Non-archimedean construction of elliptic curves and rational points

Non-archimedean construction
of elliptic curves
and rational points
Number 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
, 2014 1 / 30

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 B{F 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 Bˆ{Fˆ ãÑ PGL2pRqn ˆ PGL2pCqs ý Hn ˆ Hs
3.
, 2014 2 / 30

Quaternionic automorphic forms of level N (II)
Fix RD
0 pnq Ă B Eichler order of level n.
ΓD
0 pnq “ RD
0 pnqˆ{Oˆ
F acts discretely on Hn ˆ Hs
3.
Obtain a manifold of (real) dimension 2n ` 3s:
Y D
0 pnq “ ΓD
0 pnqz pHn
ˆ Hs
3q .
Y D
0 pnq is compact ðñ B is division.
The cohomology of Y D
0 pnq can be computed via
H˚
pY D
0 pnq, Cq – H˚
pΓD
0 pnq, Cq.
Hecke algebra TD “ ZrTq : q Ds acts on H˚pΓD
0 pnq, Zq.
Deﬁnition
f P Hn`spΓD
0 pnq, Cq eigen for TD is rational if appfq P Z, @p P TD.
, 2014 3 / 30

Elliptic curves from cohomology classes
Deﬁnition
f 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`spΓD
0 pnq, Zq a new, rational eigenclass.
Then DEf {F 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 talk
Make this conjecture (conjecturally) constructive.
, 2014 4 / 30

The case F “ Q: Cremona’s algorithm
Eichler–Shimura
X0pNq Ñ JacpX0pNqq
ş
–
H0
`
X0pNq, Ω1
˘_
H1pX0pNq, Zq
Hecke
C{Λf Ñ Ef pCq.
1 Compute H1pX0pNq, Zq (modular symbols).
2 Find the period lattice Λf by explicitly integrating
Λf “
Cż
γ
2πi
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.
, 2014 5 / 30

F ‰ Q. Existing constructions
F totally real. rF : Qs “ n, fix σ: F ãÑ R.
S2pΓ0pNqq Q f ; ˜ωf P Hn
pΓ0pNq, Cq ; Λf Ď C.
Conjecture (Oda, Darmon, Gartner)
C{Λf is isogenous to Ef ˆF Fσ.
Known to hold (when F real quadratic) for base-change of E{Q.
Exploited in very restricted cases (Dembélé, . . . ).
Explicitly computing Λf is hard –no quaternionic computations–.
F not totally real: no known algorithms. In fact:
Theorem
If F is imaginary quadratic, the lattice Λf is contained in R.
Idea
Allow for non-archimedean constructions!
, 2014 6 / 30

Non-archimedean construction
From now on: ﬁx p N.
Theorem (Tate uniformization)
There exists a rigid-analytic, Galois-equivariant isomorphism
η: ¯Fˆ
p {xqEy Ñ Ep ¯Fpq,
with qE P Fˆ
p satisfying jpEq “ q´1
E ` 744 ` 196884qE ` ¨ ¨ ¨ .
Suppose D coprime factorization N “ pDm, with D “ discpB{Fq.
§ . . . always possible when F has at least one real place.
Compute qE as a replacement for Λf .
, 2014 7 / 30

Non-archimedean path integrals on Hp
Consider Hp “ P1pCpq 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 of
şτ2
τ1
ω P Cp.
Get a PGL2pFpq-equivariant pairing
ş
: Ω1
Hp
ˆ Div0
Hp Ñ Cp.
For each Γ Ă PGL2pFpq, get induced pairing
HipΓ, Ω1
Hp
q ˆ HipΓ, Div0
Hpq
ş
// Cp
´
Φ,
ř
γ γ bDγ
¯
//
ř
γ
ż
Dγ
Φpγq.
Ω1
Hp
– space of Cp-valued boundary measures Meas0pP1pFpq, Cpq.
, 2014 8 / 30

