Talk given at the workshop "Automorphic Forms: Theory and Computation"

1. p-adic periods of abelian varieties
attached to
GL2-automorphic forms
Automorphic Forms:
Theory and Computation
London, September 6, 2016
Xavier Guitart Marc Masdeu
Universitat de Barcelona University of Warwick
Marc Masdeu p-adic periods 0 / 20

2. The Eichler–Shimura construction
f “
ř
anqn P S2pΓ0pNqq a normalized cuspidal newform.
Kf “ Qptanuně1q, a totally-real number field of degree d ě 1.
Modular curve X0pNqpCq “ Γ0pNqzH˚ ãÑ JacpX0pNqqpCq.
Hecke operators ; If ; JacpX0pNqqpCq{If – Af pCq “ Cd{Λf .
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.
The period lattice Λf is explicit. Letting fj “ σjpfq, we have:
Λf “ 2πi
Bˆż
γ
f1pzqdz, . . . ,
ż
γ
fdpzqdz
˙
: γ P H1
´
X0pNq, Z
¯F
Ă Cd
.
This allows in some cases to recover “equations” for Af .
§ Cremona (d “ 1).
§ Wang, Guardia, Gonz´alez, Gonz´alez-Jim´enez,. . . (d “ 2, 3).
Marc Masdeu p-adic periods 1 / 20

3. Hilbert modular forms (F totally real)
F totally real. rF : Qs “ r, fix σ: F ãÑ R.
Suppose f P S2pΓ0pNqq has field Kf of degree d.
S2pΓ0pNqq Q f ; ˜ωf P Hr
pΓ0pNq, Cq ; Λf Ď Cd
.
Conjecture (Oda, Darmon, Gartner)
Cd{Λf is isogenous to Aσ
f pCq, for some Af {F “attached to f”.
Proven when F real quadratic, d “ 1 and f a base change from Q.
Exploited in very restricted cases (Demb´el´e, Stein+7).
Explicitly computing Λf is hard.
§ No computations in quaternionic setting (except for Voight–Willis?).
F not totally real: conjectural construction of Guitart-M.-S¸ eng¨un.
Marc Masdeu p-adic periods 2 / 20

4. Plan of the Talk
To describe p-adic conjectural analogue of these
constructions, valid when F is a number field of
arbitrary signature.
AUTOMORPHIC FORMS:
THEORY AND COMPUTATION
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5. AUTOMORPHIC FORMS
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6. Quaternionic automorphic forms
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)
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7. Quaternionic automorphic forms (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.
H˚
pΓD
0 pnq, Qq “
à
f
H˚
pΓD
0 pnq, Qqf (irreducible factors).
Each f cuts out a field Kf , s.t. rKf : Qs “ dim H˚
pΓD
0 pnq, Cqf .
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8. Abelian varieties from cohomology classes
Conjecture (Taylor, ICM 1994)
Let f P Hn`s
pΓD
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
Remarks
Modularity says Af ; f. This is some sort of converse.
Which of the two directions is harder?
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9. THEORY
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10. 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 Qp of F.
Choose a coprime factorization N “ pDm, with D “ discpB{Fq.
Starting data: f P Hn`s
pΓD
0 ppmq, Cqnew newform.
§ Kf “ field generated by Hecke eigenvalues, d “ rKf : Qs.
We will describe a p-adic torus p¯Qˆ
p qd{Λf .
We will conjecture that Af p¯Qpq „ p¯Qˆ
p qd{Λf .
Two questions arise
1 What if Af is of dimension 2d (“QM case”)?
2 What if Af p¯Qpq doesn’t look like a torus?
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11. Ruling out the QM case
Theorem 1
Suppose that f is attached to Af {F of dimension 2d, conductor M, and
D ãÑ EndpAf q bQ, with D a quaternion division algebra.
Then f has complex multiplication.
Proof:
D “ EndpAf q bQ:
(D1 “ EndpAf q bQ (division) acts on the 4d-dim’l space H1pAf , Cq.
Therefore rD1 : Qs ď 4d. Since rD: Qs “ 4d, then D1 “ D.)
Classification of endomorphism algebras of simple abelian varieties:
1 D is totally indefinite, or
2 D is totally definite.
Lemma
If D is totally indefinite, then p | M ùñ p4d | M.
If D is totally definite, then f has complex multiplication.
We will henceforth assume that f has no CM.
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12. Proof sketch (of first part)
Lemma
If D “ EndpAf q bQ is totally indefinite, then p | M ùñ p4d | M.
A{F of GL2-type ùñ V pAq is sum of 2d two-dimensional reps.
§ vppMq “ 2dpcodimpV
Ip
ρ q ` δpq.
Need to rule out vppMq “ 2d ( ðñ codimpV
Ip
ρ q “ 1, δp “ 0).
A1 “ connected component of special fibre of the N´eron model of A.
0 Ñ T ˆ U Ñ A1
Ñ B Ñ 0. pChevalleyq
§ Suppose that vppMq “ 2d.
§ Show that T “ 0 (otherwise, dim T “ 4d, which is impossible).
§ T “ 0 ùñ A has potentially good reduction ùñ vppMq “ 2 dimpUq.
§ Conclude that dimpUq “ dimpBq “ d.
For simplicity, suppose that B is simple.
§ Then EndpBq bQ “ D ùñ !!!.
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13. Af is p-adically uniformizable
Want to generalize Tate’s uniformization:
p conductorpEq ùñ Ep¯Qpq “ ¯Qˆ
p {qZ for some q P Qp.
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 pQˆ
p qd such that
Ap¯Qpq – p¯Qˆ
p qd
{Λ.
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.
§ So the second condition above is necessary.
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14. Af is p-adically uniformizable (II)
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 pQˆ
p qd such that Ap¯Qpq – p¯Qˆ
p qd{Λ.
Proof sketch:
0 Ñ T ˆ U Ñ A1
Ñ B Ñ 0
If t “ dimpTq ą 0 then K ãÑ MtpQq, so t ě d. Hence A1 “ T.
§ So we need to show that A does not have potentially good reduction.
Look at V pAq – ‘σV σ
ρ , with ρ: GF Ñ GL2pQ q.
By assumption, V
Ip
ρ is one-dimensional.
Need to see that ρpIpq is infinite. But ρ|Ip „
ˆ
1 ‹
0 1
˙
with ‹ ‰ 0
(because detpVρq is the (unramified) cyclotomic character), so done.
Marc Masdeu p-adic periods 10 / 20

