This document summarizes a seminar on quaternionic rigid meromorphic cocycles. It discusses generalizing the construction of Darmon-Vonk classes from real quadratic fields to totally real fields using quaternion algebras. Specifically: 1) It introduces Darmon-Vonk classes in the context of real quadratic fields and proposes generalizing this to quaternion algebras over totally real fields. 2) It outlines defining cohomology classes using embeddings of orders into the quaternion algebra and studying their properties, including pairings with other cohomology classes. 3) The main result is that for certain quaternion algebras and primes, the generalized classes satisfy a conjecture relating them to class field theory, extending the Darmon

1. QUATERNIONIC RIGID MEROMORPHIC COCYCLES
ONLINE RESEARCH GROUP SEMINAR, GRONINGEN
January 20, 2021
Marc Masdeu
Universitat Autònoma de Barcelona

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
¯
“ A transcendental function ( e2πiz )
yields algebraic values
at rational arguments!
Marc Masdeu Quaternionic rigid meromorphic cocycles 1 / 19

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.
Marc Masdeu Quaternionic rigid meromorphic cocycles 1 / 19

4. CM theory
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 SL2pZq-invariant meromorphic functions:
SL2pZqzH Ñ C
Fact: every such function is a rational function in
jpzq “
1
q
` 744 ` 196884q ` 21493760q2
` ¨ ¨ ¨ , q “ e2πiz
.
Theorem (Kronecker, Weber, Takagi, Hasse)
If τ satisfies a quadratic polynomial in Zrxs, then jpτq P Qpτqab.
Moreover, we have:
Kab
« Qab
¨
ď
τPKXH
Kpjpτqq.
Marc Masdeu Quaternionic rigid meromorphic cocycles 2 / 19

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 “ P1pKpq r P1pQpq (rigid-analytic space).
Set Γ “ SL2pZr1{psq, which acts on Hp via
` a b
c d
˘
τ “ aτ`b
cτ`d .
Induces action on O “ OpHpq “ rigid analytic functions on Hp.
And on M “ MpHpq “ FracpOq “ meromorphic functions on Hp.
New Problem: E nonconstant Γ-invariant meromorphic functions:
H0pΓ, Mˆq “ Kˆ
p .
Marc Masdeu Quaternionic rigid meromorphic cocycles 3 / 19

6. Darmon–Vonk classes
Henri Darmon Jan Vonk
Marc Masdeu Quaternionic rigid meromorphic cocycles 3 / 19

7. Darmon–Vonk classes H1
pΓ, Mˆ
q
Fix an embedding K ãÑ R. Let τ P K r Q.
Write GalpK{Qq “ xσy.
For γ “
` a b
c d
˘
P Γ and w P Γτ Ă K, set:
δγpwq “
$
’
&
’
%
`1 σpwq ă a
c ă w,
´1 wa
c σpwq,
0 else.
Lemma (Darmon–Vonk)
There is a cohomology class Φˆ
τ P H1pΓ, Mˆ{Kˆ
p q such that
div Φˆ
τ pγq “
ÿ
wPΓτ
δγpwq ¨ pwq.
Obstruction to lifting Φˆ
τ to H1pΓ, Mˆq governed by H2pΓ, Kˆ
p q.
Modulo 12-torsion, H2pΓ, Kˆ
p q – H1pΓ0ppq, Kˆ
p q.
§ Related to modular forms of weight 2, level Γ0ppq (Eichler–Shimura).
Marc Masdeu Quaternionic rigid meromorphic cocycles 4 / 19

8. Darmon–Vonk classes (II)
Proposition (Darmon–Vonk)
Let p be monstruous prime, i.e. p ď 31 or p P t41, 47, 59, 71u.
Then
`
Φˆ
τ {Φˆ
pτ
˘12
lifts to a cocycle Jˆ
τ P H1pΓ, Mˆq.
The cocycle Jˆ
τ is their prosed analogue of j.
Given τ1 P HRM
p “ tz P Hp | Qpzq real quadraticu, have:
Γτ1 “ StabΓpτ1
q “ xγτ1 y.
Assume that τ1 R Γτ.
Since γτ1 τ1 “ τ1, the following quantity is well-defined
JDV
p pτ, τ1
q “ Jˆ
τ pγτ1 qpτ1
q P Kˆ
p .
Conjecture (Darmon–Vonk)
If τ1 R Γτ, the quantity JDV
p pτ, τ1q belongs to KabQpτ1qab.
Marc Masdeu Quaternionic rigid meromorphic cocycles 5 / 19