Measures and integrals
Bruhat-Tits tree of GL2pFpq, |p| “ 2.
P1pFpq – EndspT q.
Harmonic cocycles HCpAq “
tEpT q
f
Ñ A |
ř
opeq“v fpeq “ 0u
Meas0pP1pFpq, Aq – HCpAq.
So replace ω P Ω1
Hp
with
µω P Meas0pP1pFpq, Zq – HCpZq.
P1
(Fp)
U ⊂ P1
(Fp)
µ(U)
v∗
ˆv∗
e∗
T
Coleman integration: if τ1, τ2 P Hp, then
ż τ2
τ1
ω “
ż
P1pFpq
logp
ˆ
t ´ τ2
t ´ τ1
˙
dµωptq “ limÝÑ
U
ÿ
UPU
logp
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 “ limÝÑ
U
ź
UPU
ˆ
tU ´ τ2
tU ´ τ1
˙µωpUq
.
, 2014 9 / 30

The tpu-arithmetic group Γ
Choose a factorization N “ pDm.
B{F “ quaternion algebra with RampBq “ tq | Du Y tvn`1, . . . , vru.
Recall also RD
0 ppmq Ă RD
0 pmq Ă B.
Fix ιp : RD
0 pmq ãÑ M2pZpq.
Deﬁne ΓD
0 ppmq “ RD
0 ppmqˆ{Oˆ
F and ΓD
0 pmq “ RD
0 pmqˆ{Oˆ
F .
Let Γ “ RD
0 pmqr1{psˆ{OF r1{psˆ ιp
ãÑ PGL2pFpq.
Example
F “ Q and D “ 1, so N “ pM.
B “ M2pQq.
Γ0ppMq “
` a b
c d
˘
P GL2pZq: pM | c
(
{t˘1u.
Γ “
` a b
c d
˘
P GL2pZr1{psq: M | c
(
{t˘1u ãÑ PGL2pQq Ă PGL2pQpq.
, 2014 10 / 30

The tpu-arithmetic group Γ
Lemma
Assume that h`
F “ 1. Then ιp induces bijections
Γ{ΓD
0 pmq – V0pT q, Γ{ΓD
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. ΓD
0 ppmq).
Corollary
MapspE0pT q, Zq – IndΓ
ΓD
0 ppmq
Z, MapspVpT q, Zq –
´
IndΓ
ΓD
0 pmq
Z
¯2
.
, 2014 11 / 30

Cohomology
Γ “ RD
0 pmqr1{psˆ
{OF r1{psˆ ιp
ãÑ PGL2pFpq.
MapspE0pT q, Zq – IndΓ
ΓD
0 ppmq
Z, MapspVpT q, Zq –
´
IndΓ
ΓD
0 pmq
Z
¯2
.
Consider the Γ-equivariant exact sequence
0 // HCpZq // MapspE0pT q, Zq
∆ // MapspVpT q, Zq // 0
ϕ // rv ÞÑ
ř
opeq“v ϕpeqs
So get:
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
∆
Ñ
´
IndΓ
ΓD
0 pmq
Z
¯2
Ñ 0
, 2014 12 / 30

Cohomology (II)
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
∆
Ñ
´
IndΓ
ΓD
0 pmq
Z
¯2
Ñ 0
Taking Γ-cohomology,. . .
Hn`s
pΓ, HCpZqq Ñ Hn`s
pΓ, IndΓ
ΓD
0 ppmq
, Zq
∆
Ñ Hn`s
pΓ, IndΓ
ΓD
0 pmq
, Zq2
Ñ
. . . and using Shapiro’s lemma:
Hn`s
pΓ, HCpZqq Ñ Hn`s
pΓD
0 ppmq, Zq
∆
Ñ Hn`s
pΓD
0 pmq, Zq2
Ñ ¨ ¨ ¨
f P Hn`spΓD
0 ppmq, Zq being p-new ô f P Kerp∆q.
Pulling back get
ωf P Hn`s
pΓ, HCpZqq.
, 2014 13 / 30

Holomogy
Consider the Γ-equivariant short exact sequence:
0 Ñ Div0
Hp Ñ Div Hp
deg
Ñ Z Ñ 0.
Taking Γ-homology yields
Hn`s`1pΓ, Zq
δ
Ñ Hn`spΓ, Div0
Hpq Ñ Hn`spΓ, Div Hpq Ñ Hn`spΓ, Zq
Λf “
#
ˆ
ż
δpcq
ωf : c P Hn`s`1pΓ, Zq
+
Ă Cˆ
p
Conjecture A (Greenberg, Guitart–M.–Sengun)
The multiplicative lattice Λf is homothetic to qZ
E.
F “ Q: Darmon, Dasgupta–Greenberg, Longo–Rotger–Vigni.
Open in general.
, 2014 14 / 30

