Analytic construction of elliptic curves and rational points

Analytic construction of elliptic curves
and rational points
Number Theory Seminar, Bristol
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Universitat de Barcelona
2University of Warwick
3Shefﬁeld University
December 3rd, 2014
Marc Masdeu Analytic constructions December 3rd
, 2014 0 / 33

Plan
1 Elliptic curves
2 Rational points
3 The Overconvergent Method
, 2014 1 / 33

Set-up
F a number field.
Fix an ideal N Ă OF .
Finitely many iso. classes of E{F with condpEq “ N.
Problem
Given N ą 0, find all elliptic curves of conductor N with
| NmF{QpNq| ď N.
F “ Q: tables by J. Cremona (N “ 350, 000).
§ W. Stein–M. Watkins: N “ 108
(N “ 1010
for prime N) incomplete.
F “ Qp
?
5q: ongoing project, led by W. Stein (N “ 1831, first rank 2).
S. Donnelly–P. Gunnells–A. Klages-Mundt–D. Yasaki:
Cubic field of discriminant ´23 (N “ 1187).
, 2014 2 / 33

A Two-step Strategy
1 Find a list of elliptic curves of with conductor of norm ď N.
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 “Prove” that the obtained list is complete.
1 condpE{Fq “ 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.
Usually get gaps: some conductor N for which there exists
automorphic newform with rational eigenvalues taqpfquq.
§ Problem: Find elliptic curve of conductor N attached to taqpfquq.
, 2014 3 / 33

Quaternionic modular forms of level N
Suppose F has signature pr, sq, and factor N “ Dn.
B{F the quaternion algebra such that
RampBq “ tq: q | Du Y tvn`1, . . . , vru, pn ď rq.
Fix isomorphisms
v1, . . . , vn : B bFvi – M2pRq
w1, . . . ws : B bFwj – M2pCq
These yield
Bˆ
{Fˆ
ãÑ PGL2pRqn
ˆ PGL2pCqs
ý Hn
ˆ Hs
3.
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.
, 2014 4 / 33

Elliptic curves from cohomology classes
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.
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 5 / 33

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 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.
, 2014 6 / 33

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 also non-archimedean constructions!
, 2014 7 / 33

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 “ x1, τf y.
, 2014 8 / 33

Non-archimedean path integrals on Hp
Consider Hp “ P1pCpq P1pFpq.
§ 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, induce a pairing
ż
: Hi
pΓ, Ω1
Hp
q ˆ HipΓ, Div0
Hpq Ñ Cp.
Teitelbaum: Ω1
Hp
– Meas0
pP1pFpqq.
, 2014 9 / 33

(Multiplicative) Coleman Integration
Multiplicative Coleman integration on Hp can be deﬁned as:
ˆ
ż τ2
τ1
ω “ ˆ
ż
P1pFpq
ˆ
t ´ τ2
t ´ τ1
˙
dµωptq “ limÝÑ
U
ź
UPU
ˆ
tU ´ τ2
tU ´ τ1
˙µωpUq
.
Bruhat-Tits tree of GL2pFpq, |p| “ 2.
Edges giving a covering of size |p|´3.
tU is any point in U Ă P1pFpq.
Meas0
pP1pFpq, Zq – HCpZq.
So replace Ω1
Hp
with HCpZq. P1(Fp)
U ⊂ P1
(Fp)
µ(U)
v∗
ˆv∗
e∗
, 2014 10 / 33

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.
Deﬁne ΓD
0 ppmq “ RD
0 ppmqˆ{Oˆ
F and ΓD
0 pmq “ RD
0 pmqˆ{Oˆ
F .
Let Γ “ ΓD
0 pmqr1{psˆ{OF r1{psˆ ιp
ãÑ PGL2pFpq.
Lemma
Assume that h`
F “ 1. Then ιp induces bijections
Γ{ΓD
0 pmq – V0, Γ{ΓD
0 ppmq – E0
V0 (resp. E0) are the even vertices (resp. edges) of the BT tree.
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).
, 2014 11 / 33

Cohomology (I)
Γ “ RD
0 pmqr1{psˆ
{OF r1{psˆ ιp
ãÑ PGL2pFpq.
Consider the Γ-equivariant exact sequence
0 // HCpZq // MapspE0pT q, Zq
∆ // MapspVpT q, Zq // 0
ϕ // rv ÞÑ
ř
opeq“v ϕpeqs
Lemma ùñ Γ-equivariant isomorphisms
MapspE0pT q, Zq – IndΓ
ΓD
0 ppmq
Z, MapspVpT q, Zq –
´
IndΓ
ΓD
0 pmq
Z
¯2
.
So get:
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
∆
Ñ
´
IndΓ
ΓD
0 pmq
Z
¯2
Ñ 0
, 2014 12 / 33