9. Quaternionic generalisation
Joint work with Xavier Guitart (UB) and Xavier Xarles (UAB).
Xavier Guitart Xavier Xarles
We generalized the construction of Darmon–Vonk to quaternion
algebras over totally real fields, avoiding the S-arithmetic group Γ.
Marc Masdeu Quaternionic rigid meromorphic cocycles 5 / 19

10. Quaternionic generalisation: homology
B indefinite quaternion algebra split at p.
Choose splittings ι8 : B ãÑ M2pRq and ιp : B ãÑ M2pQpq.
Let R Ă B a maximal order, set Γ0p1q “ Rˆ
1 (norm-one units).
Later we’ll consider Rppq Ă R and Γ0ppq “ Rppqˆ
1 .
Notation: ∆pτq “ Div Γ0p1qτ, and ∆0pτq “ Div0
Γ0p1qτ.
If ψ: Oψ ãÑ R is an optimal embedding (Oψ Ă Kψ an order),
γψ “ ψpuq P Γ0p1q, xuy “ Oˆ
1 {tors.
If γψτψ “ τψ, then θ̄ψ “ rγψ bτψs P H1pΓ0p1q, ∆pτψqq.
If H1pΓ0p1q, Zq is torsion, then θψ P H1pΓ0p1q, ∆0pτψqq.
§ θψ considered by M.Greenberg ; quaternionic Darmon points.
§ If H1pΓ0p1q, Zq non-torsion, may use Hecke operators to lift θ̄ψ.
Marc Masdeu Quaternionic rigid meromorphic cocycles 6 / 19

11. Quaternionic generalisation: cohomology
Fix x P H, and choose (another) ψ: Oψ ãÑ R.
ι8pγψq “
` a b
c d
˘
P GL2pRq has two fixed points τ˘
8 on P1pRq.
ιppγψq fixes two points τ, τ1 P Hp.
Given γ P Γ0p1q, and w˘ “ γwτ˘
8 P Γ0p1qτ`
8, set δγpwq “ δγpw`q:
x
γx
w+
w−
C(x, γx)
C(w−
, w+
)
δγ(w) = 1
Fact: The set tw` P Γ0p1qτ`
8 | δγpwq ‰ 0u is finite.
; γ ÞÑ
ÿ
wPΓ0p1qτ`
8
δγpwq ¨ pwpq P Z1
pΓ0p1q, ∆pτqq. (wp “ gτ if w “ gτ`
8)
; rϕψs P H1pΓ0p1q, ∆pτqq (independent of the choice of x).
If H1pΓ0p1q, Zq is torsion, can lift to H1pΓ0p1q, ∆0pτqq.
Marc Masdeu Quaternionic rigid meromorphic cocycles 7 / 19

12. Pairings and overconvergent cohomology
rϕψ1 s P H1
pΓ0p1q, ∆0pτψ1 qq, θψ2 P H1pΓ0p1q, ∆0pτψ2 qq
Cross-ratio: xxQ ´ P, Q1 ´ P1yy “ pQ1´QqpP1´Pq
pQ1´PqpP1´Qq P pKψ1 Kψ2 qˆ.
H1 pΓ0ppq, Kprrptssqord,`
–
 // H1pΓ0ppq, Kprrptssq
1´Up
// // H1pΓ0ppq, Kprrptssqnilp
Φ
P
H1pΓ0ppq, Kpptqˆ{Kˆ
p q
dlogp
OO
M2pΓ0ppqq H1pΓ0p1q, ∆0pτψ2 qq
Tr
OO
Φ “
ř
ně0 Un
p pdlogp Tr ϕψ1 q.
Integration: Kprrptss ˆ ∆0pτψ2 q Ñ Kˆ
p , xf, Q ´ Py “ expp
ż Q
P
fpzqdz.
They induce corresponding pairings in H1 ˆ H1.
Marc Masdeu Quaternionic rigid meromorphic cocycles 8 / 19

13. Main result
Φ P H1
pΓ0ppq, Kprrptssq, ϕψ1 P H1
pΓ0p1q, ∆0pτψ1 qq, θψ2 P H1pΓ0p1q, ∆0pτψ2 qq
JGMX
p pψ1, ψ2q :“ xΦ, cores θψ2 y ˆ xxϕψ1 , θψ2 yy.
Conjecture
If M2pΓ0p1qq “ 0, then JGMX
p pψ1, ψ2q
¨
P Kab
ψ1
Kab
ψ2
.
Theorem (Guitart–M.–Xarles)
Assume B – M2pQq and p monstruous. Then:
JGMX
p pψ1, ψ2q
¨
“ JDV
p pψ1, ψ2q.
Extends directly to almost totally definite quaternion algebras over
totally-real number fields.
The existence of Φ doesn’t require p “monstruous”.
We insist in working with arithmetic groups Γ0p1q and Γ0ppq.
Marc Masdeu Quaternionic rigid meromorphic cocycles 9 / 19

