Slides of a talk given at the Building Bridges 4 workshop on Automorphic Forms and Related Topics. Budapest, 2018

1. Towards a theory of p-adic singular moduli
attached to global fields
Automorphic Forms and Related Topics
Building Bridges 4 – Budapest
July 17th, 2018
Marc Masdeu
Universitat Autònoma de Barcelona
Marc Masdeu p-adic singular moduli 0 / 11

2. Hilbert’s 12th
Problem
Let K{Q be a number field.
Kronecker’s Jugendtraum
Describe all abelian extensions of K via “modular functions”.
Easiest case: K “ Q.
Theorem (Kronecker–Weber (1853, 1886))
Qab “
Ť
ně1 Q
´
e
2πi
n
¯
.
The transcendental function fpzq “ e2πiz
yields algebraic values
at rational arguments!
Marc Masdeu p-adic singular moduli 1 / 11

3. K{Q imaginary quadratic: CM theory
The theory of complex multiplication is not
only the most beautiful part of mathematics
but also of the whole of science.
David Hilbert
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4. K{Q imaginary quadratic
Now, suppose K “ Qp
?
Dq with D ă 0.
Replace C with H “ tz P C | pzq ą 0u.
SL2pZq acts on H via
` a b
c d
˘
z “ az`b
cz`d .
Consider now meromorphic functions
SL2pZqzH Ñ C
Fact: every such function is a rational function in
jpzq “
1
q
` 744 ` 196884q ` 21493760q2
` ¨ ¨ ¨ , q “ e2πiz
.
Define HCM “ tτ P H | rQpτq: Qs “ 2u.
Theorem (Kronecker, Weber, Takagi, Hasse)
If τ P HCM, then jpτq P Qpτqab. Moreover, we have:
Kab
“
ď
τPKXHCM
ď
ně1
K pjpτq, ℘pτ{nqq .
Marc Masdeu p-adic singular moduli 2 / 11

5. K{Q real quadratic
From now on: K “ Qp
?
Dq real quadratic (i.e. D ą 0).
Problem: The upper-half plane does not contain real points!
In 1999, Darmon proposed to look at a p-adic analogue of H.
Fix a prime p which is inert in K.
So Kp is a quadratic extension of Qp.
Write Hp “ KpzQp.
Set Γ “ SL2pZr1{psq, which acts on Hp via FLTS.
Induces action on O “ OpHpq “ rigid analytic functions on Hp.
And on M “ MpHpq “ FracpOq “ meromorphic functions on Hp.
New Problem: tf : ΓzHp Ñ Kpu “ Kp (constants).
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6. Darmon’s solution
The problem is that H0
pΓ, Mq “ Kp.
Darmon proposed to look instead at H1
pΓ, Mˆq (note the ˆ).
If we construct a nontrivial J P H1
pΓ, Mˆq, where do we evaluate it?
Given z P HRM
p “ tz P Hp | Qpzq real quadraticu, have:
Γz “ StabΓpzq “ xγzy.
Since γzz “ z, the following quantity is well-defined
Jrzs “ Jpγzqpzq P Kˆ
p .
Theorem (Darmon–Vonk)
For every τ P K X Hp there exists Jτ P H1
pΓ, Mˆq.
Conjecture (Darmon–Vonk)
If τ1 R Γτ, the quantity Jτ rτ1s belongs to QpτqQpτ1qab.
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7. The approach of Darmon–Vonk
Henri Darmon Jan Vonk
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8. The approach of Darmon–Vonk
Fix embeddings K ãÑ R and K ãÑ Qp2 .
For γ “
` a b
c d
˘
P Γ, set:
δγpwq “
$
’&
’%
1 ¯w ă b
d ă w,
´1 ¯w ą b
d ą w,
0 else.
Lemma
The map γ ÞÑ δγ is a cocycle, yielding a class rδs “ rδτ s P H1
pΓ, ∆0pτqq.
Lemma
The class of the cocycle
γ ÞÑ Φˆ
τ pγq “
ś
wPΓτ wεpwqpz ´ wqδγpwq, εpwq “
#
0 |w|p ď 1,
´1 |w|p ą 1
defines a cohomology class in H1
pΓ, Mˆ{Kˆ
p q.
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9. Cocycles
Φˆ
τ P H1
pΓ, Mˆ{Kˆ
p q.
Consider the exact sequence
0 Ñ Kˆ
p Ñ Mˆ
Ñ Mˆ
{Kˆ
p Ñ 0.
Gives rise to the long exact sequence
¨ ¨ ¨ Ñ H1
pΓ, Mˆ
q Ñ H1
pΓ, Mˆ
{Kˆ
p q
δ
Ñ H2
pΓ, Kˆ
p q.
Modulo 12-torsion, H2
pΓ, Kˆ
p q – H1
pΓ0ppq, Kˆ
p q.
Related to modular forms of weight 2, level Γ0ppq.
Proposition (Darmon–Vonk)
Let p be a prime number in tp | p ď 31u Y t41, 47, 59, 71u. Then
`
Φˆ
τ {Φˆ
pτ
˘12
lifts to a cocycle Jˆ
τ P H1
pΓ, Mˆq.
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10. Examples
Somewhere out there is a theory that would
explain my empirical observations, and this
theory has yet to be discovered.
Marc Masdeu p-adic singular moduli 7 / 11