Recovering E from Λf
Λf “ xqf y gives us qf
?
“ qE.
Assume ordppqf q ą 0 (otherwise, replace qf ÞÑ 1{qf ).
Get
jpqf q “ q´1
f ` 744 ` 196884qf ` ¨ ¨ ¨ P Cˆ
p .
From N guess the discriminant ∆E.
§ Only ﬁnitely-many possibilities, ∆E P SpF, 12q.
jpqf q “ c3
4{∆E ; recover c4.
Recognize c4 algebraically.
1728∆E “ c3
4 ´ c2
6 ; recover c6.
Compute the conductor of Ef : Y 2 “ X3 ´ c4
48X ´ c6
864.
§ If conductor is correct, check aq’s.
, 2014 15 / 30

Example curve
F “ Qpαq, pαpxq “ x4 ´ x3 ` 3x ´ 1, ∆F “ ´1732.
N “ pα ´ 2q “ P13.
B{F 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 “ 1
13
´
´ 4656377430074α3
` 10862248656760α2
´ 14109269950515α ` 4120837170980
¯
.
c4 “ 2698473α3 ` 4422064α2 ` 583165α ´ 825127.
c6 “ 20442856268α3 ´ 4537434352α2 ´ 31471481744α ` 10479346607.
E{F : y2
`
`
α3
` α ` 3
˘
xy “ x3
`
`
`
´2α3
` α2
´ α ´ 5
˘
x2
`
`
´56218α3
´ 92126α2
´ 12149α ` 17192
˘
x
´ 23593411α3
` 5300811α2
` 36382184α ´ 12122562.
, 2014 16 / 30

The Machine
Non-archimedean
Archimedean
Ramiﬁcation
Periods Machine
H∗
H∗
f ∈ S2(Γ0(N))
Ef ?
, 2014 17 / 30

The Machine
Non-archimedean
Archimedean
Ramiﬁcation
Darmon Points
H∗
H∗
K/F quadratic
P
?
∈ Ef(Kab)
f ∈ S2(Γ0(N))
, 2014 17 / 30

The Machine
Non-archimedean
Archimedean
Ramiﬁcation
Darmon Points
H∗
H∗
Modularity
E/F
K/F quadratic
P
?
∈ E(Kab)
, 2014 17 / 30

The Machine
Darmon Points
E/F K/F quadratic
P
?
∈ E(Kab)
, 2014 17 / 30

Rational points on elliptic curves
Suppose we have E{F attached to f.
Let K{F be a quadratic extension of F.
§ Assume that N is square-free, coprime to discpK{Fq.
Hasse-Weil L-function of the base change of E to K ( psq ąą 0)
LpE{K, sq “
ź
p|N
`
1 ´ ap|p|´s
˘´1
ˆ
ź
p N
`
1 ´ ap|p|´s
` |p|1´2s
˘´1
.
Coarse version of BSD conjecture
ords“1 LpE{K, sq “ rkZ EpKq.
So ords“1 LpE{K, sq odd
BSD
ùñ DPK P EpKq of inﬁnite order.
Second goal of the talk
Find PK explicitly (at least conjecturally).
, 2014 18 / 30

Heegner Points (K{Q imaginary quadratic)
Use crucially that E is attached to f.
ωf “ 2πifpzqdz P H0
pΓ0pNq, Ω1
Hq.
Given τ P K X H, set Jτ “
ż τ
i8
ωf P C.
Well-deﬁned up to the lattice Λf “
!ş
γ ωf | γ P H1 pΓ0pNq, Zq
)
.
§ There exists an isogeny (Weierstrass uniformization)
η: C{Λf 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 L1
pE{K, 1q ‰ 0.
, 2014 19 / 30

Heegner Points: revealing the trick
Why did this work?
1 The Riemann surface Γ0pNqzH has an algebraic model X0pNq{Q.
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.
, 2014 20 / 30

Darmon’s insight
Henri Darmon
Drop hypothesis of K{F 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!
, 2014 21 / 30

