darmon points for fields of mixed signature

Darmon points for number fields of mixed signatureLondon Number Theory Seminar, UCL

February 4th, 2015

Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3

1Universitat de Barcelona

2University of Warwick

3Sheffield University

Marc Masdeu Darmon points 0 / 24

Plan

1 Two Theorems

2 Two Conjectures

3 Two Effective Conjectures

4 Two (or three) Examples


1 Two Theorems

2 Two Conjectures




Two prototypical theorems (I)

Fix an integer N ě 1 (a level/conductor)Let f P S2pΓ0pNqq

new, f “ q `ř

ně2 anpfqqn, with anpfq P Z.

f ; ωf “ 2πifpzqdz P Ω1X0pNq

, where X0pNq “ Γ0pNqzH.

Consider the lattice Λf “!

ş

γ ωf

ˇ

ˇ

ˇγ P H1pX0pNq,Zq

)

.

Theorem 1 (Eichler–Shimura–Manin)Dη : CΛf – EpCq, where EQ is an elliptic curve, condpEq “ N , and

#EpFpq “ p` 1´ appfq for all p - N.


Two prototypical theorems (II)

Let KQ be an imaginary quadratic field.Suppose that ` | N ùñ ` split in K.

§ Heegner hypothesis, to force signpLpEK, sqq “ ´1.

Given τ P K XH, set Pτ “ η

ˆż τ

i8ωf

˙

P EpCq.

Theorem 2 (Shimura, Gross–Zagier)1 Pτ P EpHτ q, where Hτ K is a ring class field attached to τ .2 TrpPτ q P EpKq nontorsion ðñ L1pEK, 1q ‰ 0.

Generalizes to CM fields (F totally real and KF totally complex).


1 Two Theorems

2 Two Conjectures




Quaternionic automorphic forms of level NF a number field of signature pr, sq.v1, . . . , vr : F ãÑ R, w1, . . . , ws : F ãÑ C.Consider a coprime factorization N “ Dn with D squarefree.Let BF be the 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 ˆ Hs3.R>0

C

H3

R>0

H

R

PGL2(R)PGL2(C)


Quaternionic automorphic forms of level N (II)

Fix RB0 pnq Ă B Eichler order of level n.ΓB0 pnq “ RB0 pnq

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

Y D0 pnq “ ΓB0 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˚pΓB0 pnq,Cq.

Hecke algebra TD “ ZrTq : q - Ds acts on H˚pΓB0 pnq,Zq.

Definitionf P Hn`spΓB0 pnq,Cq eigen for TD is rational if appfq P Z,@p P TD.


Elliptic curves from cohomology classes

Conjecture 1 (Taylor, ICM 1994)Let f P Hn`spΓB0 pnq,Zq be a new, rational eigenclass.Then DEfF of conductor N “ Dn such that

#Ef pOF pq “ 1` |p| ´ appfq @p - N.

When F is totally imaginary, instead of Ef we may get an abeliansurface AfF with QM defined over F (a.k.a. fake elliptic curve).In fact, fake elliptic curves can only arise when:

§ F is totally imaginary, and§ N is square-full: p | N ùñ p2 | N.

Our construction automatically rules out this setting.However: the conjecture above is not effective: it doesn’t give acandidate Ef .


Algebraic points from quadratic extensionsSuppose E “ Ef is 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.

Modularity of E ùñ§ Analytic continuation of LpEK, sq to C.§ Functional equation relating sØ 2´ s.

Conjecture 2 (very coarse version of BSD conjecture)Suppose that ords“1 LpEK, sq “ 1. Then DPK P EpKq of infinite order.

Known results rely on existence of Heegner points.Without Heegner points, don’t even have candidate for PK .


1 Two Theorems

2 Two Conjectures




Notation for Darmon pointsF a number field, KF a quadratic extension.

n` s “ #tv | 8F : v splits in Ku “ rkZOˆKOˆF .

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, ¨q should be p´1q#SpE,Kq.§ From now on, we assume that this is odd.§ #SpE,Kq “ 1 ùñ split automorphic forms,§ #SpE,Kq ą 1 ùñ quaternionic automorphic forms.

Fix a place ν P SpE,Kq.1 If ν “ p is finite ùñ non-archimedean construction.2 If ν is infinite ùñ archimedean construction.


Previous constructions of Darmon points

Non-archimedean

§ H. Darmon (1999): F “ Q, split.

§ M. Trifkovic (2006): F “ Qp?´dq ( ùñ KF quasi-CM), split.

§ M. Greenberg (2008): F totally real, quaternionic.

Archimedean

§ H. Darmon (2000): F totally real, split.

§ J. Gartner (2010): F totally real, quaternionic.

Generalizations§ Rotger–Longo–Vigni: lift to Jacobians of Shimura curves.§ Rotger–Seveso: cycles attached to higher weight modular forms.


Integration PairingLet Hν “ the ν-adic upper half plane. That is:

§ The Poincare upper half-plane H if ν is infinite,§ The p-adic (upper) half-plane Hp if ν “ p is finite.