Cohomology (II)
0 Ñ HCpZq Ñ IndΓ
ΓD
0 ppmq
Z
∆
Ñ
´
IndΓ
ΓD
0 pmq
Z
¯2
Ñ 0
Taking Γ-cohomology and using Shapiro’s lemma gives
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 ùñ f P Kerp∆q.
Pulling back get
ωf P Hn`s
pΓ, HCpZqq.
Next: ﬁnd θf P Hn`spΓ, Div0
Hpq against which to integrate ωf .
, 2014 13 / 33

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
Conjecture A (Greenberg, Guitart–M.–Sengun)
The multiplicative lattice Λf “
!
ˆ
ş
δpcq ωf : c P Hn`s`1pΓ, Zq
)
Ă Cˆ
p is
homothetic to qZ
E.
Known when F “ Q (Darmon, Dasgupta–Greenberg,
Longo–Rotger–Vigni), open in general.
, 2014 14 / 33

Recovering E from Λf
From N guess the discriminant ∆E.
§ Only ﬁnitely-many possibilities, ∆E P SpF, 12q.
Since ordpp∆Eq “ ordppqEq, recover qf P Cˆ
p from Λf .
Assume ordppqf q ą 0 (otherwise, replace qf ÞÑ 1{qf ).
Get
jpqf q “ q´1
f ` 744 ` 196884qf ` ¨ ¨ ¨ P Cˆ
p .
jpqf q “ c3
4{∆E ; recover c4.
Try to 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 / 33

Algorithms
Only in the cases n ` s ď 1.
§ Both H1
and H1 reduce to linear algebra for ﬁnitely-presented groups.
Use explicit presentation + word problem for ΓD
0 ppmq and ΓD
0 pmq.
§ John Voight (s “ 0).
§ Aurel Page (s “ 1).
Compute the Hecke action on H1pΓD
0 ppm, Zqq and H1pΓD
0 ppm, Zqq.
Integration pairing uses overconvergent cohomology.
, 2014 16 / 33

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 c P H2pΓ, Zq.
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 17 / 33

Plan
1 Elliptic curves
2 Rational points
, 2014 18 / 33

The Machine
Non-archimedean
Archimedean
Ramiﬁcation
Periods Machine
H∗
H∗
f ∈ S2(Γ0(N))
Ef ?
, 2014 19 / 33

The Machine
Non-archimedean
Archimedean
Ramiﬁcation
Darmon Points
H∗
H∗
K/F quadratic
P
?
∈ Ef(Kab)
f ∈ S2(Γ0(N))
, 2014 19 / 33

The Machine
Non-archimedean
Archimedean
Ramiﬁcation
Darmon Points
H∗
H∗
Modularity
E/F
K/F quadratic
P
?
∈ E(Kab)
, 2014 19 / 33

The Machine
Darmon Points
E/F K/F quadratic
P
?
∈ E(Kab)
, 2014 19 / 33

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 20 / 33

Heegner Points (K{Q imaginary quadratic)
Use crucially that E is attached to f.
ωf “ 2πifpzqdz “ 2πi
ÿ
ně1
ane2πinz
dz 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 21 / 33

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 22 / 33

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 23 / 33

New notation
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).
The triple pE, K, pq determine uniquely the quaternion algebra B:
RampBq “ SpE, Kq tpu.
, 2014 24 / 33

History
H. Darmon (’99): F “ Q, quasi-CM, B – M2pFq.
M. Trifkovic (’06): F img. quadratic ( ùñ quasi-CM)), B – M2pFq.
M. Greenberg (’08): F totally real, B arbitrary.
, 2014 25 / 33

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.
Have the exact sequence
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 26 / 33

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.
We predict also the exact number ﬁeld over which Pψ is deﬁned.
Include a Shimura reciprocity law like that of Heegner points.
, 2014 27 / 33

Non-archimedean cubic Darmon point (I)
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 28 / 33

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 29 / 33

Thank you !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
, 2014 29 / 33

Plan
1 Elliptic curves
2 Rational points
, 2014 30 / 33

Overconvergent Method (I)
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 31 / 33

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.
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 32 / 33

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 H1pΓD
0 ppmq, Dq.
2 Ψ is a lift of f.
3 Ψ is a Up-eigenclass.
Corollary
The explicitly computed ˜Φ “ Ψ knows the above integrals.
, 2014 33 / 33

Thank you (again) !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
, 2014 33 / 33

Analytic construction of elliptic curves and rational points

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