD0 pmq ‹ΓD

0 ppmqΓD0 pmq:

¨ ¨ ¨ Ñ Hn`s`1pΓ,Zqδ1Ñ Hn`spΓ

D0 ppmq,Zq

βÑ Hn`spΓ

D0 pmq,Zq2 Ñ ¨ ¨ ¨

f new at p ùñ βpfq “ 0.§ f “ δ1pcf q, for some cf P Hn`s`1pΓ,Zq.

Conjecture (rephrased)The element

Lf “ż

δpcf qωf .

satisfies (up to a rational multiple) logppqEq “ Lf .


Algorithms

Only in the cases n` s ď 1.

§ Both H1 and H1: fox calculus (linear algebra for finitely-presentedgroups).

Use explicit presentation + word problem for ΓD0 ppmq and ΓD

0 pmq.

§ John Voight (s “ 0).

§ Aurel Page (s “ 1).

Need the Hecke action on H1pΓD0 ppmq,Zq and H1pΓ

D0 ppmq,Zq.

§ Shapiro’s lemma ùñ enough to work with ΓD0 pmq.

Integration pairing uses the overconvergent method.


Overconvergent Method

Starting data: cohomology class φ “ ωf P H1pΓ,Ω1

Hpq.

Goal: to compute integralsşτ2τ1φγ , for γ P Γ.

Recall thatż τ2

τ1

φγ “

ż

P1pFpq

logp

ˆ

t´ τ1

t´ τ2

˙

dµγptq.

Expand the integrand into power series and change variables.§ We are reduced to calculating the moments:

ż

Zp

tidµγptq for all γ P Γ.

Note: Γ Ě ΓD0 pmq Ě ΓD

0 ppmq.Technical lemma: All these integrals can be recovered from#

ż

Zp

tidµγptq : γ P ΓD0 ppmq

+

.


Overconvergent Method (II)

D “ tlocally analytic Zp-valued distributions on Zpu.§ ϕ P D maps a locally-analytic function h on Zp to ϕphq P Zp.§ D is naturally a ΓD

0 ppmq-module.

The map ϕ ÞÑ ϕp1Zpq induces a projection:

H1pΓD0 ppmq,Dq

ρ // H1pΓD0 ppmq,Zpq.

PΦ

//

P

φ

Theorem (Pollack-Stevens, Pollack-Pollack)There exists a unique Up-eigenclass Φ lifting φ.

Moreover, Φ is explicitly computable by iterating the Up-operator.


Overconvergent Method (III)

But we wanted to compute the moments of a system of measures. . .

PropositionConsider the map ΓD

0 ppmq Ñ D:

γ ÞÑ”

hptq ÞÑ

ż

Zp

hptqdµγptqı

.

1 It satisfies a cocycle relation ùñ induces a classΨ P H1

´

ΓD0 ppmq,D

¯

.

2 Ψ is a lift of φ.3 Ψ is a Up-eigenclass.

CorollaryThe explicitly computed Φ “ Ψ knows the above integrals.


Recovering E from Λf

Λf “ xqf y gives us qf?“ qE .

Assume ordppqf q ą 0 (otherwise, replace qf ÞÑ 1qf ).Get

jpqf q “ q´1f ` 744` 196884qf ` ¨ ¨ ¨ P Cˆp .

From N guess the discriminant ∆E .§ Only finitely-many possibilities, ∆E P SpF, 12q.

jpqf q “ c34∆E ; recover c4.

Recognize c4 algebraically.1728∆E “ c3

4 ´ c26 ; recover c6.

Compute the conductor of Ef : Y 2 “ X3 ´ c448X ´

c6864 .

§ If conductor is correct, check aq’s.


Example curve (joint with X. Guitart and H. Sengun)

F “ Qpαq, pαpxq “ x4 ´ x3 ` 3x´ 1, ∆F “ ´1732.N “ pα´ 2q “ p13.BF ramified only at all infinite real places of F .There is a rational eigenclass f P S2pΓ0p1,Nqq.From f we compute ωf P H1pΓ,HCpZqq and Λf .

qf?“ qE “ 8 ¨ 13` 11 ¨ 132 ` 5 ¨ 133 ` 3 ¨ 134 ` ¨ ¨ ¨ `Op13100q.

jE “ 113

´

´ 4656377430074α3 ` 10862248656760α2 ´ 14109269950515α ` 4120837170980¯

.

c4 “ 2698473α3 ` 4422064α2 ` 583165α´ 825127.c6 “ 20442856268α3´ 4537434352α2´ 31471481744α` 10479346607.

EF : y2 ``

α3 ` α` 3˘

xy “ x3`

``

´2α3 ` α2 ´ α´ 5˘

x2

``

´56218α3 ´ 92126α2 ´ 12149α` 17192˘

x

´ 23593411α3 ` 5300811α2 ` 36382184α´ 12122562.


