Slides for a 30-minute talk given during the Building Bridges workshop "Automorphic Forms and Related Topics" in Sarajevo, July 2016

1. Computing p-adic periods
of abelian surfaces
from automorphic forms
Building Bridges
Automorphic Forms and Related Topics
Xavier Guitart 1 Marc Masdeu 2
1Universitat de Barcelona
2University of Warwick
Marc Masdeu p-adic periods 0 / 15

2. The Eichler–Shimura construction
f P S2pΓ0pNqq a (normalized) newform.
Kf “ Qptanuně1q, a totally-real number field of degree d ě 1.
Eichler–Shimura construction
X0pNq // JacpX0pNqq
ş
–
H0
pX0pNq,Ω1
q
_
H1pX0pNq,Zq
Hecke// // Cd{Λf – Af pCq.
1 The complex torus Af pCq has a model Af over Q.
§ conductorpAf q “ Nd
.
§ EndpAf q bQ Ą Kf .
§ LpAf , sq “
ś
σ : Kf ÑR Lpσ
f, sq.
2 The period lattice Λf is explicit:
Λf “ 2πi
Bˆż
γ
σ1
fpzqdz, . . . ,
ż
γ
σd
fpzqdz
˙
: γ P H1
´
X0pNq, Z
¯F
Ă Cd
.
3 This allows in some cases to recover Af .
§ Cremona (d “ 1).
§ Wang, Guardia, Gonz´alez, Gonz´alez-Jim´enez,. . . (d “ 2, 3).
Marc Masdeu p-adic periods 1 / 15

3. Quaternionic automorphic forms of level N
F a number field, h´
F “ 1, signature pr, sq, and fix N Ă OF .
Fix embeddings v1, . . . , vr : F ãÑ R, w1, . . . , ws : F ãÑ C.
Choose factorization N “ Dn, with D square free.
Let B{F be a 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 Bˆ{Fˆ ãÑ PGL2pRqn ˆ PGL2pCqs ý Hn ˆ Hs
3.
R>0
C
H3
R>0
H
R
PGL2(R)
PGL2(C)
Marc Masdeu p-adic periods 2 / 15

4. 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 an orbifold of (real) dimension 2n ` 3s:
Y D
0 pnq “ ΓD
0 pnqz pHn
ˆ Hs
3q .
Cohomology of Y D
0 pnq can be computed (assume B ‰ M2pFq) 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.
Hn`s
pΓD
0 pnq, Qq “
à
f
Hn`s
pΓD
0 pnq, Qqf (irreducible factors).
Each f cuts out a field Kf , s.t. rKf : Qs “ dim Hn`spΓD
0 pnq, Cqf .
Marc Masdeu p-adic periods 3 / 15

5. Abelian varieties from cohomology classes
Conjecture (Taylor, ICM 1994)
Let f P Hn`spΓD
0 pnq, Cqf a new, cuspidal eigenclass, Kf “ Qptanuq.
Set d “ rKf : Qs. Then D a simple abelian variety Af {F such that either:
1 dimpAf q “ d, conductor pDnqd, and EndpAf q bQ Ě Kf , such that
LpAf , sq “
ź
σ :Kf ãÑC
Lpσ
f, sq, or
2 dimpAf q “ 2d, conductor pDnq2d, and EndpAf q bQ Ě D,
(D a quaternion division algebra over Kf ) such that
LpAf , sq “
ź
σ :Kf ãÑC
Lpσ
f, sq2
Remark
Modularity says Af ; f. This is a converse.
Which one is harder?
Marc Masdeu p-adic periods 4 / 15

6. A p-adic construction of Af
Assumption
From now on suppose that D p N.
Denote by ¯Qp “ alg. closure of the p-adic completion of F.
Choose a coprime factorization N “ pDm, with D “ discpB{Fq.
Starting data: f P Hn`spΓD
0 ppmq, Cqnew
f eigenform.
§ Kf “ field generated by Hecke eigenvalues, d “ rKf : Qs.
We will describe a p-adic lattice Λf Ă p¯Qˆ
p qd.
We will conjecture that Af p¯Qpq „ p¯Qˆ
p qd{Λf .
Two questions arise
1 What if Af is of dimension 2d (“fake case”)?
2 What if Af p¯Qpq doesn’t look like a torus?
Marc Masdeu p-adic periods 5 / 15

