p-adic integration and elliptic curves over number fields

p-adic integration and elliptic curvesover number fields

p-adic Methods in Number Theory, Milano

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Institut fur Experimentelle Mathematik

2University of Warwick

3Sheffield University

October 22, 2014

Marc Masdeu p-adic integration and elliptic curves October 22, 2014 0 / 22

The Machine

Darmon Points

E/F K/F quadratic

P?∈ E(Kab)


The Machine

Non-archimedean

Archimedean

Ramification

Darmon Points

H∗ H∗

Modularity

E/F

K/F quadratic

P?∈ E(Kab)


The Machine

Non-archimedean

Archimedean

Ramification

Darmon Points

H∗ H∗

K/F quadratic

P?∈ Ef (Kab)

f ∈ S2(Γ0(N))


The Machine

Non-archimedean

Archimedean

Ramification

Darmon Points

H∗ H∗

f ∈ S2(Γ0(N))

???


Set-up

K a number field.Fix an ideal N Ă OK .Finitely many iso. classes of EK with condpEq “ N:

ProblemGiven N ą 0, find all elliptic curves of conductor N with

|NmKQpNq| ď N.

K “ Q: Tables by J. Cremona (N “ 350, 000).§ W. Stein–M. Watkins: N “ 108 (N “ 1010 for prime N) incomplete.

K “ Qp?

5q: ongoing project, led by W. Stein (N “ 1831, first rank 2).S. Donnelli–P. Gunnells–A. Kluges-Mundt–D. Yasaki:Cubic field of discriminant ´23 (N “ 1187).


StrategyTwo steps

1 Find a list of elliptic curves of with conductor of norm ď N .2 “Prove” that the obtained list is complete.

1 For (1) the process is as follows:1 List Weierstrass equations of small height.2 Compute their conductors (Tate’s algorithm).3 Compute isogeny graph of the curves in the list.4 Twist existing curves by small primes to get other curves.

2 For (2) use modularity conjecture:1 condpEKq “ N ùñ D automorphic form of level N.2 Compute the fin. dim. space S2pΓ0pNqq, with its Hecke action.3 Match all rational eigenclasses to curves in the list.

3 In this talk: assume modularity when needed.4 One is left with some gaps: some conductor N for which there exists

automorphic newform with rational eigenvalues taqpfquq.§ Problem: Find the elliptic curve attached to taqpfquq.


Goals of the talk

1 Recall the existing analytic constructions of elliptic curves

2 Propose a conjectural p-adic construction

3 Show an example


The case K “ Q: Eichler–Shimura

X0pNq Ñ JacpX0pNqq

ş

–H0pX0pNq,Ω

1X0pNq

q_

H1pX0pNq,ZqHecke CΛf .

Theorem (Manin)There is an isogeny

η : CΛf Ñ Ef pCq.

1 Compute H1pΓ0pNq,Zq (modular symbols).2 Find the period lattice Λf by explicitly integrating

Λf “

C

ż

γ2πi

ÿ

ně1

anpfqe2πinz : γ P H1pX0pNq,Zq

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 .


K ‰ Q. Existing constructionsK totally real. rK : Qs “ n, fix σ : K ãÑ R.

S2pΓ0pNqq Q f ; ωf P HnpX0pNq,Cq; Λf Ď C.

Conjecture (Oda, Darmon, Gartner)CΛf is isogenous to Ef bK Kσ.

Known to hold (when F real quadratic) for base-change of EQ.Exploited in very restricted cases (Dembele, . . . ).Explicitly computing Λf is hard –no quaternionic computations–.

K not totally real: no known algorithms. In fact:

TheoremIf K is imaginary quadratic, the lattice Λf is contained in R.

IdeaTry instead a non-archimedean construction!


Non-archimedean construction

From now on: assume there is some p such that p ‖ N.Replace the role of R (and C) with Kp (and its extensions).

Theorem (Tate uniformization)Let EK be an elliptic curve of conductor N, and let p ‖ N. There exists arigid-analytic, Galois-equivariant isomorphism

η : Kˆp ΛE Ñ EpKpq,

where ΛE “ qZE , with qE P Kp satisfyingjpEq “ q´1

E ` 744` 196884qE ` ¨ ¨ ¨ .

