The document discusses a conjectural p-adic construction of elliptic curves attached to modular forms over number fields. It proposes using p-adic path integrals to define periods of the elliptic curves in terms of cohomology classes of certain arithmetic groups. Specifically, the period lattice would be generated by integrals of a modular form's cohomology class against cycles representing the boundary of the Bruhat-Tits tree. This aims to make the modularity conjecture constructive in a non-archimedean setting.

1. p-adic integration and elliptic curves
over number fields
p-adic Methods in Number Theory, Milano
Xavier Guitart 1 Marc Masdeu 2 Mehmet Haluk Sengun 3
1Institut f ¨ ur Experimentelle Mathematik
2University of Warwick
3Sheffield University
October 22, 2014
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 0 / 22

2. The Machine
E/F K/F quadratic
Darmon Points
P
?2
E(Kab)
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 1 / 22

3. The Machine
E/F
Modularity
H H
Darmon Points
Ramification
Archimedean
Non-archimedean
K/F quadratic
P
?2
E(Kab)
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 1 / 22

4. The Machine
H H
Darmon Points
Ramification
Archimedean
Non-archimedean
K/F quadratic
P
?2
Ef (Kab)
f 2 S2(0(N))
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 1 / 22

5. The Machine
f 2 S2(0(N))
H H
Darmon Points
Ramification
Archimedean
Non-archimedean
???
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 1 / 22

6. Set-up
K a number field.
Fix an ideal N € OK.
Finitely many iso. classes of E{K with condpEq N:
Problem
Given N ¡ 0, find all elliptic curves of conductor N with
|NmK{QpNq| ¤ N:
K Q: Tables by J. Cremona (N 350; 000).
Ÿ W. Stein–M. Watkins: N 108 (N 1010 for prime N) incomplete.
?
5q: ongoing project, led by W. Stein (N 1831, first rank 2).
K Qp
S. Donnelli–P. Gunnells–A. Kluges-Mundt–D. Yasaki:
Cubic field of discriminant 23 (N 1187).
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 2 / 22

7. Strategy
Two steps
1 Find a list of elliptic curves of with conductor of norm ¤ N.
2 “Prove” that the obtained list is complete.
1 For (1) the process is as follows:
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 For (2) use modularity conjecture:
1 condpE{Kq N ùñ D automorphic form of level N.
2 Compute the fin. dim. space S2p0pNqq, with its Hecke action.
3 Match all rational eigenclasses to curves in the list.
3 In this talk: assume modularity when needed.
4 One is left with some gaps: some conductor N for which there exists
automorphic newform with rational eigenvalues taqpfquq.
Ÿ Problem: Find the elliptic curve attached to taqpfquq.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 3 / 22

8. Goals of the talk
1 Recall the existing analytic constructions of elliptic curves
2 Propose a conjectural p-adic construction
3 Show an example
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 4 / 22

9. The case K Q: Eichler–Shimura
X0pNq Ñ JacpX0pNqq
³
H0pX0pNq;
1
X0pNqq_
H1pX0pNq; Zq
Hecke C{f :
Theorem (Manin)
There is an isogeny
: C{f Ñ Ef pCq:
1 Compute H1p0pNq; Zq (modular symbols).
2 Find the period lattice f by explicitly integrating
f
C»
2i
¸
n¥1
G
anpfqe2inz :
P H1pX0pNq; Zq
:
3 Compute c4pf q; c6pf q P C by evaluating Eistenstein series.
4 Recognize c4pf q; c6pf q as integers ; Ef : Y 2 X3
c4
48X
c6
864 .
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 5 / 22

10. K Q. Existing constructions
K totally real. rK: Qs n, fix : K ãÑ R.
S2p0pNqq Q f ; ~!f P Hn
pX0pNq;Cq ; f „ C:
Conjecture (Oda, Darmon, Gartner)
C{f is isogenous to Ef bK K.
Known to hold (when F real quadratic) for base-change of E{Q.
Exploited in very restricted cases (Demb´ el ´ e, . . . ).
Explicitly computing f is hard –no quaternionic computations–.
K not totally real: no known algorithms. In fact:
Theorem
If K is imaginary quadratic, the lattice f is contained in R.
Idea
Try instead a non-archimedean construction!
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 6 / 22

11. Non-archimedean construction
From now on: assume there is some p such that p k N.
Replace the role of R (and C) with Kp (and its extensions).
Theorem (Tate uniformization)
Let E{K be an elliptic curve of conductor N, and let p k N. There exists a
rigid-analytic, Galois-equivariant isomorphism
: K
p {E Ñ Ep K
pq;
where E qZE
, with qE P K
p satisfying
jpEq q1
E 744 196884qE .
Suppose D coprime factorization N pDm, with D discpB{Kq.
Ÿ Always possible when K has at least one real place.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 7 / 22