Tables: Imaginary quadratic fields

|∆K | fKpxq NmpNq pDm c4pEq, c6pEq3 r1,´1s 196 p3r ´ 2q7p´6r ` 2q28p1q ´131065r,

474493313 r1,´1s 196 p´3r ` 1q7p6r ´ 4q28p1q ´131065r,

474493314 r1, 0s 130 p3r ´ 2q13p´r ´ 3q10p1q ´264r ` 257,

´6580r ` 25834 r1, 0s 130 p´3r ´ 2q13p´3r ´ 1q10p1q 264r ` 257,

6580r ` 25837 r2,´1s 44 prq2p3r ` 1q22p1q 648r ` 481,

´28836r ` 44477 r2,´1s 44 pr ´ 1q2p3r ´ 4q22p1q ´648r ` 1129,

28836r ´ 243898 r2, 0s 99 pr ` 1q3p´4r ` 1q33p1q 444r ` 25,

14794r ´ 162638 r2, 0s 99 pr ´ 1q3p´4r ´ 1q33p1q ´444r ` 25,

´14794r ´ 162638 r2, 0s 99 p´r ´ 3q11p3q9p1q ´444r ` 25,

´14794r ´ 162638 r2, 0s 99 pr ´ 3q11p3q9p1q 444r ` 25,

14794r ´ 16263


Tables: cubic p1, 1q fields|∆K | fKpxq NmpNq pDm c4pEq, c6pEq

23 r1, 0,´1s 185 pr2 ` 1q5p3r2 ´ r ` 1q37p1q 643318r2 ´ 1128871r ` 852306,

925824936r2 ´ 1624710823r ` 1226456111

31 r´1, 1, 0s 129 p´r ´ 1q3p´3r2 ´ 2r ´ 1q43p1q ´4787r2 ` 10585r ` 3349,

1268769r2 ´ 371369r ` 424764

44 r1, 1,´1s 121 p2r ´ 1q11pr2 ` 2q11p1q 4097022r2 ´ 6265306r ` 7487000,

14168359144r2 ´ 21861492432r `

26039140708

44 r1, 1,´1s 121 p2r ´ 1q11pr2 ` 2q11p1q 1774r2 ´ 1434r ´ 1304,

´42728r2 ´ 123104r ´ 54300

44 r1, 1,´1s 121 p2r ´ 1q11pr2 ` 2q11p1q 4097022r2 ´ 6265306r ` 7487000,

14168359144r2 ´ 21861492432r `

26039140708

59 r´1, 2, 0s 34 p´r2 ´ 1q2p´r2 ´ 2r ´ 2q17p1q 262r2 ` 513r ` 264,

´2592r2 ` 448r ` 13231

59 r´1, 2, 0s 34 p´r2 ´ 2r ´ 2q17p´r2 ´ 1q2p1q 16393r2 ` 20228r ´ 12524,

4430388r2 ´ 5579252r ` 1619039

59 r´1, 2, 0s 46 p´2r2 ` r ´ 2q23p´r2 ´ 1q2p1q 18969r2 ` 8532r ` 41788,

4216716r2 ` 1911600r ` 9298151

59 r´1, 2, 0s 74 p´r2 ´ 1q2p2r2 ` 2r ` 1q37p1q 33054r2 ` 15049r ` 72776,

9702640r2 ` 4400116r ` 21401723

59 r´1, 2, 0s 88 p´r2 ´ 1q2pr ´ 2q11pr2 ` r ` 1q4 16609r2 ` 7084r ` 37332,

3522136r2 ` 1613876r ` 7760395

59 r´1, 2, 0s 187 p2r2 ` r ` 2q17pr ´ 2q11p1q ´32r2 ´ 848r ` 432,

´7600r2 ` 23368r ´ 8704

76 r´2,´2, 0s 117 p2r2 ´ r ´ 3q13p´r2 ` 2r ` 1q9p1q 48r ` 16,

´128r2 ´ 224r ´ 216

83 r´2, 1,´1s 65 pr ` 1q5p´2r ` 1q13p1q 3089r2 ` 1086r ` 4561,

333604r2 ` 117840r ` 493059

83 r´2, 1,´1s 65 pr ` 1q5p´2r ` 1q13p1q 304r2 ` 112r ` 449,

6616r2 ` 2328r ` 9791

83 r´2, 1,´1s 65 p´2r ` 1q13pr ` 1q5p1q 3089r2 ` 1086r ` 4561,

333604r2 ` 117840r ` 493059

83 r´2, 1,´1s 65 p´2r ` 1q13pr ` 1q5p1q 4499473r2 ` 1589254r ` 6650137,

18573712184r2`6560420272r`27451337687

83 r´2, 1,´1s 106 prq2p2r2 ´ 3r ` 3q53p1q 2329r2 ` 822r ` 3441,

´34264r2 ´ 12104r ´ 50645

87 r1, 2,´1s 123 pr2 ´ r ` 1q3pr2 ` 4q41p1q 1424r2 ` 3792r ` 2384,

´245696r2 ´ 201800r ´ 4144

87 r1, 2,´1s 129 pr2 ´ r ` 1q3p´3r ` 1q43p1q ´752r2 ` 2272r ` 1009,

27496r2 ´ 152144r ´ 63977

87 r1, 2,´1s 129 pr2 ´ r ` 1q3p´3r ` 1q43p1q ´752r2 ` 2272r ` 1009,

27496r2 ´ 152144r ´ 63977

104 r´2,´1, 0s 143 pr2 ` r ´ 1q11p2r ` 1q13p1q 12r2 ` 12r ` 25,

´144r2 ´ 90r ´ 125

107 r´2, 3,´1s 40 p´r2 ´ 1q5pr2 ´ r ` 3q4prq2 3880r2 ´ 984r ` 10473,

405820r2 ´ 105348r ` 1142075

107 r´2, 3,´1s 135 p´r2 ´ 1q5p3q27p1q 16r2 ´ 16r,184r ´ 296

108 r´2, 0, 0s 34 p2r ` 1q17prq2p1q 184r2 ` 212r ` 265,

´5010r2 ´ 6306r ´ 7773

108 r´2, 0, 0s 85 p´r2 ´ 1q5p2r ` 1q17p1q 2224r2 ` 2816r ` 3520,

´229056r2 ´ 288672r ´ 363768

108 r´2, 0, 0s 85 p2r ` 1q17p´r2 ´ 1q5p1q 2224r2 ` 2816r ` 3520,

´229056r2 ´ 288672r ´ 363768

108 r´2, 0, 0s 125 p´r2 ´ 1q5pr2 ´ 2r ´ 1q25p1q 496r2,

22088

108 r´2, 0, 0s 145 p´r2 ´ 1q5pr ` 3q29p1q 144r2 ` 176r ` 240,

3816r2 ` 4752r ` 6088

108 r´2, 0, 0s 155 p´r2 ´ 1q5pr2 ` 3q31p1q 16r2 ` 20r ` 17,

´606r2 ´ 762r ´ 929

116 r´2, 0,´1s 34 p´2r ` 1q17p´r ` 1q2p1q 846760r2 ` 589024r ` 998761,

1781332252r2 ` 1239131712r ` 2101097467

116 r´2, 0,´1s 34 p´r ` 1q2p´2r ` 1q17p1q 4592r2 ` 3192r ` 5417,

274400r2 ` 190876r ` 323659

116 r´2, 0,´1s 38 p´r ` 1q2p2r ` 1q19p1q 82921r2 ` 57626r ` 97746,

54599355r2 ` 37980374r ` 64400978

116 r´2, 0,´1s 38 p2r ` 1q19p´r ` 1q2p1q 1081r2 ` 746r ` 1266,

66555r2 ` 46310r ` 78482

116 r´2, 0,´1s 58 p´r ` 1q2pr2 ` r ´ 3q29p1q 22024r2 ` 15320r ` 25977,

´4956678r2 ´ 3447968r ´ 5846447

135 r´1, 3, 0s 55 pr2 ´ r ` 2q11pr2 ` 1q5p1q 4139r2 ´ 19599r ` 5885,

2077971r2 ´ 1764501r ` 352796

135 r´1, 3, 0s 88 pr2 ´ r ` 2q11p2q8p1q ´1751r2 ´ 1226r ` 577,

´131901r2 ´ 120528r ` 52524

139 r2, 1,´1s 46 pr ´ 3q23p´rq2p1q 22560r2 ` 19560r ` 1033,

´8413992r2 ` 2336724r ` 7421723

139 r2, 1,´1s 57 pr ´ 1q3p´2r ` 1q19p1q 18r2 ` 61r ` 39,296r ` 239

139 r2, 1,´1s 57 p´2r ` 1q19pr ´ 1q3p1q 258r2 ` 541r ` 279,

´17136r2 ´ 9280r ` 3767

140 r´2, 2, 0s 25 pr2 ` 1q5pr ` 1q5p1q 1488r2 ` 992r ` 3968,

110440r2 ` 88352r ` 287144

140 r´2, 2, 0s 70 pr2 ` r ` 1q7pr ` 1q5prq2 139012r2 ` 106502r ` 360441,