Suppose D coprime factorization N “ pDm, with D “ discpBKq.§ Always possible when K has at least one real place.


Quaternionic modular forms of level NSuppose K has signature pr, sq, and fix N “ pDm.BK the quaternion algebra such that

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

Fix isomorphisms v1, . . . , vn : B bKvi –M2pRq andw1, . . . ws : B bKwj –M2pCq, yielding

BˆKˆ ãÑ PGL2pRqn ˆ PGL2pCqs ýHn ˆ Hs3.

Fix RD0 ppmq Ă RD

0 pmq Ă B Eichler orders of level pm and m.ΓD

0 ppmq “ RD0 ppmq

ˆOˆK acts discretely on Hn ˆ Hs3.Obtain a manifold of (real) dimension 2n` 3s:

Y D0 ppmq “ ΓD

0 ppmqz pHn ˆ Hs3q .

Y D0 ppmq is compact ðñ B is division (assume it, for simplicity).


Group cohomology

The cohomology of Y D0 ppmq can be computed via

H˚pY D0 ppmq,Cq – H˚pΓD

0 ppmq,Cq.

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

f P Hn`spΓD0 ppmq,Cq eigen for TD is rational if aqpfq P Z,@q P TD.

Conjecture (Taylor, ICM 1994)

Let f P Hn`spΓD0 ppmq,Zq be a new, rational eigenclass. Then there is

an elliptic curve EfK of conductor N such that

#Ef pOKqq “ 1` |q| ´ appfq @q - N.

GoalMake this (conjecturally) constructive.


Non-archimedean path integralsHp “ P1pCpqr P1pKpq has a rigid-analytic structure.PGL2pKpq acts on Hp through fractional linear transformations:

`

a bc d

˘

¨ z “az ` b

cz ` d, z P Hp.

We consider rigid-analytic 1-forms ω P Ω1Hp

.Given two points τ1 and τ2 in Hp, define:

ż τ2

τ1

ω “ Coleman integral.

Get a PGL2pKpq-equivariant pairingż

: Ω1HpˆDiv0 Hp Ñ Cp.

For each Γ Ă PGL2pKpq, induce a pairingż

: H ipΓ,Ω1Hpq ˆHipΓ,Div0 Hpq Ñ Cp.


Coleman Integration

Coleman integration on Hp can be defined as:ż τ2

τ1

ω “

ż

P1pKpq

logp

ˆ

t´ τ2

t´ τ1

˙

dµωptq “ limÝÑU

ÿ

UPUlogp

ˆ

tU ´ τ2

tU ´ τ1

˙

resApUqpωq.

Bruhat-Tits tree of GL2pKpq, |p| “ 2.Hp having the Bruhat-Tits as retract.Annuli ApUq for a covering of size |p|´3.tU is any point in U Ă P1pKpq.

P1(Kp)

U ⊂ P1(Kp)


The tpu-arithmetic group Γ

Recall we have chosen a factorization N “ pDm.BK “ chosen quaternion algebra of discriminant D.

Recall also RD0 ppmq Ă RD

0 pmq Ă B.Define ΓD

0 ppmq “ RD0 ppmq

ˆ and ΓD0 pmq “ RD

0 pmqˆ.

SetΓ “ ΓD

0 pmq ‹ΓD0 ppmq

ΓD0 pmq,

ΓD0 pmq “ wpΓ

D0 pmqw

´1p .

Fix an embedding ιp : R0 ãÑM2pZpq.

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

ΓΓD0 pmq – V0, ΓΓD

0 ppmq – E0

V0 (resp. E0) are the even vertices (resp. edges) of the BT tree.


Cohomology (I)

Γ “ RD0 pmqr1ps

ˆOKr1psˆ ιp

ãÑ PGL2pKpq.

Consider the Γ-equivariant exact sequence

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

f // rv ÞÑř

opeq“v fpeqs

Have Γ-equivariant isomorphisms

MapspE0pT q,Zq – IndΓΓD0 ppmq

Z, MapspVpT q,Zq –´

IndΓΓD0 pmq

Z¯2.