14. Relation to Stark–Heegner points
Another construction of classes in H1pΓ0ppq, Kprrptssq`.
Let E{Q be an elliptic curve of conductor p discpBq.
; fE P M2pΓ0ppqq (quaternionic modular form).
ΦE PH1 pΓ0ppq, Kprrptssqord,`
–

// H1pΓ0ppq, Kprrptssq // // H1pΓ0ppq, Kprrptssqnilp
fE P M2pΓ0ppqq H1pΓ0ppq, Kpptqˆ{Kˆ
p q
dlogp
OO
Control Theorem (Stevens): lift fE to ΦE P H1pΓ0ppq, Kprrptssq`.
Tate uniformization: ηp : Cˆ
p Ñ EpCpq.
Conjecture (Darmon, Greenberg)
JGMX
p pE, ψq :“ ηp pxΦE, cores θψyq P EpKab
ψ q Ă EpCpq.
Marc Masdeu Quaternionic rigid meromorphic cocycles 10 / 19

15. Examples
The fun of the subject
seems to me to be in
the examples.
Marc Masdeu Quaternionic rigid meromorphic cocycles 10 / 19

16. Large genus example
Let p “ 37 (not monstruous). dim H1pΓ0ppq, Qq “ 1 ` 2 ¨ 2 “ 5.
Consider K1 “ Qp
?
3 ¨ 19q, K2 “ Qp
?
61q.
We get
J37pψ1, ψ2q “ 17662185365697700344487705388816
`9911534933329042169401982379602
˜
1 `
?
57
2
¸
` Op3720
q
We check that J37pψ1, ψ2q is the image of an element
α P M “ Qp
?
´3,
?
´19,
?
57q such that ι37pαq » J37pψ1, ψ2q.
Ideal generated by α is supported over primes of norms 11 and 17,
as predicted by an analogue of the Gross–Zagier factorisation
theorem (they could only be 11, 17, 23, 31, 37, 67, 79).
Marc Masdeu Quaternionic rigid meromorphic cocycles 11 / 19

17. Quaternionic example
B “
´
6,´1
Q
¯
, discriminant 6. Set p “ 5
R “ x1, i, 1 ` i ` j, i ` ky.
OK1 “ Zp1`
?
53
2 q and ψ1p1`
?
53
2 q “ 1{2 ´ 3{2i ´ 1{2j.
; γψ1
“ 51{2 ` 21{2i ` 7{2j.
OK2 “ Zp
?
23q and ψ2p
?
23q “ 2i ` j.
; γψ2 “ 1151 ` 480i ` 240j.
We get
J5pψ1, ψ2q “ 50971141466526826096289662898361868496463698468806135561183036939036
`9674029354607221223815165708202713711819464972332940921086896674730
1 `
?
53
2
`Op597
q
The period J5pψ1, ψ2q satisfies, modulo roots of unity, the polynomial
fpxq “ 41177889x4
`7867012x3
`33058502x2
`7867012x`41177889.
One has L “ K1K2pfq is the compositum of the narrow class fields
of K1 and K2, as predicted by the conjecture.
Marc Masdeu Quaternionic rigid meromorphic cocycles 12 / 19

18. Exchange of primes (global object?)
Idea: Compare Jp with discpBq “ q ¨ r vs Jq with discpBq “ p ¨ r.
K1 “ Qp
?
53q (h`
K1
“ 1), K2 “ Qp
?
23q (h`
K2
“ 2).
Calculate J3p53, 23q P Q32 using B of discriminant 10.
2263329212681251489468 ` 6644010739654744556634 1`
?
53
2
` Op346
q
Calculate J5p53, 23q P Q52 using B of discriminant 6.
76500603105371600283097752216081 ` 71876326310173029976331143056540 1`
?
53
2
` Op547
q
Compositum M of the narrow class fields is generated by a root of
x8
´ 4x7
` 84x6
´ 238x5
` 1869x4
´ 3346x3
` 7260x2
´ 5626x ` 3497.
There is ι3 : M ãÑ Q32 and ι5 : M ãÑ Q52 .
We check that there is α P M and units u1, u2 in K1K2 Ă M such
that:
ι3pαu1q “ J3p53, 23q and ι5pαu2q “ J5p53, 23q.
Ideal generated by α is supported over primes of norms 2 and 31.
Marc Masdeu Quaternionic rigid meromorphic cocycles 13 / 19