´100613641r2 ´ 77548384r ´ 260995189

140 r´2, 2, 0s 95 pr2 ` 1q5pr2 ` 2r ` 3q19p1q 16r2 ` 16r,

´64r2 ` 240r ´ 120

140 r´2, 2, 0s 95 pr2 ` 1q5pr2 ` 2r ` 3q19p1q 64r2 ´ 64r ` 48,

´824r2 ´ 368r ` 616

172 r3,´1,´1s 45 pr ´ 2q5pr2 ´ r ´ 1q9p1q ´1072r2 ´ 80r ` 1872,

´49976r2 ´ 48864r ` 25920

175 r´3, 2,´1s 27 prq3pr2 ´ r ` 2q9p1q ´384r2 ` 816r ´ 416,

5904r2 ´ 32472r ` 31816

199 r´1, 4,´1s 21 p´r2 ` r ´ 2q7pr2 ´ r ` 3q3p1q 98529r2 ` 22348r ´ 12672,

´41881233r2 ` 130193546r ´ 31313977

199 r´1, 4,´1s 21 p´r2 ` r ´ 2q7pr2 ´ r ` 3q3p1q ´112647r2 ´ 62978r ` 24321,

´60304454r2 ´ 96556295r ` 29529884

199 r´1, 4,´1s 33 pr ´ 2q11pr2 ´ r ` 3q3p1q 2802r2 ` 3055r ´ 996,

´398780r2 ` 635911r ´ 139543

199 r´1, 4,´1s 49 p´r2 ` r ´ 2q7p´r2 ´ 3q7p1q 6447r2 ´ 31223r ` 7758,

3699375r2 ´ 3171676r ` 577928

199 r´1, 4,´1s 49 p´r2 ´ 3q7p´r2 ` r ´ 2q7p1q 6447r2 ´ 31223r ` 7758,

3699375r2 ´ 3171676r ` 577928

199 r´1, 4,´1s 77 pr ` 1q7pr ´ 2q11p1q 12952r2 ´ 10791r ` 49899,

2866751r2 ´ 2163173r ` 10872899

199 r´1, 4,´1s 99 pr ´ 2q11pr2 ` 1q9p1q ´120r2 ` 576r ´ 143,

380r2 ` 4776r ´ 1281

200 r2, 2,´1s 14 pr ` 1q2pr2 ´ r ` 1q7p1q ´401r2 ´ 3756r ´ 2274,

182521r2 ´ 243668r ´ 235802

200 r2, 2,´1s 14 pr2 ´ r ` 1q7pr ` 1q2p1q ´241r2 ` 404r ` 366,

5649r2 ` 3068r ´ 394

200 r2, 2,´1s 65 p´r2 ´ r ´ 1q13p´r2 ` r ´ 3q5p1q ´1176r2 ´ 1944r ´ 767,

75636r2 ´ 142236r ´ 124561

204 r´3, 1,´1s 21 pr2 ` r ` 1q7prq3p1q ´48r2 ` 96r ´ 32,

´288r2 ` 1008r ´ 872

204 r´3, 1,´1s 21 pr2 ` r ` 1q7prq3p1q 262r2 ´ 326r ´ 44,

´1784r2 ´ 5128r ` 11612

211 r´3,´2, 0s 21 pr ` 2q7p´rq3p1q 22010896r2 ` 41672992r ` 34877233,

296072400488r2 ` 560550677168r `

469139740087

212 r´2, 4,´1s 35 pr2 ´ r ` 1q7pr2 ´ r ` 3q5p1q 29888r2 ´ 13952r ` 112113,

10054302r2 ´ 4693580r ` 37714701

216 r´2, 3, 0s 34 prq2pr2 ` r ` 5q17p1q 307r2 ` 194r ` 1057,

´11235r2 ´ 6786r ´ 37821

216 r´2, 3, 0s 34 pr2 ` r ` 5q17prq2p1q 307r2 ` 194r ` 1057,

´11235r2 ´ 6786r ´ 37821

|∆K | fKpxq NmpNq pDm c4pEq, c6pEq

216 r´2, 3, 0s 38 p´2r2 ´ 2r ´ 7q19prq2p1q 16r2 ` 81,

´216r2 ´ 192r ´ 601

231 r3, 0,´1s 33 p´r ` 1q3pr2 ´ r ` 2q11p1q 465r2 ´ 1011r ` 1189,

25273r2 ´ 54957r ` 64546

231 r3, 0,´1s 33 pr2 ´ r ` 2q11p´r ` 1q3p1q 465r2 ´ 1011r ` 1189,

25273r2 ´ 54957r ` 64546

231 r3, 0,´1s 51 pr2 ` 1q17prq3p1q ´47r2 ´ 50r ` 145,

938r2 ´ 291r ` 4

239 r´3,´1, 0s 24 pr ` 1q3p2q8p1q 9r2 ` 18r ` 25,

143r2 ` 236r ` 268

239 r´3,´1, 0s 57 p´r2 ´ r ` 1q19pr ` 1q3p1q 1170r2 ` 1953r ` 2098,

108233r2 ` 180929r ` 194227

243 r´3, 0, 0s 10 pr ´ 2q5pr ´ 1q2p1q 27576r2 ` 39771r ` 57360,

11428272r2 ` 16482420r ` 23771763

243 r´3, 0, 0s 10 pr ´ 1q2pr ´ 2q5p1q 27576r2 ` 39771r ` 57360,

11428272r2 ` 16482420r ` 23771763

243 r´3, 0, 0s 22 pr ` 2q11pr ´ 1q2p1q 2002130917752r2 ` 2887572455827r `

4164600133648,

7076846143946804016r2 `

10206578310238918020r `

14720433182250993839

243 r´3, 0, 0s 34 pr2 ` 2q17pr ´ 1q2p1q 10167352r2 ` 14663859r ` 21148944,

´80986535280r2 ´ 116802795708r ´

168458781921

243 r´3, 0, 0s 46 p´2r ` 1q23pr ´ 1q2p1q 19946163r2 ` 28767345r ` 41489691,

222892996797r2 ` 321467328855r `

463636116909

255 r´3, 0,´1s 15 p´r2 ´ 1q5pr ´ 1q3p1q 248r2 ´ 320r ´ 263,

´2556r2 ´ 4104r ` 16523

255 r´3, 0,´1s 15 p´r2 ´ 1q5prq3p1q 19r2 ´ r ` 88,

279r2 ` 908

255 r´3, 0,´1s 51 pr2 ´ 2q17pr ´ 1q3p1q ´32r2 ` 240r ´ 336,

´3416r2 ` 2400r ` 7392

255 r´3, 0,´1s 51 pr2 ´ 2q17pr ´ 1q3p1q 80r2 ´ 80r ´ 128,

288r2 ´ 2088r ` 2888

255 r´3, 0,´1s 65 pr2 ´ r ` 1q13pr ` 1q5p1q 3r2 ´ 105r ` 88,

´909r2 ` 1116r ´ 1576

268 r5,´3,´1s 14 pr2 ´ 2q7pr ´ 1q2p1q ´285113701784r2 ´ 52062773310r `

950706811227,

144006413291532359r2 `

50857254178772568r ´ 433038348784793416

300 r´3,´3,´1s 9 p´r2 ` 2r ` 2q3prq3p1q 26r2 ` 46r ` 4,

504r2 ` 504r ` 460

300 r´3,´3,´1s 9 p´r2 ` 2r ` 2q3prq3p1q 26r2 ` 46r ` 4,

504r2 ` 504r ` 460

300 r´3,´3,´1s 33 pr2 ´ r ´ 1q11prq3p1q 11072r2 ` 17760r ` 12865,

4675808r2 ` 7475664r ` 5398495

300 r´3,´3,´1s 90 p´r2 ` 2r ` 2q3p´r ´ 3q30p1q ´71,´1837

307 r2, 3,´1s 10 pr ´ 1q5p´rq2p1q ´1450479r2 ´ 118958r ` 338681,

´1778021804r2´7601175244r´3506038549

307 r2, 3,´1s 45 pr ´ 1q5pr2 ´ 2r ` 5q9p1q r2 ` 154r ` 81,

´1744r2 ´ 1756r ´ 441

324 r´4,´3, 0s 4 pr ´ 2q2p´r ´ 1q2p1q 345255874728r2 ` 758120909880r `

628931968401,

686899433218582980r2 `

1508309811434747772r `

1251283596457392135

324 r´4,´3, 0s 22 pr2 ´ r ´ 1q11pr ´ 2q2p1q 808464801r2 ` 1775245884r ` 1472731953,

77832295537635r2 ` 170905971571164r `141782435639127

324 r´4,´3, 0s 84 pr2 ` 3r ` 3q7p´r2 ´ 3r ´ 2q12p1q 143742984r2 ` 315634200r ` 261847993,

