Non-archimedean construction of elliptic curves and rational points

Non-archimedean constructionof elliptic curves

and rational pointsNumber Theory Seminar, Sheffield

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Universitat de Barcelona

2University of Warwick

3Sheffield University

December 9th, 2014

Marc Masdeu Non-archimedean constructions December 9th , 2014 0 / 30

Plan

1 Quaternionic automorphic forms and elliptic curves

2 Darmon points

3 The overconvergent method


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 squarefree.v1, . . . , vr : F ãÑ R, w1, . . . , ws : F ãÑ C.Let BF be the quaternion algebra such that

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

R>0

C

H3

R>0

H

R

PGL2(R)PGL2(C)

Fix isomorphisms

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

These yield BˆFˆ ãÑ PGL2pRqn ˆ PGL2pCqs ýHn ˆ Hs3.


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 ˆ Hs3.

Obtain a manifold of (real) dimension 2n` 3s:

Y D0 pnq “ ΓD

0 pnqz pHn ˆ Hs3q .

Y D0 pnq is compact ðñ B is division.

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.

Definitionf P Hn`spΓD

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


Elliptic curves from cohomology classes

Definitionf P Hn`spΓD

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

Conjecture (Taylor, ICM 1994)

f P Hn`spΓD0 pnq,Zq a new, rational eigenclass.

Then DEfF of conductor N “ Dn such that

#Ef pOF pq “ 1` |p| ´ appfq @p - N.

To avoid fake elliptic curves, assume N is not square-full: Dp ‖ N.

First Goal of the talkMake this conjecture (conjecturally) constructive.


The case F “ Q: Cremona’s algorithm

Eichler–Shimura

X0pNq Ñ JacpX0pNqq

ş

–H0

`

X0pNq,Ω1˘_

H1pX0pNq,ZqHecke CΛf Ñ 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. Existing constructions

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, . . . ).Explicitly computing Λf is hard –no quaternionic computations–.

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

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.

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 ` ¨ ¨ ¨ .

Suppose D coprime factorization N “ pDm, with D “ discpBF q.§ . . . always possible when F has at least one real place.

Compute qE as a replacement for Λf .


Non-archimedean path integrals on Hp

Consider Hp “ P1pCpqr P1pFpq.It is the right 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

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 refinent (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

.


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.

Marc Masdeu Non-archimedean constructions December 9th, 2014 10 / 30

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 – IndΓΓD0 ppmq

Z, MapspVpT q,Zq –´

IndΓΓD0 pmq

Z¯2.


Cohomology

Γ “ RD0 pmqr1ps

ˆOF r1psˆ ιp

ãÑ PGL2pFpq.

MapspE0pT q,Zq – IndΓΓD0 ppmq

Z, MapspVpT q,Zq –´

IndΓΓD0 pmq

Z¯2.

Consider the Γ-equivariant exact sequence

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

ϕ // rv ÞÑř

opeq“v ϕpeqs

So get:

0 Ñ HCpZq Ñ IndΓΓD0 ppmq

Z ∆Ñ

´

IndΓΓD0 pmq

Z¯2Ñ 0


Cohomology (II)

0 Ñ HCpZq Ñ IndΓΓD0 ppmq

Z ∆Ñ

´

IndΓΓD0 pmq

Z¯2Ñ 0

Taking Γ-cohomology,. . .

Hn`spΓ,HCpZqq Ñ Hn`spΓ, IndΓΓD0 ppmq

,Zq ∆Ñ Hn`spΓ, IndΓ

ΓD0 pmq

,Zq2 Ñ ¨ ¨ ¨

. . . and using Shapiro’s lemma:

Hn`spΓ,HCpZqq Ñ Hn`spΓD0 ppmq,Zq

∆Ñ Hn`spΓD

0 pmq,Zq2 Ñ ¨ ¨ ¨

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

Pulling back getωf P H

n`spΓ,HCpZqq.


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.Open in general.


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

F “ Qpαq, pαpxq “ x4 ´ x3 ` 3x´ 1, ∆F “ ´1732.N “ pα´ 2q “ P13.BF of 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.


The Machine

Non-archimedean

Archimedean

Ramification

Periods Machine

H∗ H∗

f ∈ S2(Γ0(N))

Ef ?


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∗

Modularity

E/FK/F quadratic

P?∈ E(Kab)


The Machine

Darmon Points

E/F K/F quadratic

P?∈ E(Kab)


Rational points on elliptic curves

Suppose we have EF attached to f .Let KF be a quadratic extension of F .

§ Assume that N is square-free, coprime to discpKF q.

Hasse-Weil L-function of the base change of E to K (<psq ąą 0)

LpEK, sq “ź

p|N

`

1´ ap|p|´s˘´1

ˆź

p-N

`

1´ ap|p|´s ` |p|1´2s

˘´1.

Coarse version of BSD conjecture

ords“1 LpEK, sq “ rkZEpKq.

So ords“1 LpEK, sq oddBSDùñ DPK P EpKq of infinite order.

Second goal of the talkFind PK explicitly (at least conjecturally).


Heegner Points (KQ imaginary quadratic)

Use crucially that E is attached to f .

ωf “ 2πifpzqdz P H0pΓ0pNq,Ω1Hq.

Given τ P K XH, set Jτ “ż τ

i8ωf P C.

Well-defined up to the lattice Λf “!

ş

γ ωf | γ P H1 pΓ0pNq,Zq)

.§ There exists an isogeny (Weierstrass uniformization)

η : CΛf Ñ EpCq.

§ Set Pτ “ ηpJτ q P EpCq.Fact: Pτ P EpHτ q, where Hτ K is a ring class field attached to τ .

