non-archimedean construction of elliptic curves and rational points
TRANSCRIPT
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
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Plan
1 Quaternionic automorphic forms and elliptic curves
2 Darmon points
3 The overconvergent method
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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.
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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.
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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.
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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 .
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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!
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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 .
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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.
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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
.
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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
ˆOˆF and ΓD0 pmq “ RD
0 pmqˆOˆF .
Let Γ “ RD0 pmqr1ps
ˆOF 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.
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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.
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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 Ƅ
´
IndΓΓD0 pmq
Z¯2Ñ 0
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Cohomology (II)
0 Ñ HCpZq Ñ IndΓΓD0 ppmq
Z Ƅ
´
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.
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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.
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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.
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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.
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The Machine
Non-archimedean
Archimedean
Ramification
Periods Machine
H∗ H∗
f ∈ S2(Γ0(N))
Ef ?
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The Machine
Non-archimedean
Archimedean
Ramification
Darmon Points
H∗ H∗
K/F quadratic
P?∈ Ef (Kab)
f ∈ S2(Γ0(N))
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The Machine
Non-archimedean
Archimedean
Ramification
Darmon Points
H∗ H∗
Modularity
E/FK/F quadratic
P?∈ E(Kab)
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The Machine
Darmon Points
E/F K/F quadratic
P?∈ E(Kab)
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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).
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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.
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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.
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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!
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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).
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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.
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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.
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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.
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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.
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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 ).
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Thank you !
Bibliography, code and slides at:http://www.warwick.ac.uk/mmasdeu/
Marc Masdeu Non-archimedean constructions December 9th, 2014 27 / 30
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
+
.
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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.
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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.
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Thank you !(now, for real)
Bibliography, code and slides at:http://www.warwick.ac.uk/mmasdeu/
Marc Masdeu Non-archimedean constructions December 9th, 2014 30 / 30