4700399015844r2 ` 10321245891900r `

8562435635987

327 r´3,´2,´1s 9 prq3pr ` 1q3p1q 13r2 ` 22r ` 25,

144r2 ` 225r ` 242

327 r´3,´2,´1s 15 p´r ` 1q5prq3p1q 1645r2 ´ 2647r ´ 2984,

55543r2 ´ 6268r ´ 298328

327 r´3,´2,´1s 15 prq3p´r ` 1q5p1q 1645r2 ´ 2647r ´ 2984,

55543r2 ´ 6268r ´ 298328

335 r1, 4,´1s 25 pr2 ´ r ` 3q5p´r ` 1q5p1q ´951r2 ` 1190r ` 57,

61922r2 ´ 78025r ´ 346

335 r1, 4,´1s 25 pr2 ´ r ` 3q5p´r ` 1q5p1q 10r2 ´ 11r ` 10,

52r2 ´ 271r ´ 29

335 r1, 4,´1s 65 p´r2 ` 2r ´ 4q13p´r ` 1q5p1q 61r2 ´ 77r ` 247,

101r2 ´ 107r ` 380

351 r´3, 3, 0s 33 pr ´ 2q11prq3p1q 16r2 ` 144r ´ 128,

1824r2 ´ 72r ´ 1160

356 r7, 1,´1s 14 p´r ´ 2q7p12r2 ´ r ` 3

2q2p1q 1577904r2 ` 58258032r ` 83210433,

157810225239r2 ` 783843846012r `

817040026548

356 r7, 1,´1s 26 p´r ` 2q13p´12r2 ` r ´ 5

2q2p1q ´353192r2 ´ 495936r ` 44233,

´560380445r2 ´ 897785708r ´ 94909392

356 r7, 1,´1s 26 p´r ` 2q13p´12r2 ` r ´ 5

2q2p1q 88412r2 ` 1393648r ` 1878333,

112777386r2 ` 1758482408r ` 2367346473

356 r7, 1,´1s 196 prq7pr ´ 3q28p1q 4182384r2 ´ 3886864r ´ 15048991,

´37671142504r2 ´ 30349104360r `

38274580847

364 r´2, 4, 0s 21 pr ´ 1q3p´r ´ 1q7p1q ´368r2 ´ 3712r ` 1840,

´72736r2 ` 343360r ´ 146264

364 r´2, 4, 0s 26 p´r2 ´ 1q13p´rq2p1q ´266582r2 ` 148350r ´ 10479,

´274275343r2 ` 306719520r ´ 83736937

379 r´4, 1,´1s 6 pr ´ 1q3p´r ` 2q2p1q 1418236432r2 ` 1053691808r ` 3254778265,

137488390576232r2 ` 102148264969648r `315528648990403

379 r´4, 1,´1s 6 p´r ` 2q2pr ´ 1q3p1q 1418236432r2 ` 1053691808r ` 3254778265,

137488390576232r2 ` 102148264969648r `315528648990403

379 r´4, 1,´1s 21 pr ´ 1q3pr ` 1q7p1q 15373338r2 ` 11421763r ` 35281005,

´155147444344r2 ´ 115268221468r ´

356055251669

379 r´4, 1,´1s 21 pr ` 1q7pr ´ 1q3p1q 15373338r2 ` 11421763r ` 35281005,

´155147444344r2 ´ 115268221468r ´

356055251669

379 r´4, 1,´1s 27 pr ´ 1q3pr2 ` 1q9p1q 1532208r2 ` 1138368r ` 3516337,

1280550616r2 ` 951396864r ` 2938796535

379 r´4, 1,´1s 34 pr ´ 3q17p´r ` 2q2p1q 90342993r2 ` 67121158r ` 207332433,

2363568298948r2 ` 1756034817652r `

5424265343699

439 r5,´2,´1s 15 p´r ` 1q3pr ´ 2q5p1q ´439r2 ` 1212r ´ 1252,

27743r2 ´ 76494r ` 78935

439 r5,´2,´1s 15 pr ´ 2q5p´r ` 1q3p1q ´439r2 ` 1212r ´ 1252,

27743r2 ´ 76494r ` 78935

440 r´8, 2, 0s 10 p´r2 ´ 2r ´ 5q5p´12r2 ´ r ´ 2q2p1q ´349392832r2 ´ 1512227664r` 3500497481,

´12893566003280r2 ´ 143880769408104r `276285496852283

440 r´8, 2, 0s 10 p´ 12r2 ´ r ´ 2q2p´r

2 ´ 2r ´ 5q5p1q ´349392832r2 ´ 1512227664r` 3500497481,

´12893566003280r2 ´ 143880769408104r `276285496852283

440 r´8, 2, 0s 26 p2r ´ 3q13p´12r2 ´ r ´ 2q2p1q

9532r2 ´ 6046r ` 8769,

´ 4195612

r2 ` 835646r ´ 810505

451 r8,´5,´1s 26 p2r ´ 3q13p´r ` 2q2p1q 34296r2 ` 4776r ´ 189951,

5707476r2 ` 13155804r ´ 1647297

459 r´8, 3, 0s 22 p 12r2 ´ 1

2r ` 1q11p´

12r2 ´ 1

2r ´ 2q2p1q 16r2 ´ 104r ` 121,

´240r2 ` 1260r ´ 1357

459 r´8, 3, 0s 33 p´ 12r2 ´ 1

2r ` 1q11p

12r2 ` 1

2r ` 3q3p1q ´ 19

2r2 ` 15

2r ` 21,

´36r2 ` 96r ´ 37

459 r´8, 3, 0s 33 pr2 ` r ` 5q11p12r2 ` 1

2r ` 3q3p1q

1788292

r2 ` 2705212

r ` 472861,

´83966694r2 ´ 127020222r ´ 444049333

459 r´8, 3, 0s 34 p 12r2 ` 3

2r ´ 3q17p´

12r2 ´ 1

2r ´ 2q2p1q

1252r2 ` 79

2r ´ 59,

282r2 ´ 2430r ` 2691

459 r´8, 3, 0s 44 p 12r2 ´ 1

2r ` 1q11pr ´ 1q4p1q

312r2 ´ 105

2r ` 44,

411r2 ´ 1452r ` 1256

459 r´8, 3, 0s 44 p 12r2 ´ 1

2r ` 1q11pr ´ 1q4p1q

1032r2 ´ 55

2r ´ 60,

237r2 ` 1374r ´ 2984

460 r´3, 5,´1s 6 p´rq3pr ´ 1q2p1q 38808r2 ` 63978r ´ 55637,

28650959r2 ` 29220772r ´ 29738968

460 r´3, 5,´1s 25 p2r2 ´ r ` 10q5p´r2 ´ 4q5p1q 36772r2 ´ 83396r ` 37921,

32322356r2 ´ 98725758r ` 49331449

460 r´3, 5,´1s 26 pr2 ´ r ` 1q13pr ´ 1q2p1q 973808r2 ´ 7106166r ` 4086627,

´8777739333r2`7426503436r´1197128148

515 r´4,´1,´1s 14 p´r ` 2q2pr2 ´ 2r ´ 1q7p1q ´7341361r2 ´ 9117211r ´ 13098483,

´14436506787r2 ´ 17928648161r ´

25757667905

519 r7,´4,´1s 39 p´r2 ` 3q13p´r ` 2q3p1q ´280r2 ´ 960r ´ 751,

54220r2 ` 11272r ´ 242353

547 r´4,´3,´1s 14 p´r ` 1q7pr2 ´ 2r ´ 2q2p1q 14509048r2 ` 24346088r ` 21671521,

200457117220r2 ` 336365736396r `

299413898447

652 r5, 7,´1s 14 p´ 12r2 ` r ` 1

2q7p´

12r2 ` r ´ 9

2q2p1q 18r2 ´ 36r ` 147,

405r2 ´ 648r ` 3294

687 r3, 4,´1s 9 prq3pr ` 1q3p1q ´7r2 ` 38r ` 25,

´18r2 ´ 423r ´ 244

743 r´3, 5, 0s 9 p´r ` 1q3prq3p1q 736r2 ` 416r ` 3913,

´110256r2 ´ 62192r ´ 586373

755 r2, 5,´1s 10 p´2r2 ` 3r ´ 11q5p´rq2p1q ´1634r2 ` 10769r ` 4135,

110372r2 ` 1174880r ` 412903

815 r´9,´7, 0s 9 pr ` 1q3p´r ` 3q3p1q 26678105835217r2 ` 83793885354406r `

76443429630973,

717286463675094140331r2 `

2252941797094015980448r `

2055312234304678362824

1196 r´7, 5,´1s 14 p´rq7p´r ` 1q2p1q ´12r2 ´ 4r ` 25,

´4r2 ´ 134r ` 181


Tables: quartic p2, 1q fields (I)|∆K | fKpxq NmpNq pDm c4pEq, c6pEq

643 r1,´2, 0,´1s 175 pr3 ´ r2 ´ r ´ 1q7p2r3 ´ r2 ´ 2q25p1q ´1783r3 ` 1032r2 ` 522r ` 3831,