As a set, we have (up to taking connected component):

HνpKνq “ P1pKνqr P1pFνq.

Hν comes equipped with an analytic structure (complex- or rigid-).When ν is infinite, there is a natural PGL2pRq-equivariant pairing

Ω1H ˆDiv0 HÑ C “ Kν ,

which sendspω, pτ2q ´ pτ2qq ÞÑ

ż τ2

τ1

ω P C.

Analogously, Coleman integration gives a natural pairing

Ω1HνˆDiv0 Hν Ñ Kν .


Rigid one-forms and measuresThe assignment

µ ÞÑ ω “

ż

P1pFpq

dz

z ´ tdµptq

induces an isomorphism Meas0pP1pFpq,Zq – Ω1Hp,Z.

Inverse given by ω ÞÑ“

U ÞÑ µpUq “ resApUq ω‰

.

Theorem (Teitelbaum)ż τ2

τ1

ω “

ż

P1pFpq

log

ˆ

t´ τ1

t´ τ2

˙

dµptq “ limÝÑU

ÿ

UPUlog

ˆ

tU ´ τ1

tU ´ τ2

˙

µpUq.

If the residues of ω are all integers, have a multiplicative refinement:

ˆ

ż τ2

τ1

ω “ limÝÑU

ź

UPU

ˆ

tU ´ τ2

tU ´ τ1

˙µpUq

P Kˆp .


The group Γ

BF “ quaternion algebra with RampBq “ SpE,Kqr tνu.Write N “ Dn (or N “ pDn if ν “ p).Fix Eichler order RB0 pnq Ă B, and a splitting ιν : RB0 pnq ãÑM2pFνq.Let Γ “ RB0 pnqr1νs

ˆOF r1νsˆ ιν

ãÑ PGL2pFνq.

Example (archimedean)F real quadratic, SpE,Kq “ tνu.This gives B “M2pF q.Γ “ Γ0pNq “

`

a bc d

˘

P GL2pOF q : c P N(

OˆF Ă PGL2pOF q.

Example (non-archimedean)F “ Q, N “ pM , SpE,Kq “ tpu.This gives B “M2pQq.Γ “

`

a bc d

˘

P GL2pZr1psq : M | c(

t˘1u ãÑ PGL2pQq Ă PGL2pQpq.


Overview of the construction

Need to assume that F has narrow class number 1.We attach to E a cohomology class

ΦE P Hn`s

`

Γ,Ω1Hν

˘

.

We attach to each embedding ψ : K ãÑ B a homology class

Θψ P Hn`s

`

Γ,Div0 Hν

˘

.

§ Well defined up to the image of Hn`s`1pΓ,ZqδÑ Hn`spΓ,Div0 Hνq.

Cap-product and integration on the coefficients yield an element:

Jψ “ ˆ

ż

Θψ

ΦE P Kˆν .

Jψ is well-defined up to the lattice L “!

ˆş

δpθqΦE : θ P Hn`s`1pΓ,Zq)

.


Conjectures

Conjecture 1 (Oda, Yoshida, Greenberg, Guitart-M-Sengun)There is an isogeny η : Kˆ

ν LÑ EpKνq.

Dasgupta–Greenberg, Rotger–Longo–Vigni: some non-arch. cases.Completely open in the archimedean case.

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

Pψ “ ηpJψq P EpKνq.

Conjecture 2 (Darmon, Greenberg, Trifkovic, Gartner, G-M-S)1 The local point Pψ is global, defined over EpHψq.2 For all σ P GalpHψKq, σpPψq “ Precpσq¨ψ (Shimura reciprocity).3 TrHψKpPψq is nontorsion if and only if L1pEK, 1q ‰ 0.


Cohomology class attached to E

Recall that SpE,Kq and ν determine:

Γ “ ιν`

RB0 pnqr1νsˆ˘

Ă PGL2pFνq.

Theorem (Darmon, Greenberg, Trifkovic, Gartner, G.–M.–S.)There exists a unique (up to sign) class

ΦE P Hn`s

`

Γ,Ω1Hν

˘

such that:

1 TqΦE “ aqΦE for all q - N.2 UqΦE “ aqΦE for all q | N.3 ΦE is “integrally valued”.

Idea: relate Hn`spΓ,Ω1Hνq with H˚pΓB0 pNq,Cq (as Hecke modules).


Homology classes attached to orders O Ă KLet ψ : O ãÑ RB0 pnq be an embedding of an order O of K.

§ Which is optimal: ψpOq “ RB0 pnq X ψpKq.Consider the group Oˆ1 “ tu P Oˆ : NmKF puq “ 1u.

§ rankpOˆ1 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.

KˆFˆψ

ãÑ BˆFˆινãÑ PGL2pFνq ýHν .

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

Hn`s`1pΓ,Zqδ // Hn`spΓ,Div0 Hνq // Hn`spΓ,DivHνq

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 Hνq.§ Well defined up to δpHn`s`1pΓ,Zqq.