Theorem (Gross-Zagier)

PK “ TrHτ KpPτ q nontorsion ðñ L1pEK, 1q ‰ 0.


Heegner Points: revealing the trick

Why did this work?

1 The Riemann surface Γ0pNqzH has an algebraic model X0pNqQ.

2 There is a morphism φ defined over Q:

φ : JacpX0pNqq Ñ E.

3 The CM point pτq ´ p8q P JacpX0pNqqpHτ q gets mapped to:

φppτq ´ p8qq “ Pτ P EpHτ q.


Darmon’s insight

Henri Darmon

Drop hypothesis of KF being CM.§ Simplest case: F “ Q, K real quadratic.

However:§ There are no points on JacpX0pNqq attached to such K.§ In general there is no morphism φ : JacpX0pNqq Ñ E.§ When F is not totally real, even the curve X0pNq is missing!


New notation

Still assume p ‖ N “ condpEq.The triple pE,K, pq determines uniquely the quaternion algebra B:

RampBq “ SpE,Kqr tpu.

Set n` s “ #tv | 8F : v splits in Ku.KF is CM ðñ n` s “ 0.

§ If n` s “ 1 we call KF quasi-CM.

SpE,Kq “!

v | N8F : v not split in K)

.

Sign of functional equation for LpEK, sq should be p´1q#SpE,Kq.§ From now on, we assume that #SpE,Kq is odd.

Assume there is a finite prime p P SpE,Kq.§ If p was an infinite place ùñ archimedean case (not today).


Homology classes attached to K

Let ψ : O ãÑ RD0 pmq be an embedding of an order O of K.

§ Which is optimal: ψpOq “ RD0 pmq X ψpKq.

Consider the group Oˆ1 “ tu P Oˆ : NmKF puq “ 1u.§ rankpOˆ

1 q “ rankpOˆq ´ rankpOˆF q “ n` s.

Choose a basis u1, . . . , un`s P Oˆ1 for the non-torsion units.§ ; ∆ψ “ ψpu1q ¨ ¨ ¨ψpun`sq P Hn`spΓ,Zq.

Kˆ acts on Hp through Kˆ ψãÑ Bˆ

ιpãÑ GL2pFpq.

§ Let τψ be the (unique) fixed point of Kˆ on Hp.

Hn`s`1pΓ,Zqδ // Hn`spΓ,Div0 Hpq // Hn`spΓ,DivHpq

deg// Hn`spΓ,Zq

Θψ ? // r∆ψ b τψs

// r∆ψs

Fact: r∆ψs is torsion.§ Can pull back a multiple of r∆ψ b τψs to Θψ P Hn`spΓ,Div0 Hpq.§ Well defined up to δpHn`s`1pΓ,Zqq.


Conjectures

Jψ “ ˆ

ż

Θψ

ωf P Kˆp Λf .

Conjecture A (restated)There is an isogeny β : Kˆ

p Λf Ñ EpKpq.

The Darmon point attached to E and ψ : K Ñ B is:

Pψ “ βpJψq P EpKpq.

Conjecture B (Darmon, Greenberg, Trifkovic, G-M-S)1 The local point Pψ is global, and belongs to EpKabq.2 Pψ is nontorsion if and only if L1pEK, 1q ‰ 0.


Non-archimedean cubic Darmon point

F “ Qprq, with r3 ´ r2 ´ r ` 2 “ 0.F has signature p1, 1q and discriminant ´59.Consider the elliptic curve EF given by the equation:

EF : y2 ` p´r ´ 1qxy ` p´r ´ 1q y “ x3 ´ rx2 ` p´r ´ 1qx.

E has conductor NE “`

r2 ` 2˘

“ p17q2, where

p17 “`

´r2 ` 2r ` 1˘

, q2 “ prq .

Consider K “ F pαq, where α “?´3r2 ` 9r ´ 6.

The quaternion algebra BF has discriminant D “ q2:

B “ F xi, j, ky, i2 “ ´1, j2 “ r, ij “ ´ji “ k.


Non-archimedean cubic Darmon point (II)

The maximal order of K is generated by wK , a root of the polynomial

x2 ` pr ` 1qx`7r2 ´ r ` 10

16.

One can embed OK in the Eichler order of level p17 by:

wK ÞÑ p´r2 ` rqi` p´r ` 2qj ` rk.

We obtain γψ “ 6r2´72 ` 2r`3

2 i` 2r2`3r2 j ` 5r2´7

2 k, and

τψ “ p12g`8q`p7g`13q17`p12g`10q172`p2g`9q173`p4g`2q174`¨ ¨ ¨

After integrating we obtain:

Jψ “ 16`9¨17`15¨172`16¨173`12¨174`2¨175`¨ ¨ ¨`5¨1720`Op1721q,

which corresponds to:

Pψ “ ´108ˆ

ˆ

r ´ 1,α` r2 ` r

2

˙

P EpKq.


What’s next

Equations for abelian surfaces of GL2-type.

Computing in H2 and H2 (sharblies?)

Reductive groups other than GL2.

Higher class numbers ( ùñ Γ non-transitive on T ).


Thank you !

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


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

Overconvergent Method

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

Hpq.

Goal: to compute integralsşτ2τ1

Φγ , for γ P Γ.Recall that

ż 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.

P

f

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 Ψ: ΓD

0 ppmq Ñ D:

γ ÞÑ”

hptq ÞÑ

ż

Zp

hptqdµγptqı

.

1 Ψ belongs to H1´

ΓD0 ppmq,D

¯

.

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

Corollary

The explicitly computed Φ “ Ψ knows the above integrals.


Thank you !(now, for real)

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


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

Non-archimedean construction of elliptic curves and rational points

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