116369r3 ´ 62909r2 ´ 30125r ´ 248439

688 r´1,´2, 0, 0s 11 p´r3 ` r2 ` r ` 2q11p1qp1q 200r3 ` 284r2 ` 376r ` 136,

´5184r3 ´ 7280r2 ´ 10024r ´ 3672

688 r´1,´2, 0, 0s 19 p2r3 ´ 3q19p1qp1q 552r3 ` 764r2 ` 1064r ` 392,

´11536r3 ´ 16160r2 ´ 22584r ´ 8312

731 r´1, 0, 2,´1s 80 pr2 ` 1q5p1qp2q16 ´848r3 ` 1529r2 ` 456r ´ 420,

45471r3 ´ 164824r2 ` 11648r ` 72230

775 r´1,´3, 0,´1s 176 p´r3 ` r2 ` 1q11p2q16p1q ´ 62772r3 ´ 2939r2 ´ 5696r ´ 3239

2,

´528578r3 ´ 495324r2 ´ 959488r ´ 272875

976 r´1, 0, 3,´2s 44 pr ´ 2q11p1qpr3 ´ r2 ` r ` 2q4 ´42r3 ´ 21r2 ` 20r ` 10,

´10860r3 ´ 12344r2 ` 6618r ` 4899

976 r´1, 0, 3,´2s 65 pr3 ´ 2r2 ` 4rq13p1qpr3 ´ 2r2 ` 3r ` 1q5 72r3 ` 20r2 ´ 40r ´ 4,

´1456r3 ` 3800r2 ´ 176r ´ 1200

1107 r´1,´2, 0,´1s 99 pr ´ 1q3p´2r ` 1q33p1q 105488r3 ` 90125r2 ` 152590r ` 66821,

120373437r3 ` 96189249r2 ` 171765105r `67816591

1156 r1,´1,´2,´1s 19 pr3 ´ r2 ´ 2r ´ 3q19p1qp1q ´816481030r3´882631565r2´203810962r`392346684,

´68032828897760r3 ´ 73544780430596r2 ´16982427384164r ` 32692074898043

1156 r1,´1,´2,´1s 19 pr ` 2q19p1qp1q ´384131503r3´ 415253582r2´ 95887519r`184588047,

82379129020040r3 ` 89053403394404r2 `

20563566138596r ´ 39585957243581

1192 r´1, 1, 2,´1s 38 pr2 ` 2q19p1qpr3 ´ r2 ` 2rq2 9504r3 ` 11111r2 ´ 4762r ´ 5690,

´2387028r3 ` 7298060r2 ` 2454128r ´

3005365

1255 r´1,´3,´1, 0s 170 pr3 ´ r ´ 2q2p´2r3 ` 2r2 ` 3q85p1q 517916r3 ` 904037r2 ` 1060116r ` 296716,

´1433064139r3 ´ 2501458160r2 ´

2933309166r ´ 820990264

1423 r´1,´2, 1,´1s 98 pr ´ 1q2p2r3 ´ r2 ` 2r ´ 2q49p1q 39690531r3 ` 20246442r2 ` 70104884r `

26465314,

702653466524r3 ` 356968363314r2 `

1240909503739r ` 466012978440

1423 r´1,´2, 1,´1s 98 pr3 ´ r2 ` 2r ´ 1q7pr3 ´ 2r2 ` 2r ´ 1q14p1q 54577r3 ` 27699r2 ` 96525r ` 36260,

1735232r3 ` 881975r2 ` 3066920r` 1151600

1588 r2, 0,´3,´1s 56 p´r3 ` r2 ` 3r ` 1q7pr3 ´ r2 ´ 3rq8p1q 94560r3 ` 111816r2 ´ 39672r ´ 86639,

747493992r3 ` 883740564r2 ´ 313920684r ´685060489

1588 r2, 0,´3,´1s 152 pr3 ´ 3r ´ 1q19pr3 ´ r2 ´ 3rq8p1q 3496200r3`4469800r2´803168r´2816543,

26973722420r3 ` 32247663708r2 ´

10621228512r ´ 24308855297

1600 r´4, 0,´2, 0s 11 p 12r2 ´ r ´ 1q11p1qp1q 12r3 ` 48r2 ` 56r ` 20,

´284r3 ´ 460r2 ´ 472r ´ 1024

1600 r´4, 0,´2, 0s 11 p 12r2 ` r ´ 1q11p1qp1q 276r3 ` 490r2 ` 336r ` 628,

´18172r3 ´ 32652r2 ´ 22424r ´ 40464

1600 r´4, 0,´2, 0s 19 p 12r3 ´ 1

2r2 ´ r ´ 1q19p1qp1q ´44r3 ` 112r2 ´ 56r ` 148,

´1660r3 ` 2572r2 ´ 2056r ` 3136

1600 r´4, 0,´2, 0s 19 p´ 12r3 ´ 1

2r2 ` r ´ 1q19p1qp1q 44r3 ` 112r2 ` 56r ` 148,

1660r3 ` 2572r2 ` 2056r ` 3136

1732 r´1, 3, 0,´1s 13 pr ´ 2q13p1qp1q 3455801r3`1359008r2´3314187r`836393,

7590438778r3 ´ 14215787438r2 ´

23508658710r ` 9402560739

1732 r´1, 3, 0,´1s 182 pr3 ´ r ` 3q7pr2 ´ r ´ 2q26p1q ´17184648r3 ´ 14365296r2 ` 9302744r ´

813151,

93038140030r3 ´ 219828160822r2 ´

331159079722r ` 135298016971

1823 r´2, 3, 0,´1s 114 p´r3 ` r ´ 3q3pr3 ` r2 ` 2q38p1q 233810r3 ´ 9696r2 ´ 336273r ` 159951,

´70457084r3´ 403468159r2´ 171041003r`342434077

1879 r1,´3,´2,´1s 140 p 12r3 ´ 2r ´ 1

2q7pr

3 ´ r2 ´ r ´ 2q20p1q ´2436r3 ´ 3240r2 ´ 2688r ` 1045,

´49029102r3 ´ 65262564r2 ´ 54075240r `21032621


Tables: quartic p2, 1q fields (II)|∆K | fKpxq NmpNq pDm c4pEq, c6pEq

2051 r1, 3,´1,´1s 15 pr3 ´ r2 ` 2q5p1qp´r ` 1q3 ´489r3 ` 1228r2 ´ 1242r ` 18,

46792r3 ´ 100917r2 ` 73440r ` 47160

2068 r1, 3,´2,´1s 7 pr ´ 2q7p1qp1q 26909497r3 ` 20141314r2 ´ 35624307r ´11296953,

247303058576r3 ´ 3168333376r2 ´

656295560992r ´ 182979737393

2068 r1, 3,´2,´1s 13 p´r3 ` r2 ` 2r ´ 1q13p1qp1q 34500648r3 ` 3814392r2 ´ 84122424r ´23737447,

´77408488074r3 ´ 354426093238r2 ´

415474468618r ´ 92161502469

2068 r1, 3,´2,´1s 56 pr ´ 2q7pr3 ´ 2r ` 1q8p1q ´3576591826r3 ´ 1882130113r2 `

6123537074r ` 1835712204,

321001991693952r3 ` 322520099276304r2 ´281263304453488r ´ 100176319060369

2068 r1, 3,´2,´1s 182 pr ´ 2q7p´r3 ` r2 ´ 2q26p1q ´1994707423r3 ´ 282234694r2 `

4755878517r ` 1346474783,

´8733155599162r3 ´ 54136988565986r2 ´

71594660083402r ´ 16347374680241

2092 r´2,´3, 1,´1s 8 prq2p1qpr ´ 1q4 ´3r3 ` 29r2 ´ r ` 75,

231r3 ´ 61r2 ` 497r ´ 287

2096 r2,´2,´2, 0s 28 pr3 ´ r ´ 1q7p1qpr3 ` r2 ´ 2q4 116r3 ´ 390r2 ` 402r ´ 94,

7354r3 ` 222r2 ´ 29620r ` 17640

2116 r´2, 0, 1,´1s 5 pr2 ` 1q5p1qp1q 129712r3 ` 31248r2 ` 168480r ` 209073,

´109612390r3´ 26402860r2´ 142375012r´176669575

2116 r´2, 0, 1,´1s 130 pr2 ` 1q5pr ` 2q26p1q 105064r3 ` 25312r2 ` 136464r ` 169353,