New notation
Still assume p N “ condpEq.
The triple pE, K, pq determines uniquely the quaternion algebra B:
RampBq “ SpE, Kq tpu.
Set n ` s “ #tv | 8F : v splits in Ku.
K{F is CM ðñ n ` s “ 0.
§ If n ` s “ 1 we call K{F quasi-CM.
SpE, Kq “
!
v | N8F : v not split in K
)
.
Sign of functional equation for LpE{K, 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).
, 2014 22 / 30

Homology classes attached to K
Let ψ: O ãÑ RD
0 pmq be an embedding of an order O of K.
§ Which is optimal: ψpOq “ RD
0 pmq X ψpKq.
Consider the group Oˆ
1 “ tu P Oˆ : NmK{F puq “ 1u.
§ rankpOˆ
1 q “ rankpOˆ
q ´ rankpOˆ
F q “ n ` s.
Choose a basis u1, . . . , un`s P Oˆ
1 for the non-torsion units.
§ ; ∆ψ “ ψpu1q ¨ ¨ ¨ ψpun`sq P Hn`spΓ, Zq.
Kˆ acts on Hp through Kˆ ψ
ãÑ Bˆ ιp
ãÑ GL2pFpq.
§ Let τψ be the (unique) ﬁxed point of Kˆ
on Hp.
Hn`s`1pΓ, Zq
δ // Hn`spΓ, Div0
Hpq // Hn`spΓ, Div Hpq
deg
// Hn`spΓ, Zq
Θψ
? // r∆ψ bτψs // r∆ψs
Fact: r∆ψs is torsion.
§ Can pull back a multiple of r∆ψ bτψs to Θψ P Hn`spΓ, Div0
Hpq.
§ Well deﬁned up to δpHn`s`1pΓ, Zqq.
, 2014 23 / 30

Conjectures
Jψ “ ˆ
ż
Θψ
ωf P Kˆ
p {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 L1pE{K, 1q ‰ 0.
, 2014 24 / 30

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 E{F given by the equation:
E{F : y2
` p´r ´ 1q xy ` p´r ´ 1q y “ x3
´ rx2
` p´r ´ 1q x.
E has conductor NE “
`
r2 ` 2
˘
“ p17q2, where
p17 “
`
´r2
` 2r ` 1
˘
, q2 “ prq .
Consider K “ Fpαq, where α “
?
´3r2 ` 9r ´ 6.
The quaternion algebra B{F has discriminant D “ q2:
B “ Fxi, j, ky, i2
“ ´1, j2
“ r, ij “ ´ji “ k.
, 2014 25 / 30

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´7
2 ` 2r`3
2 i ` 2r2`3r
2 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
`Op1721
q,
which corresponds to:
Pψ “ ´108 ˆ
ˆ
r ´ 1,
α ` r2 ` r
2
˙
P EpKq.
, 2014 26 / 30

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 ).
, 2014 27 / 30

Thank you !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
, 2014 27 / 30

Overconvergent Method
Starting data: cohomology class Φ “ ωf P H1pΓ, Ω1
Hp
q.
Goal: to compute integrals
şτ2
τ1
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
ti
dµγptq for all γ P Γ.
Note: Γ Ě ΓD
0 pmq Ě ΓD
0 ppmq.
Technical lemma: All these integrals can be recovered from
#ż
Zp
ti
dµγptq: γ P ΓD
0 ppmq
+
.
, 2014 28 / 30

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 ϕp1Zp q induces a projection:
H1pΓD
0 ppmq, Dq
ρ
// H1pΓD
0 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.
, 2014 29 / 30

Overconvergent Method (III)
But we wanted to compute the moments of a system of measures. . .
Proposition
Consider the map Ψ: ΓD
0 ppmq Ñ D:
γ ÞÑ
”
hptq ÞÑ
ż
Zp
hptqdµγptq
ı
.
1 Ψ belongs to H1
´
ΓD
0 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.
, 2014 30 / 30

Thank you !
(now, for real)
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
, 2014 30 / 30

Non-archimedean construction of elliptic curves and rational points

Recommended

Recommended

More Related Content

What's hot

What's hot (20)

Similar to Non-archimedean construction of elliptic curves and rational points

Similar to Non-archimedean construction of elliptic curves and rational points (20)

More from mmasdeu

More from mmasdeu (14)

Recently uploaded

Recently uploaded (20)

Non-archimedean construction of elliptic curves and rational points