12. Quaternionic modular forms of level N
Suppose K has signature pr; sq, and fix N pDm.
B{K the quaternion algebra such that
RampBq tq: q | Du Y tvn1; : : : ; vru; pn ¤ rq:
Fix isomorphisms v1; : : : ; vn : B bKvi M2pRq and
w1; : : :ws : B bKwj M2pCq, yielding
n
PGL2pCq
B{K ãÑ PGL2pRq
s ý Hn
Hs
3:
Fix RD
0 ppmq € RD
0 pmq € B Eichler orders of level pm and m.
D0
ppmq RD
K acts discretely on Hn Hs
0 ppmq{O
3.
Obtain a manifold of (real) dimension 2n 3s:
Y D
0 ppmq D0
ppmqz pHn
Hs
3q :
Y D
0 ppmq is compact ðñ B is division (assume it, for simplicity).
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 8 / 22

13. Group cohomology
The cohomology of Y D
0 ppmq can be computed via
HpY D
0 ppmq;Cq HpD0
ppmq;Cq:
Hecke algebra TD ZrTq : q - Ds acts on HpD0
ppmq; Zq.
f P HnspD0
ppmq;Cq eigen for TD is rational if aqpfq P Z; @q P TD.
Conjecture (Taylor, ICM 1994)
Let f P HnspD0
ppmq;Zq be a new, rational eigenclass. Then there is
an elliptic curve Ef {K of conductor N such that
#Ef pOK{qq 1 |q| appfq @q - N:
Goal
Make this (conjecturally) constructive.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 9 / 22

14. Non-archimedean path integrals
Hp P1pCpq r P1pKpq has a rigid-analytic structure.
PGL2pKpq acts on Hp through fractional linear transformations:
a b
c d
z
az b
cz d
; z P Hp:
We consider rigid-analytic 1-forms ! P
1
Hp .
Given two points 1 and 2 in Hp, define:
» 2
1
! Coleman integral.
Get a PGL2pKpq-equivariant pairing
»
:
1
Hp Div0 Hp Ñ Cp:
For each € PGL2pKpq, induce a pairing
»
: Hi
p;
1
Hpq Hip; Div0 Hpq Ñ Cp:
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 10 / 22

15. Coleman Integration
Coleman integration on Hp can be defined as:
» 2
!
1
»
logp
P1pKpq
t 2
t 1
d!ptq lim
ÝÑU
¸
UPU
logp
tU 2
tU 1
resApUqp!q:
Bruhat-Tits tree of GL2pKpq, |p| 2.
Hp having the Bruhat-Tits as retract.
Annuli ApUq for a covering of size |p|3.
tU is any point in U € P1pKpq.
P1(Kp)
U P1(Kp)
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 11 / 22

16. The tpu-arithmetic group
Recall we have chosen a factorization N pDm.
B{K chosen quaternion algebra of discriminant D.
Recall also RD
0 ppmq € RD
0 pmq € B.
Define D0
0 ppmq and D0
ppmq RD
0 pmq.
pmq RD
Set
D0
pmq D0
ppmq
{D0
pmq; {D0
pmq wpD0
pmqw1
p :
Fix an embedding p : R0 ãÑ M2pZpq.
Lemma
Assume that h
K 1. Then p induces bijections
{D0
pmq V0; {D0
ppmq E0
V0 (resp. E0) are the even vertices (resp. edges) of the BT tree.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 12 / 22

17. Cohomology (I)
0 pmqr1{ps{OKr1{ps p ãÑ PGL2pKpq:
RD
Consider the -equivariant exact sequence
0/HCpZq/ MapspE0pT q;Zq
/ MapspVpT q;Zq/ 0
f/ rvÞÑ
°
opeqv fpeqs
Have -equivariant isomorphisms
MapspE0
pT q;Zq Ind
D0
ppmq Z; MapspVpT q; Zq
Ind
D0
2
pmq Z
:
So get:
0 Ñ HCpZq Ñ Ind
D0
ppmq Z Ñ
Ind
D0
2
pmq Z
Ñ 0
Taking -cohomology and using Shapiro’s lemma gives
Hns
p;HCpZqq Ñ Hns
pD0
ppmq; Zq
Ñ
Hns
pD0
pmq; Zq
2
Ñ
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 13 / 22

18. Cohomology (II)
Taking -cohomology and using Shapiro’s lemma gives
Hns
p;HCpZqq Ñ Hns
pD0
ppmq; Zq
Ñ
Hns
pD0
pmq; Zq
2
Ñ
f P HnspD0
ppmq; Zq being p-new ùñ f P Kerpq.
Pulling back, get !f P Hnsp;HCpZqq.
!f P Hns
p; Meas0pP1
pKpqqq:
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 14 / 22