78278092r3 ` 18855232r2 ` 101675032r `126166043

2116 r´2, 0, 1,´1s 130 pr3 ´ r2 ` r ` 1q13pr3 ` rq10p1q 105064r3 ` 25312r2 ` 136464r ` 169353,

78278092r3 ` 18855232r2 ` 101675032r `126166043

2183 r´1, 1, 3,´2s 126 p´r3 ` 2r2 ´ 4rq7pr3 ´ 2r2 ` 4r ` 1q18p1q ´330539r3 ´ 223654r2 ` 72664r ` 52816,

421344240r3 ` 649688112r2 ´ 51218957r ´170790474

2191 r´1, 0, 3,´1s 70 p´r3 ´ 2r ´ 2q5p´2r3 ` r2 ´ 5r ´ 2q14p1q ´928r3 ` 6929r2 ´ 312r ´ 2120,

´885775r3 ` 1164640r2 ` 179150r ´ 336602

2191 r´1, 0, 3,´1s 80 p´r3 ´ 2r ´ 2q5p2q16p1q ´408r3 ` 2689r2 ´ 105r ´ 821,

120899r3 ` 70135r2 ´ 44492r ´ 24989

2243 r´1,´3,´1,´1s 75 pr ´ 1q5p´r3 ` 2r2 ´ r ` 2q15p1q 586900359r3 ` 694528587r2 ` 929522310r `

268803085,

63399246832324r3 ` 75025661482408r2 `

100410590521972r ` 29037147633615

2243 r´1,´3,´1,´1s 75 pr3 ´ r2 ´ 2r ´ 2q5p´r3 ` r2 ` 2r ` 3q15p1q 586900359r3 ` 694528587r2 ` 929522310r `

268803085,

63399246832324r3 ` 75025661482408r2 `

100410590521972r ` 29037147633615

2243 r´1,´3,´1,´1s 105 p´r3 ` 2r2 ` 2q7p´r3 ` 2r2 ´ r ` 2q15p1q 4336158r3`5131353r2`6867535r`1985981,

´22914354769r3 ´ 27116483373r2 ´

36291344215r ´ 10494880213

2243 r´1,´3,´1,´1s 105 pr ´ 1q5pr2 ´ r ` 1q21p1q 920025r3 ` 1088737r2 ` 1457115r ` 421377,

3942374598r3 ` 4665343442r2 `

6243862193r ` 1805625754

2284 r´4, 2, 2,´2s 22 p´r2 ` r ` 1q11p1qp12r3 ´ r2 ` 1q2 ´4322076r3 ` 3371584r2 ´ 4531104r ´

14171719,

´293858698818r3 ` 229234508344r2 ´

308070583688r ´ 963537590781

2327 r´2,´1,´1, 0s 48 pr2 ´ 1q3p2q16p1q 60947675662300r3 ` 95467421346487r2 `

88590894936957r ` 77819621400035,

1595218950381053851625r3 `

2498724287457442364789r2 `

2318740987988175420378r `

2036818157516553727423

2327 r´2,´1,´1, 0s 66 pr2 ´ 1q3p´r3 ` 2q22p1q 24654r3 ` 41044r2 ` 36631r ` 33971,

13602419r3 ` 21481224r2 ` 19830770r `17549287

2327 r´2,´1,´1, 0s 78 pr2 ´ 1q3pr3 ´ r2 ` rq26p1q 1632339r3`2556895r2`2372706r`2084241,

5442997756r3 ` 8525820467r2 `

7911705090r ` 6949764691

2443 r´1,´3, 0, 0s 63 pr3 ´ 2q3p´r ´ 2q21p1q 51601r3 ` 81980r2 ` 123695r ` 33695,

´38870055r3 ´ 59926714r2 ´ 92404714r ´25325630

2443 r´1,´3, 0, 0s 117 p´r3 ` r2 ´ r ` 2q13p´r2 ` 1q9p1q ´26624r3 ´ 78583r2 ` 147974r ` 56321,

41156101r3 ´ 906363r2 ´ 80062921r ´

24803969

2480 r´2,´2, 0, 0s 17 p´r3 ` r2 ` r ` 1q17p1qp1q 8r3 ´ 12r2 ´ 12r ` 17,

212r3 ` 628r2 ´ 818r ´ 887

2480 r´2,´2, 0, 0s 19 p´r2 ` r ´ 1q19p1qp1q ´648r3 ` 524r2 ´ 408r ` 1636,

29224r3 ´ 23272r2 ` 18616r ´ 73216

2608 r´2,´2,´2, 0s 50 p´r ` 1q5pr3 ´ r2 ´ rq10p1q ´18122952r3 ` 23309952r2 ` 6270652r `

28184369,

´178706675384r3 ` 229835084602r2 `

61821736238r ` 277904169213

2696 r1,´3, 0,´1s 24 pr3 ´ 2q3pr3 ´ 3q8p1q 25999152r3 ` 20125515r2 ` 35704342r ´

14654974,

´282591287516r3 ´ 218749239468r2 ´

388079405968r ` 159288610195

2816 r´1,´4,´2, 0s 15 pr2 ´ r ´ 1q5p1qpr3 ´ r2 ´ 2r ´ 1q3 134184108r3 ´ 165313100r2 ´ 203588440r ´

41893502,

2470282983044r3 ´ 3964870336170r2 ´

2128766125800r ´ 223343175430

2859 r´3, 3,´1,´1s 7 p´r3 ` r ´ 1q7p1qp1q ´4976r3 ` 12905r2 ´ 15523r ` 9529,

1469059r3´3794717r2`4539759r´2782843

3119 r´4,´3,´2,´1s 23 p 23r3 ´ r2 ´ 1

3r ´ 1

3q23p1qp1q 16743632r3 ` 25416768r2 ` 30512064r `

26598352,

´406345115512r3 ´ 616830291616r2 ´

740486023984r ´ 645505557528

3188 r2,´4, 1,´1s 24 p´r3 ` r2 ´ r ` 3q3p´r3 ` r2 ´ r ` 4q8p1q 2788172026368r3 ` 1423837175512r2 `

4939120830288r ´ 3691304019543,

´10952993228320557238r3 ´

5593370421245480720r2 ´

19402732864546458324r `

14500836945256233797

3216 r3, 0,´1,´2s 5 p´r ´ 1q5p1qp1q 16r3 ´ 40r2 ` 48r ´ 20,

104r3 ´ 376r2 ` 816r ´ 608

3271 r´1,´1, 3, 0s 110 pr3 ` r2 ` 3r ` 2q5p´r3 ` r2 ´ 2r ` 2q22p1q 228r3 ´ 115r2 ` 220r ` 132,