15. 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 discrete Γ Ă PGL2pQpq, get induced pairing (cap product)
Hi
pΓ, Ω1
Hp,Zq ˆ HipΓ, Div0
Hpq
x¨,¨y
// Cˆ
p
´
φ,
ř
γ γ bDγ
¯
//
ř
γ ˆ
ż
Dγ
φpγq.
Marc Masdeu p-adic periods 11 / 20

16. 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 12 / 20

17. 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`s
pΓ, Ω1
Hp,Zqf .
Conjecture (Guitart–M.–S¸ eng¨un)
Set Λf “ xωf , δpcqy: 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 13 / 20

18. COMPUTATION
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19. Equations from periods: elliptic curves
Λf “ xqy gives us q
?
“ qE.
Assume vppqq ą 0 (otherwise, replace q ÞÑ q´1).
Get
jpqq “ q´1
` 744 ` 196884q ` ¨ ¨ ¨ P Qˆ
p .
From N guess the discriminant ∆E.
§ Only finitely-many possibilities. . .
jpqq “ 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.
Marc Masdeu p-adic periods 14 / 20

20. Example curve (joint with X. Guitart and H. S¸ eng¨un)
F “ Qpαq “ NumberField/4.2.1732.1
§ fαpxq “ x4
´ x3
` 3x ´ 1, discpFq “ ´1732, signature p2, 1q.
N “ pα ´ 2q13 “ p13.
B{F ramified only at all infinite real places of F.
There is a rational eigenclass f P S2pΓOF
0 pNqq.
From f we compute ωf P H1
pΓ, Ω1
Hp,Zq and Λf “ xqf y.
qE
?
“ qf “ 8 ¨ 13 ` 11 ¨ 132 ` 5 ¨ 133 ` 3 ¨ 134 ` ¨ ¨ ¨ ` Op13100q.
jE “ 1
13
´
´ 4656377430074α3
` 10862248656760α2
´ 14109269950515α ` 4120837170980
¯
.
c4 “ ´7936α3 ` 24320α2 ´ 35328α ` 20225.
c6 “ 3717634α3 ´ 9135590α2 ` 12165066α ´ 4343229.
E{F : y2
`
`
α3
` 2
˘
xy ` pα ` 1q y “
x3
`
`
´α3
´ α ´ 3
˘
x2
`
`
166α3
´ 506α2
` 736α ´ 421
˘
x
´ 4208α3
` 10456α2
´ 13994α ` 5180
Marc Masdeu p-adic periods 15 / 20

21. Example surface (joint with X. Guitart)
F “ Qpαq “ NumberField/3.1.23.1
§ 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.
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22. Equations from periods: abelian surfaces
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.
Write Xf : y2 “ xpx ´ 1qpx ´ x1qpx ´ x2qpx ´ x3q, and define
λ1 “ 1 ´ x´1
1 , λ2 “ p1 ´ x2q´1
, λ3 “ x3.
Teiteilbaum’s thesis: D 3 power series in the variables p1, p2, p3:
λk “ λkppq “
ÿ
pi,jqPZ2
a
pkq
i,j pi
1pj
2p
pi´jq
3 ,
and from this one can compute the absolute Igusa invariants of Xf .
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 17 / 20

23. 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 MdpQpq 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 MdpZq.
Remark
This is equivalent to
Y logppV1q “ logppV2qZ, Y ordppV1q “ ordppV2qZ.
Marc Masdeu p-adic periods 18 / 20

24. 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 19 / 20

25. 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 recognized I2, I4, and I6:
I2 “ 576α2
´ 712α ` 840,
I4 “ 7396α2
´ 11208α ` 9636,
I6 “ 2882256α2
´ 4646648α ` 3543824.
; Xf via Mestre’s algorithm (Magma 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 20 / 20

26. Thank you !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
Marc Masdeu p-adic periods 20 / 20