19. Holomogy cycles
Consider the -equivariant s.e.s:
0 Ñ Div0 Hp Ñ DivHp
deg
Ñ Z Ñ 0:
Taking -homology yields
Hns1p;Zq
Ñ
Hnsp; Div0 Hpq Ñ Hnsp; DivHpq Ñ Hnsp;Zq
Let LppEq
logppqEq
ordppqEq (p-adic L-invariant).
Conjecture
The set
f
#»
pcq
!f : c P Hns1p; Zq
+
€ Cp
is infinite and contained in the line ZLppEq.
Known when K Q (Darmon, Dasgupta–Greenberg,
Longo–Rotger–Vigni), open in general.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 15 / 22

20. Lattice: explicit construction
Start f P HnspD0
ppm; Zqq.
Duality yields ^ f P HnspD0
ppm;Zqq.
Mayer–Vietoris exact sequence for D0
pmq D0
ppmq
^
D0
pmq:
Ñ Hns1p;Zq
1
Ñ HnspD0
ppmq;Zq

21. Ñ
HnspD0
2
Ñ
pmq; Zq
^ f new at p ùñ

22. p ^ fq 0.
Ÿ ^ f 1pcf q, for some cf P Hns1p;Zq.
Conjecture
The element
Lf
»
pcf q
!f :
is a nonzero multiple of LppEq.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 16 / 22

23. Algorithms
Only when n s ¤ 1.
Use explicit presentation for D0
ppmq and D0
pmq.
Ÿ s 0 ùñ J. Voight.
Ÿ s 1 ùñ A. Page.
Compute the Hecke action on H1pD0
ppm; Zqq and H1pD0
ppm;Zqq.
Integration pairing uses overconvergent cohomology.
Ÿ Lift f to overconvergent class F P HnspD0
ppmq;Dq.
Ÿ Use F to to recover moments the measures !f p
q.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 17 / 22

24. Overconvergent Method (I)
Starting data: cohomology class !f P H1p;
1
Hp
q.
Goal: to compute integrals
³2
1
, for
P .
Recall that » 2
1
»
P1pKpq
logp
t 1
t 2
d
ptq:
Expand the integrand into power series and change variables.
Ÿ We are reduced to calculating the moments:
»
Zp
tid
ptq for all
P .
Note: … D0pmq … D0
ppmq.
Technical lemma: All these integrals can be recovered from
#»
Zp
tid
ptq :
P D0
ppmq
+
:
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 18 / 22

25. Overconvergent Method (II)
D tlocally analytic Zp-valued distributions on Zpu.
D0
Ÿ ' P D maps a locally-analytic function h on Zp to 'phq P Zp.
Ÿ D is naturally a ppmq-module.
The map 'ÞÑ 'p1Zpq induces a projection:
: H1
pD0
ppmq;Dq Ñ H1
pD0
ppmq;Zpq:
Theorem (Pollack-Stevens, Pollack-Pollack)
There exists a unique Up-eigenclass F lifting f.
Moreover, F is explicitly computable by iterating the Up-operator.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 19 / 22

26. Overconvergent Method (III)
But we wanted to compute the moments of a system of measures. . .
Proposition
Consider the map G: D0
ppmq Ñ D:
ÞÑ
hptqÞÑ
»
Zp
:
hptqd
ptq
1 G belongs to H1pD0
ppmq;Dq.
2 G is a lift of f.
3 G is a Up-eigenclass.
Corollary
The explicitly computed F knows the above integrals.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 20 / 22

27. Recovering E from Lf
Start with q log1
p pLf q P Cp
, assume ordppqq ¡ 0.
Get
jpqq q1
744 196884q P Cp
:
From N guess the discriminant E.
34
Ÿ Only finitely-many possibilities, E P SpK; 12q.
From j c{ recover c4 P Cp.
26
Try to 34
recognize c4 algebraically.
From 1728 c crecover c6.
Compute the conductor of Ef : Y 2 X3
c4
48X
c6
864 .
Ÿ If conductor is correct, check ap’s.
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 21 / 22

28. Example curve
K Qpq; ppxq x4 x3 3x 1, K 1732.
N p 2q P13.
B{K of ramified only at all infinite real places of K.
There is a rational eigenclass f P S2p0p1;Nqq.
From f we compute !f P H1p;
1
³
Hp
!f 8 13 11 132 5 133 3 134 Op13100q.
c q and c P H2p; Zq.
qE
jE 1
13
46563774300743 108622486567602 14109269950515 4120837170980
.
c4 26984733 44220642 583165 825127.
c6 204428562683 45374343522 3147148174410479346607.
E{F : y2
3
3
xy x3
23
2
5
x2
562183
921262
12149 17192
x
235934113
53008112
36382184 12122562:
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 22 / 22

29. Thank you !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
Marc Masdeu p-adic integration and elliptic curves October 22, 2014 22 / 22