19. Example over real quadratic
F “ Qp
?
5q, and let w “ 1`
?
5
2 .
B{F of discriminant 2OF given by B “
´
´w,´2
F
¯
.
R “ x1, i, 2w ` 2i ` j, 2w ` 2 ` 2wi ` ky maximal order.
p “ p´3w ` 2q, of norm 11.
K1 “ Fp
?
1 ´ 2wq, and ψ1 : OK1 ãÑ R given by
?
1 ´ 2w ÞÑ pw ´ 2qi ´ j.
; γψ1 “ w ´ 2 ` p2w ´ 3qi ` pw ´ 1qj.
K2 “ Fp
?
9 ´ 14wq, and ψ2 : OK2 ãÑ R, given by
?
9 ´ 14w ÞÑ p´3w ` 2qi ` pw ´ 2qk.
; γψ2 “ ´55w ` 88 ` p´50w ` 81qi ` p34w ´ 55qk.
Obtain
Jppψ1, ψ2q “ 2650833861085011569846208847449970229624664608755690791954838 ` Op11
59
q,
Jppψ1, ψ2q satisfies the polynomial
25420x4
´ 227820x3
` 2200011x2
´ 27566220x ` 372174220.
This generates a quadratic unramified extension of K1K2.
Marc Masdeu Quaternionic rigid meromorphic cocycles 14 / 19

20. Tables for D “ 6, p “ 5
+ 8 12 53 77 92 93
8 - - 3, 5 2, 3 5
12 - 5 ? 2 -
53 - 5 ? 3, 23, 31 2, 5, 41
77 3, 5 ? ? ? ?
92 2, 3 2 3, 23, 31 ? ?
93 5 - 2, 5, 41 ? ?
- 8 12 53 77 92 93
8 H - 3, 5 2, 3 2, 5
12 H 2,5 ? H H
53 - 2, 5 3, 5 2, 3, 23, 31 2, 5, 41
77 3, 5 ? 3, 5 ? ?
92 2, 3 H 2, 3, 23, 31 ? ?
93 2, 5 H 2, 5, 41 ? ?
Marc Masdeu Quaternionic rigid meromorphic cocycles 15 / 19

21. Tables for D “ 10, p “ 3
+ 5 8 53 77 92
5 - - 3 2, 3
8 - - 3, 5 2, 3
53 - - 3, 5, 31 2, 3, 23, 31
77 3 3, 5 3, 5, 31 ?
92 2, 3 2, 3 2, 3, 23, 31 ?
- 5 8 53 77 92
5 - - 3 3
8 - - 3, 5 2, 3
53 - - ? 3, 23, 31
77 3 3, 5 ? ?
92 3 2, 3 3, 23, 31 ?
Marc Masdeu Quaternionic rigid meromorphic cocycles 16 / 19

22. Tables for D “ 22, p “ 3
+ 8 29 44 77
8 - - 2
29 - 3 2
44 - 3 2, 11
77 2 2 2, 11
- 8 29 44 77
8 - H ?
29 - ? ?
44 H ? ?
77 ? ? ?
Marc Masdeu Quaternionic rigid meromorphic cocycles 17 / 19

23. A cartoon
Darmon – Vonk
H∗
H∗
2020
Non-archimedean
Ramification
?
Marc Masdeu Quaternionic rigid meromorphic cocycles 18 / 19

24. Final remarks
The cohomology classes we constructed span (via Shapiro’s
isomorphism) all of H1pΓ, Mˆ{Kˆ
p q.
§ c.f. recent work of Lennart Gehrmann.
Darmon–Vonk already observe that Jppψ1, ψ2q is related to
Jppψ2, ψ1q´1. We conjecture that the same holds for this new
classes (but we don’t have a proof).
We haven’t been able to relate JppE, τq with Jppτ, Eq, but we expect
a similar relation to the above to hold.
Function field analogue?
Marc Masdeu Quaternionic rigid meromorphic cocycles 19 / 19

25. Thank you !
http://www.mat.uab.cat/˜masdeu/
Marc Masdeu Quaternionic rigid meromorphic cocycles 19 / 19