So get:

0 Ñ HCpZq Ñ IndΓΓD0 ppmq

Z ∆Ñ

´

IndΓΓD0 pmq

Z¯2Ñ 0

Taking Γ-cohomology and using Shapiro’s lemma gives

Hn`spΓ,HCpZqq Ñ Hn`spΓD0 ppmq,Zq

∆Ñ Hn`spΓD

0 pmq,Zq2 Ñ ¨ ¨ ¨


Cohomology (II)

Taking Γ-cohomology and using Shapiro’s lemma gives

Hn`spΓ,HCpZqq Ñ Hn`spΓD0 ppmq,Zq

∆Ñ Hn`spΓD

0 pmq,Zq2 Ñ ¨ ¨ ¨

f P Hn`spΓD0 ppmq,Zq being p-new ùñ f P Kerp∆q.

Pulling back, get ωf P Hn`spΓ,HCpZqq.

ωf P Hn`spΓ,Meas0pP1pKpqqq.


Holomogy cyclesConsider the Γ-equivariant s.e.s:

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

Let LppEq “logppqEq

ordppqEq(p-adic L-invariant).

ConjectureThe set

Λf “

#

ż

δpcqωf : c P Hn`s`1pΓ,Zq

+

Ă Cp

is infinite and contained in the line ZLppEq.

Known when K “ Q (Darmon, Dasgupta–Greenberg,Longo–Rotger–Vigni), open in general.


Lattice: explicit construction

Start f P Hn`spΓD0 ppm,Zqq.

Duality yields f P Hn`spΓD0 ppm,Zqq.

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

0 ppmqΓD

0 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.

ConjectureThe element

Lf “ż

δpcf qωf .

is a nonzero multiple of LppEq.


Algorithms

Only when n` s ď 1.

Use explicit presentation for ΓD0 ppmq and ΓD

0 pmq.§ s “ 0 ùñ J. Voight.§ s “ 1 ùñ A. Page.

Compute the Hecke action on H1pΓD0 ppm,Zqq and H1pΓ

D0 ppm,Zqq.

Integration pairing uses overconvergent cohomology.§ Lift f to overconvergent class F P Hn`spΓD

0 ppmq,Dq.§ Use F to to recover moments the measures ωf pγq.


Overconvergent Method (I)Starting data: cohomology class Φ “ ωf P H

1pΓ,Ω1Hpq.

Goal: to compute integralsşτ2τ1

Φγ , for γ P Γ.Recall that

ż τ2

τ1

Φγ “

ż

P1pKpq

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:

ρ : H1pΓD0 ppmq,Dq Ñ H1pΓD

0 ppmq,Zpq.

Theorem (Pollack-Stevens, Pollack-Pollack)There exists a unique Up-eigenclass F lifting f .

Moreover, F 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 G : ΓD

0 ppmq Ñ D:

γ ÞÑ”

hptq ÞÑ

ż

Zp

hptqdµγptqı

.

1 G belongs to H1pΓD0 ppmq,Dq.

2 G is a lift of f .3 G is a Up-eigenclass.

CorollaryThe explicitly computed F knows the above integrals.


Recovering E from Lf

Start with q “ log´1p pLf q P Cˆp , assume ordppqq ą 0.

Getjpqq “ q´1 ` 744` 196884q ` ¨ ¨ ¨ P Cˆp .

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

From j “ c34∆ recover c4 P Cp.

Try to recognize c4 algebraically.From 1728∆ “ c3

4 ´ c26 recover c6.

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

c6864 .

§ If conductor is correct, check ap’s.


Example curveK “ Qpαq, pαpxq “ x4 ´ x3 ` 3x´ 1, ∆K “ ´1732.N “ pα´ 2q “ P13.BK of ramified only at all infinite real places of K.There is a rational eigenclass f P S2pΓ0p1,Nqq.From f we compute ωf P H1pΓ,Ω1

Hpq and c P H2pΓ,Zq.

qE “ ˆş

δc ωf “ 8 ¨ 13` 11 ¨ 132 ` 5 ¨ 133 ` 3 ¨ 134 ` ¨ ¨ ¨ `Op13100q.jE “ 1

13

´

´ 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.


Thank you !

Bibliography, code and slides at:http://www.warwick.ac.uk/mmasdeu/


http://www.warwick.ac.uk/mmasdeu/

p-adic integration and elliptic curves over number fields

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