7. Ruling out the “fake case”
Theorem 1
Suppose that f is attached to Af {F of dimension 2d, conductor M, and
D ãÑ EndpAq bQ, with D a quaternion division algebra.
Then f has complex multiplication.
Proof:
A dimension argument shows D “ EndpAq bQ.
Classification of endomorphism algebras of simple abelian varieties:
1 D is totally indefinite, or
2 D is totally definite.
Lemma
Suppose that D is totally indefinite. Then p | M ùñ p4d | M.
Lemma
Suppose that D is totally definite. Then f has complex multiplication.
Marc Masdeu p-adic periods 6 / 15

8. Af has purely multiplicative reduction
Want to generalize Tate’s uniformization:
p conductorpEq ùñ Ep¯Qpq “ ¯Qˆ
p {qZ for some q P Qp.
If conductorpE1q “ p2 and conductorpE2q “ M with p M, then
E1 ˆ E2 has conductor p2M, but E1 ˆ E2 isn’t p-adically uniformized.
Theorem 2
Suppose that A{F is an abelian variety of dimension d, such that
pd conductorpAq, and
EndpAq bQ Ą K, where K is totally real of degree d.
Then D a discrete lattice Λ Ă pFˆ
p qd such that
Ap¯Qpq – p¯Qˆ
p qd
{Λ.
Marc Masdeu p-adic periods 7 / 15

9. Integration on Hp
Consider Hp “ P1pQp2 q P1pQpq.
It is a p-adic analogue to H:
§ It has a rigid-analytic structure.
§ Action of PGL2pQpq by fractional linear transformations.
§ Rigid-analytic 1-forms ω P Ω1
Hp
.
‹ Denote by Ω1
Hp,Z the forms having Z-valued residues.
§ Coleman integration ; make sense of
şτ2
τ1
ω P Cp.
Darmon constructed a PGL2pQpq-equivariant pairing
ˆ
ż
: Ω1
Hp,Z ˆ Div0
Hp Ñ Qˆ
p2 Ă Cˆ
p .
For each Γ Ă PGL2pQpq, get induced pairing (cap product)
HipΓ, Ω1
Hp,Zq ˆ HipΓ, Div0
Hpq
ˆ
ż
// Cˆ
p
´
φ,
ř
γ γ bDγ
¯
//
ř
γ ˆ
ż
Dγ
φpγq.
Marc Masdeu p-adic periods 8 / 15

10. 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 PGL2pQpq.
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.
Marc Masdeu p-adic periods 9 / 15

11. The conjecture
Theorem 3
Hn`s
pΓD
0 ppmq, Zqp´new
f – Hn`s
pΓ, Ω1
Hp,Zqf .
0 Ñ Div0
Hp Ñ Div Hp
deg
Ñ Z Ñ 0 (Γ-equivariant).
Hn`s`1pΓ, Zq
δ
Ñ Hn`spΓ, Div0
Hpq Ñ Hn`spΓ, Div Hpq Ñ Hn`spΓ, Zq
Set ωf to be a fixed basis of Hn`spΓ, Ω1
Hp,Zqf .
Conjecture (Guitart–M.–S¸ eng¨un)
Set
Λf “
#
ˆ
ż
δpcq
ωf : c P Hn`s`1pΓ, Zq
+
Ă p¯Qˆ
p qd
.
Then Λf Ă pQˆ
p qd, and Af p¯Qpq – p¯Qˆ
p qd{Λf .
(Greenberg–Stevens, Dasgupta–Greenberg, Longo–Rotger–Vigni,
Greenberg–Seveso): F “ Q.
(Spiess): F totally real, B “ M2pFq, Qp “ Qp and d “ 1.
Marc Masdeu p-adic periods 10 / 15

12. Example surface
F “ Qpαq “ NumberField/3.1.23.1
§ F “ Qpαq, with fαpxq “ x3
´ x2
` 1, discpFq “ ´23, signature p1, 1q.
B “ Fxi, jy, i2 “ 9α2 ´ 3α ´ 11, j2 “ ´2α2
§ discpBq “ d “ p8α2
´ 10α ´ 1q821.
p “ p´2α2 ` αq7
For l “ pα2 ` α ´ 2q11, the operator Tl has charpoly x2 ´ 2x ´ 19.
; f such that Kf » Qp
?
5q.
Integration pairing gives Λf “
´
A0 B0
C0 D0
¯
“ ZrTls ¨ p A0 B0 q:
A0 “ 7´4
¨ 27132321333884163473566078077966608077268973477 pmod 752
q
B0 “ 397745278075295216478310410412961033205591801491513 pmod 760
q.
Marc Masdeu p-adic periods 11 / 15