1 Two Theorems

2 Two Conjectures




Example curveF “ Qprq with r4 ´ r3 ` 3r ´ 1 “ 0.F has signature p2, 1q and discriminant ´1732.N “ pr ´ 2q “ p “ p13.BF ramified only at all infinite real places of F .There is a rational eigenclass f P S2pΓ

B0 pNqq.

From f we compute ωf P H1pΓ,Ω1Hpq and Λf “ qZf .

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

13

´

´ 4656377430074r3 ` 10862248656760r2 ´ 14109269950515r ` 4120837170980¯

.

EF : y2 ``

r3 ` r ` 3˘

xy “ x3`

``

´2r3 ` r2 ´ r ´ 5˘

x2

``

´56218r3 ´ 92126r2 ´ 12149r ` 17192˘

x

´ 23593411r3 ` 5300811r2 ` 36382184r ´ 12122562.


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:

ψ : 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`¨ ¨ ¨

This yields Θψ P H1pΓ,Div0 Hpq.After integrating we get:

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.


Archimedean cubic Darmon pointLet F “ Qprq with r3 ´ r2 ` 1 “ 0.F signature p1, 1q and discriminant ´23.Consider the elliptic curve EF given by the equation:

EF : y2 ` pr ´ 1qxy ``

r2 ´ r˘

y “ x3 ``

´r2 ´ 1˘

x2 ` r2x.

E has prime conductor NE “`

r2 ` 4˘

of norm 89.K “ F pαq, with α2 ` pr ` 1qα` 2r2 ´ 3r ` 3 “ 0.

§ K has class number 1, thus we expect the point to be defined over K.SpE,Kq “ tσu, where σ : F ãÑ R is the real embedding of F .

§ Therefore the quaternion algebra B is just M2pF q.The arithmetic group to consider is

Γ “ Γ0pNEq Ă PGL2pOF q.

Γ acts naturally on the symmetric space HˆH3:

HˆH3 “ tpz, x, yq : z P H, x P C, y P Rą0u.


Archimedean cubic Darmon point (II)E ; ωE , an automorphic form with Fourier-Bessel expansion:

ωEpz, x, yq “ÿ

αPδ´1OFα0ą0

apδαqpEqe´2πipα0z`α1x`α2xqyH pα1yq ¨

ˆ

´dx^dzdy^dzdx^dz

˙

Hptq “

ˆ

í

2eiθK1p4πρq,K0p4πρq,

i

2eíθK1p4πρq

˙

t “ ρeiθ.

§ K0 and K1 are hyperbolic Bessel functions of the second kind:

K0pxq “

ż 8

0

e´x coshptqdt, K1pxq “

ż 8

0

e´x coshptq coshptqdt.

ωE is a 2-form on Γz pHˆH3q.The cocycle ΦE is defined as (γ P Γ):

ΦEpγq “

ż γÖ

OωEpz, x, yq P Ω1

H with O “ p0, 1q P H3.


Archimedean cubic Darmon point (III)Consider the embedding ψ : K ãÑM2pF q given by:

α ÞÑ

ˆ

´2r2 ` 3r r ´ 3r2 ` 4 2r2 ´ 4r ´ 1

˙

Let γψ “ ψpuq, where u is a fundamental norm-one unit of OK .γψ fixes τψ “ ´0.7181328459824` 0.55312763561813i P H.

§ Construct Θ1ψ “ rγψ b τψs P H1pΓ,DivHq.Θ1ψ is equivalent to a cycle

ř

γi bpsi ´ riq taking values in Div0H.

Jψ “ÿ

i

ż si

ri

ΦEpγiq “ÿ

i

ż γi¨O

O

ż si

ri

ωEpz, x, yq.

We obtain, summing over all ideals pαq of norm up to 400, 000:

Jψ “ 0.0005281284234` 0.0013607546066i; Pψ P EpCq.

Numerically (up to 32 decimal digits) we obtain:

Pψ?“ ´10ˆ

`

r ´ 1, α´ r2 ` 2r˘

P EpKq.


Conclusion

For all quadratic KF , proposed an analytic construction ofcandidate PK P EpKνq whenever ords“1 LpEK, sq “ 1.

Heegner points are a very special case of the construction.

The TruthTM covers much more than the CM paradigm.

Along the way, give the first systematic method to find equationsfor elliptic curves using automorphic forms over non-totally-realfields.

We have extensive numerical evidence.§ However, nothing beyond quasi-CM!


What’s next

Equations for abelian surfaces of GL2-type (in progress).

Computing in H2 and H2, maybe using sharblies?

Higher class numbers (technical and computational difficulties).

Reductive groups other than GL2? (Please help!)


Thank you !

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


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

Cohomology

Γ “ RB0 pnqr1psˆOF r1ps

ˆ ιpãÑ 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 getΦf P H

n`spΓ,HCpZqq.


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.


Overconvergent MethodStarting data: cohomology class Φ 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.


darmon points for fields of mixed signature

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