6359r3 ´ 2608r2 ´ 6398r ´ 1760

3275 r´9, 6, 2,´1s 19 p´ 19r3 ´ 2

9r2 ´ 8

9r ´ 7

3q19p1qp1q

233r3 ´ 20

3r2 ` 34

3r ´ 14,

4969r3 ´ 1690

9r2 ` 1979

9r ´ 263

33275 r´9, 6, 2,´1s 19 pr ´ 2q19p1qp1q 3r3 ` 30r2 ´ 27r ´ 10,

3329r3 ´ 632

9r2 ´ 7136

9r ` 2693

33284 r´2, 0,´1,´1s 6 pr ´ 1q3p1qprq2 2016r3 ` 1720r2 ` 1160r ` 2161,

´290488r3 ´ 248004r2 ´ 169132r ´ 313401

3407 r´3, 1,´2,´1s 84 p 12r3 ´ 2r ` 1

2q7pr

2 ´ rq12p1q28129013

2r3 ` 15057426r2 ` 3048856r `

407549212

,

´125734882980r3 ´ 134611455788r2 ´

27256382584r ´ 182171881573

3475 r´11, 8,´2,´1s 11 p´ 17r3 ´ 2

7r2 ´ 4

7r ` 1

7q11p1qp1q

617r3 ´ 214

7r2 ´ 351

7r ` 905

7,

´ 16327r3 ` 2420

7r2 ` 6940

7r ´ 10751

73475 r´11, 8,´2,´1s 11 p´rq11p1qp1q ´16r3 ´ 8r2 ` 24r ´ 95,

90087r3 ` 5948

7r2 ´ 8124

7r ` 59025

7


Tables: quartic p2, 1q fields (III)|∆K | fKpxq NmpNq pDm c4pEq, c6pEq

3559 r´2,´1, 3,´2s 20 pr2 ´ 1q5p1qp´r2 ` r ´ 2q4 266r3 ´ 251r2 ` 481r ` 402,

´8721r3 ` 6359r2 ´ 17561r ´ 13594

3571 r3, 5,´5,´1s 45 pr2 ´ 3q3pr2 ` r ´ 5q15p1q ´247r3 ` 1481r2 ` 5186r ` 1954,

313052r3 ` 544864r2 ´ 374892r ´ 240173

3632 r2,´2, 0,´2s 13 pr3 ´ r2 ´ r ´ 1q13p1qp1q 110352r3 ` 24580r2 ` 54624r ´ 99300,

124669648r3 ` 27763200r2 ` 61709112r ´112178880

3632 r2,´2, 0,´2s 14 p´r ´ 1q7p1qp´rq2 2474r3 ` 522r2 ` 1234r ´ 2217,

523532r3 ` 116757r2 ` 258994r ´ 471065

3632 r2,´2, 0,´2s 26 pr3 ´ r2 ´ 3q13p1qp´rq2 10028r3 ` 2232r2 ` 4964r ´ 9023,

´3562482r3´793354r2´1763358r`3205557

3723 r´1, 3, 1,´1s 7 pr3 ´ r2 ` 2r ` 2q7p1qp1q 381r3 ´ 208r2 ´ 592r ` 201,

´9752r3 ´ 3598r2 ` 7307r ´ 1656

3723 r´1, 3, 1,´1s 17 pr ´ 2q17p1qp1q 168r3 ´ 1126r2 ´ 1303r ` 504,

´24313r3 ` 42209r2 ` 63347r ´ 22837

3775 r´11, 7, 0,´1s 19 p 38r3 ´ 1

4r2 ` 1

4r ` 3

8q19p1qp1q 36r3 ` 8r2 ` 32r ` 253,

´662r3 ´ 228r2 ´ 448r ´ 5011

3775 r´11, 7, 0,´1s 19 p 38r3 ´ 1

4r2 ` 1

4r ` 27

8q19p1qp1q ´17r3 ` 30r2 ´ 70r ` 8,

4892r3 ´ 607r2 ` 1283r ´ 1633

23888 r3,´6, 0,´2s 3 p 1

2r3 ´ 1

2r2 ´ 1

2r ´ 5

2q3p1qp1q 12362r3 ` 8406r2 ` 22518r ´ 13842,

7035016r3`4781484r2`12812832r´7875996

3899 r´3, 1, 2,´2s 23 pr3 ´ 2r2 ` r ` 1q23p1qp1q ´14r3 ` 14r2 ´ 25r ´ 21,

381r3 ´ 249r2 ` 364r ` 978

3967 r1, 5,´2,´1s 13 p 12r3 ´ 2r ` 1

2q13p1qp1q

33212r3 ` 1456r2 ´ 2668r ´ 1081

2,

´163448r3 ` 28056r2 ` 583644r ` 107371

3967 r1, 5,´2,´1s 17 p 12r3 ´ 2r ` 5

2q17p1qp1q ´ 3537

2r3 ´ 125r2 ` 4948r ` 1841

2,

´99064r3 ´ 306744r2 ´ 273576r ´ 41165

4108 r´2,´2, 0,´1s 52 pr2 ´ r ` 1q13p´r3 ` 2r2 ´ r ` 2q4p1q ´52r3 ` 56r2 ` 316r ` 177,

3676r3 ´ 2050r2 ´ 1438r ` 1283

4192 r´2,´2, 1, 0s 28 pr2 ` r ` 1q7pr2 ` r ` 2q4p1q 68388r3 ´ 97900r2 ´ 25440r ` 47889,

50048814r3 ´ 57380110r2 ´ 27657416r `22526745

4192 r´2,´2, 1, 0s 44 pr3 ` r ´ 1q11pr2 ` r ` 2q4p1q 993568r3 ´ 1182928r2 ´ 521264r ` 485673,

´2157501576r3 ` 964037714r2 `

2148667444r ` 353613881

4204 r´4,´2, 0, 0s 20 p´r ` 1q5p´rq4p1q 145360531282796r3 ` 161931312392192r2 ´390193058066092r ´ 440654493862007,

´1159392135670300645002r3 `

9949620873463783912066r2 ´

2497558503469317783050r ´

17611520739674724691341

4319 r2,´1,´4,´1s 42 prq2p´r3 ` 2r2 ` 3r ´ 1q21p1q 2626337501r3`4156522706r2`229413693r´

2033846625,

694511908654437r3`1099155960247844r2`60666438159866r ´ 537832894958445

4384 r´4, 0, 3,´2s 10 p 12r3 ´ r2 ` 5

2r ´ 2q5p1qp

12r3 ` 1

2r ` 2q2 ´39342r3 ` 91445r2 ` 10340r ´ 83032,

´8399230r3 ´ 58605841r2 ` 42062128r `73787052

4423 r1, 4,´3,´1s 50 p´r ` 2q5p´r2 ´ r ` 2q10p1q ´4642767r3 ´ 1724885r2 ` 13234188r `

2913911,

´19031399895r3 ´ 11910891879r2 `

44594523793r ` 10072542896

4423 r1, 4,´3,´1s 50 p´r ` 2q5p´r2 ´ r ` 2q10p1q 3516856r3`2151917r2´8338704r´1880324,

´9366159063r3 ` 477887546r2 `

34591729866r ` 7408649776

4564 r1,´5, 0,´1s 5 p 12r3 ` r ´ 3

2q5p1qp1q ´280r3 ` 240r2 ` 64r ` 1449,

10942r3 ´ 8954r2 ´ 1978r ´ 55513

4568 r´1,´3, 2,´1s 12 pr2 ` 2q3p1qpr2 ` 3q4 10845937505r3 ` 4588202505r2 `

28221044093r ` 7621698413,

´1760006389370257r3 ´

744542896235865r2 ´ 4579522784006957r ´1236798376628657

4652 r2, 5,´3,´1s 44 p 12r3 ´ 3

2r ` 2q11p´

12r3 ` 3

2r ´ 1q4p1q ´1938032413r3 ` 62742964314r2 `

143570326721r ` 41574563255,

14844318169935843r3 `

50626339684931473r2 `

49275897864569564r ` 11502761970012547

4663 r2,´5, 2,´1s 11 p´2r3 ` r2 ´ 3r ` 9q11p1qp1q 4296r3 ` 1705r2 ` 10968r ´ 6148,