13. Equations from periods: abelian surfaces
Writing Af “ JacpXf q
Suppose that Af is principally polarizable, so that Af “ JacpXf q for a
genus-2 hyperelliptic curve Xf . Can we find an equation for Xf ?
Expect Af p¯Qpq – pQˆ
p q2{Λf .
Λf “ x
` A
B
˘
,
` B
D
˘
y ; p1 “ pBDq´1{2, p2 “ pABq´1{2, p3 “ B1{2.
Teiteilbaum’s thesis: D 3 power series in the variables p1, p2, p3:
ιkppq “
ÿ
pi,jqPZ2
a
pkq
i,j pi
1pj
2p
pi´jq
3 .
that express the absolute Igusa invariants of Xf in terms of p.
From N “ pD guess the discriminant I10 “ u ¨ 2a ¨ N2, with u P Oˆ
F .
ι1 “ I5
2 {I10 ; I2, ι2 “ I3
2 I4{I10 ; I4, ι3 “ I2
2 I6{I10 ; I6.
Mestre’s algorithm:
; genus-2 hyperelliptic curve Xf with invariants pI2 : I4 : I6 : I10q.
Marc Masdeu p-adic periods 12 / 15

14. Moving inside the isogeny class
Problem
Af is determined up to isogeny, so we should allow for “isogenous” Λf .
Recall that Cˆ
p {qZ
1 „ Cˆ
p {qZ
2 ðñ D y, z P Z‰0 with qy
1 “ qz
2.
What is the right analogue in higher dimension?
Theorem (Kadziela)
Let V1, V2 P MdˆdpQpq whose columns generate Λ1 and Λ2. Then
p¯Qˆ
p qd{Λ1 is isogenous to p¯Qˆ
p qd{Λ2 if and only if
V Y
1 “ Z
V2, for some Y, Z P MdˆdpZq.
Marc Masdeu p-adic periods 13 / 15

15. Example surface (II)
Recall the periods
A0 “ 7´4
¨ 27132321333884163473566078077966608077268973477 pmod 752
q
B0 “ 397745278075295216478310410412961033205591801491513 pmod 760
q.
Guess Kadziela matrices Y “
ˆ
´1 ´1
´1 0
˙
Z “
ˆ
1 1
1 0
˙
.
New set of periods:
A “ 7´1
¨ 180373636240760651045145390062543188665673147874 ` Op755
q
B “ 101858856942719452845868815022429183828273612324 ` Op756
q
Invariants:
ι1 “
I5
2
I10
“ 7´2
¨ 383000380988298534086703050832398358583029537 ` Op751
q
ι2 “
I3
2 I4
I10
“ 7´2
¨ 216286438165031483296107998530348655636952080 ` Op751
q
ι3 “
I2
2 I6
I10
“ 7´2
¨ 17712448343391292208503851621997332642044090 ` Op750
q.
Marc Masdeu p-adic periods 14 / 15

16. Example surface (III)
The discriminant of the X should have support t2, p, du. In this case, the
fundamental unit of F is α, so we try discriminants of the form
I10 “ αa
2b
p´2α2
` αq2
p8α2
´ 10α ´ 1q2
.
For a “ ´12 and b “ 12 we obtain recognized I2, I4, and I6:
I2 “ 576α2
´ 712α ` 840,
I4 “ 7396α2
´ 11208α ` 9636,
I6 “ 2882256α2
´ 4646648α ` 3543824.
; Xf via Mestre’s algorithm (gives an awful model!).
Xf is a twist of the curve:
X1
f : y2
` px3
` p´α2
´ 1qx2
´ α2
x ` 1qy “
p´α2
` 1qx4
´ 2α2
x3
` p´α2
´ 3α ´ 1qx2
` p´3α ´ 2qx ´ α ´ 1.
(can be checked by comparing the invariants of Xf and X1
f ).
Marc Masdeu p-adic periods 15 / 15

17. Thank you !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
Marc Masdeu p-adic periods 15 / 15