11. Examples
p “ 2
J1`
?
5
2
„
´3 `
?
21
2

“
´37 ˘ 48
?
´3
91
pmod 21000
q.
p “ 7
J1`
?
5
2
r2
?
6s “
1
17
´
3 ˘ 8
?
2 ˘ 12
?
´1 ˘ 2
?
´2
¯
pmod 7400
q.
p “ 5
J?
3r
1 `
?
17
2
s
?
P Qp
?
´1,
?
3,
?
17q,
satisfies the polynomial
194481x8 ` 1100736x7 ` 20364174x6 ` 71994624x5 ` 840839779x4 `
71994624x3 ` 20364174x2 ` 1100736x ` 194481 pmod 5200q
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12. Interlude: Stark–Heegner points
Harold Stark Kurt Heegner
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13. Interlude: Stark–Heegner points
In 1999, Darmon constructed cocycles
¯Φˆ
E P H1
pΓ, Mˆ
{Kˆ
p q
attached to an elliptic curve of conductor p.
Tate: EpKpq – Kˆ
p {qZ
E.
Showed that ¯Φˆ
E can be lifted to
Φˆ
E P H1
pΓ, Mˆ
{qZ
Eq.
Conjecture (Darmon, 1999)
ΦErτs P EpQpτqab
q.
Notation: ∆pτq “ Div Γ0τ, and ∆0pτq “ Div0
Γ0τ.
For τ “ 8, write ∆ “ ∆p8q “ Div P1
pQq (and similar for ∆0).
Note also that Φˆrτs “ xΦˆ, Θτ y, with Θτ “ rγτ bτs P H1pΓ0, ∆pτqqs.
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14. Darmon–Vonk classes were invented in 20th
century!
Set Mτ “ tf P M | divpfq Ă Γτu.
Recall div ¯Φˆ
τ “ rδτ s.
¯Φˆ
τ H1
pΓ, Mˆ
τ {Kˆ
p q
rδτ s H1
pΓ, DivΓτq
ϕDIT
τ H1
pΓ0, ∆pτqq
:torsion
H1pΓ0, ∆ b∆pτqq H1pΓ0, ∆pτqq
´
∆0 b∆pτq
¯
Γ0
´
∆ b∆pτq
¯
Γ0
rγτ b τs rpγτ 8 ´ 8q bτs
div–
η
Shapiro–
Borel–Serre–
δ
Marc Masdeu p-adic singular moduli 10 / 11

15. Conclusion
The cocycle Φˆ
τ “comes from” the Stark–Heegner class rγτ bτs.
The quantity computed by Darmon–Vonk is “just”
Φˆ
τ rzs “ xpη´1
˝ δqΘτ , Θτ1 y,
Can interpret as a “winding number” between cycles Θτ and Θτ1 .
In order to generalize to other arithmetic groups, need to find
analogues of ϕDIT
τ P H1
pΓ0, ∆pτqq.
With Xavier Guitart (UB) and Xavier Xarles (UAB), we are trying to
find analogues in number fields and function fields.
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16. Thank you !
http://www.mat.uab.cat/~masdeu/
Marc Masdeu p-adic singular moduli 11 / 11