´3722961r3 ´ 1477666r2 ´ 9510026r `

5330364

4775 r´9,´9, 2,´1s 11 p´ 512r3 ` 2

3r2 ´ 5

6r ` 13

4q11p1qp1q

3074r3 ´ 499r2 ´ 2771

2r ´ 2953

4,

´ 690643

r3 ` 1467853

r2 ´ 1827233

r ´ 87279

4775 r´9,´9, 2,´1s 11 p 16r3 ` 1

3r2 ` 1

3r ´ 1

2q11p1qp1q 247r3 ` 539r2 ´ 163r ´ 335,

´41241r3 ´ 13659r2 ´ 21597r ´ 27719

4832 r´2,´4,´1, 0s 17 p´r3 ` r2 ` 2r ` 5q17p1qp1q ´24r3 ` 17r2 ` 12r ` 82,

580r3 ´ 347r2 ´ 524r ´ 1770

4907 r´1,´4,´2,´1s 11 pr3 ´ r2 ´ 3r ´ 3q11p1qp1q ´191405r3 ´ 287504r2 ´ 336559r ´ 76491,

1214356660r3 ` 1824081112r2 `

2135314036r ` 485335595

4944 r´1,´4,´1, 0s 17 pr3 ´ 4q17p1qp1q 316049736r3 ` 586633069r2 ` 772824316r `170272429,

´23749113529508r3 ´ 44081717952580r2 ´58072797643568r ´ 12794882314805

4979 r1,´3,´1,´1s 13 p´r3 ` r2 ` r ` 1q13p1qp1q 32r3 ´ 128r2 ` 144r ´ 32,

´1464r3 ` 3856r2 ´ 1824r ` 240


Surfaces (joint with X. Guitart and H. Sengun)

This all generalizes to higer dimensional (e.g. 2-dim’l) components.The pairing

H1pΓ0pNq,Zq ˆH1pΓ0pNq,Zq Ñ Cˆpyields, by taking bases of the irreducible factors, a lattice Λ Ă pCˆp q2.Should correspond to the Cp-points of an abelian surface A split at p.From the lattice Λ one can compute the p-adic L-invariant Lp of aMumford–Schottky genus 2 curve.

§ Lp P TbZ Qp.§ Corresponding to a hyperelliptic curve X with JacX “ A.

We can use formulas of Teitelbaum (1988) to recover a Weierstrassequation for X from Lp.From this equation ; approximate Igusa invariants of X.Algebraic recognition algorithms ; actual Igusa invariants.


Toy example: an abelian surface over F “ Q

Consider the Shimura curve Xp15 ¨ 11q, which has genus g “ 9.One of the factors J of JacXp15 ¨ 11q is two-dimensional.T2 acts on J with characteristic polynomial P2pxq “ x2 ` 2x´ 1.We compute a basis tφ1, φ2u of H1pΓ0p15 ¨ 11q,ZqT2“P2 and a“pseudo-dual basis” tθ1, θ2u of H1pΓ0p15 ¨ 11q,ZqT2“P2 .The integration pairing yields a symmetric matrix

ˆ

ş

θ1φ1

ş

θ2φ1

ş

θ1φ2

ş

θ2φ2

˙

“

ˆ

A BB D

˙

.

A “ 3 ¨ 114 ` 3 ¨ 115 ` 4 ¨ 116 ` ¨ ¨ ¨ Òp1124q

B “ 4` 7 ¨ 11` 7 ¨ 112 ` 4 ¨ 113 ` ¨ ¨ ¨ Òp1120q

D “ 9 ¨ 114 ` 9 ¨ 115 ` 8 ¨ 116 ` 9 ¨ 117 ` ¨ ¨ ¨ Òp1121q


Toy example: an abelian surface over F “ Q (II)

This allows to recover the 11-adic L-invariant:

L11pJ2q “ 3 ¨ 11` 8 ¨ 112 ` 3 ¨ 114 ` 9 ¨ 115 ` ¨ ¨ ¨

` p7 ¨ 112 ` 3 ¨ 113 ` 5 ¨ 114 ` 2 ¨ 115 ` ¨ ¨ ¨ q ¨ T2 P TbQp.

We recover the Igusa–Clebsch invariants

pI2 : I4 : I6 : I10q “ p2584 : ´75356 : 37541976 : 2123453113q

Mestre’s algorithm (together with model reduction) yields thehyperelliptic curve

y2 “ ´x6 ` 4x4 ´ 10x3 ` 16x2 ´ 9

After twisting (by ´1 in this case) we get a curve whose first fewEuler factors match with those obtained by the T-action on J .


Example surface over cubic p1, 1q field

Let F “ Qprq, where r3 ´ r2 ` 2r ´ 3 “ 0.Let p7 “ pr

2 ´ r ` 1q.Let BF be the totally definite quaternion algebra of disc. q3 “ prq.

JacXBpΓ0ppqq has a two-dimensional factor J .T2 acts on J with characteristic polynomial P2pxq “ x2 ` x´ 10.A similar calculation as before recovers the 7-adic L-invariant:

L7pJq “ 4 ¨ 7` 2 ¨ 72 ` 5 ¨ 73 ` 3 ¨ 74 ` 3 ¨ 75 ` ¨ ¨ ¨ `Op7300q

` p72 ` 6 ¨ 74 ` 2 ¨ 75 ` 2 ¨ 76 ` ¨ ¨ ¨ `Op7300qq ¨ T2 P TbQ7.

Sadly, haven’t yet been able to recover Igusa–Clebsch invariants.§ (We have about a dozen more examples over cubic and quartic

fields. . . )


Beyond degree 1

F “ quartic totally-complex field, N “ p (for simplicity).

In this setting Γ “ SL2pOF r1psq, which acts on H23.

The relevant groups are now H2pΓ,Div0 Hpq and H2pΓ,Ω1Hpq.

§ H2pSL2pOF q,DivHpq & H2pΓ0ppq,Dq – H2pSL2pOF q, coIndDq.

§ The algorithms of J. Voight and A. Page do not extend to this situation.

What did the modular symbols algorithm teach us?

§ Exploit the cusps. . .

§ . . . use sharblies!


Sharblies and overconvergent leezarbs

Have a short exact sequence of GL2pF q-modules

0 Ñ ∆0 Ñ DivP1pF qdegÑ ZÑ 0, ∆0 “ Div0 P1pF q

Applying the functors p´q bV or Homp´,W q and taking homologyand cohomology yields connecting homomorphims

H2pΓ0ppq, V q

δ

ˆ H2pΓ0ppq,W qX // H0pΓ0ppq, V bW q

H1pΓ0ppq,∆0 bV q ˆ H1pΓ0ppq,Homp∆0,W qq

X //

δ

OO

H0pΓ0ppq, Xq

ev˚

OO

X “ ∆0 bV bHomp∆0,W qevÑ V bW, γ bv bφ ÞÑ v bφpγq.

This diagram is “compatible”:

θ X δpφq “ ev˚pδθ X φq.

Reduced to H1pΓ0ppq,∆0 bV q and H1pΓ0ppq,Homp∆0,W qq.


Sharblies and overconvergent leezarbs (II)

Reduced to H1pΓ0ppq,∆0 bV q and H1pΓ0ppq,Homp∆0,W qq.

Sharblies were invented by Szczarba and Lee (szczarb-lee) tocompute with H1pΓ0ppq,∆

0 bV q.§ Natural generalization of modular symbols to higher rank groups.

‹ Ash–Rudolph, Ash–Gunnells, . . .§ Used to compute structure as hecke modules.

‹ Ash, Gunnells, Hajir, Jones, McConnell, Yasaki, . . .

§ They give an acyclic a resolution of ∆0 bV .§ The analogue of continued fractions algorithm is “sharbly reduction”.

In order to compute H1pΓ0ppq,Homp∆0,W qq, we introduce a dualversion of sharblies, the leezarbs.

§ Leezarbs are an acyclic resolution of Homp∆0,W q.§ Can reuse the sharbly reduction algorithm to work with leezarbs.§ This is work in progress with X. Guitart and A. Page, stay tuned!


Thank you !and

Congratulations to the people of CatalunyaWho have realized that, in the course of human events,

it has become necessary to dissolve the political bandswhich have connected them with Spain.


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