Explains a paper of Bertolini-Darmon on the rationality of Stark-Heegner (Darmon) points.

1. Traces of rationality
of Darmon points
[BD09] Bertolini–Darmon, “The rationality of Stark–Heegner points over genus fields of real quadratic fields”
Seminari de Teoria de Nombres de Barcelona
Barcelona, January 2017
Marc Masdeu
University of Warwick
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2. Introduction
E{Q an elliptic curve of conductor N “ pM, with p M.
K{Q a real quadratic field, satisfying:
§ p is inert in K
§ | M ùñ split in K.
These conditions imply that ords“1 LpE{K, sq is odd.
BSD predicts DPK P EpKq nontorsion. . .
. . . but Heegner points are not readily available, since K is real!
§ For example H X K “ H.
Darmon’s insight: consider Hp “ P1pCpq P1pQpq instead.
§ Constructed a point Pψ P EpCpq for each ψ: OK ãÑ M2pQq.
Conjecture (2001): Pψ P EpKabq, and behave like Heegner points.
In particular, he conjectured that
ř
ψ Pψ P EpKq.
§ This is (a particular case of) what Bertolini–Darmon proved in 2009.
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3. Plan
1 Define Darmon points (a.k.a. Stark–Heegner points).
2 State the main result of [BD09].
3 Sketch the proof.
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4. Measure-valued modular symbols and integration
Let Γ “
` a b
c d
˘
P M2pZr1{psq X SL2pQq: N | c
(
.
Meas0
pP1pQpqq “ the Γ-module of measures on P1pQpq having total
measure 0.
From previous talks: E ; f P S2pΓ0pNqq ; If P Symbnew
Γ0ppMqpZq`.
Proposition
D! µ “ µf P H1pΓ, Meas0
pP1pQpqqq satisfying:
µγpZpq “ If t8 Ñ γ8u for all γ P Γ0pNq Ă Γ.
Recall q “ qE P pZp the Tate period attached to E{Qp.
There is a Γ-equivariant pairing Meas0
pP1pQpqq ˆ Div0
Hp Ñ Cp:
pµ, τ2 ´ τ1q ÞÑ
ż
P1pQpq
logq
ˆ
t ´ τ2
t ´ τ1
˙
dµptq.
Cap product induces a pairing
x¨, ¨y: H1
pΓ, Meas0
pP1
pQpqqq ˆ H1pΓ, Div0
Hpq Ñ Cp.
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5. Indefinite integrals
x¨, ¨y: H1
pΓ, Meas0
pP1
pQpqqq ˆ H1pΓ, Div0
Hpq Ñ Cp.
Consider the short exact sequence of Γ-modules
0 Ñ Div0
Hp
ι
Ñ Div Hp
deg
Ñ Z Ñ 0.
Taking Γ-homology, get a long exact sequence
¨ ¨ ¨ Ñ H2pΓ, Zq
δ
Ñ H1pΓ, Div0
Hpq
ι˚
Ñ H1pΓ, Div Hpq
deg˚
Ñ H1pΓ, Zq Ñ ¨ ¨
Fact: H1pΓ, Zq “ Γab is a finite abelian group, say of size eΓ.
So Θ P H1pΓ, Div Hpq ; Θ0 P H1pΓ, Div0
Hpq such that
ι˚pΘ0q “ eΓΘ.
Define xµf , Θy :“ 1
eΓ
xµf , Θ0y, which is an “indefinite integral”
x¨, ¨y: H1
pΓ, Meas0
pP1
pQpqqq ˆ H1pΓ, Div Hpq Ñ Cp.
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6. Darmon points
Recall K{Q our real quadratic field, of discriminant D.
Cl`
pKq the narrow class group of K.
Consider optimal embeddings
ψ: OK ãÑ M0pMq “ t
` a b
c d
˘
P M2pZq : M | cu.
We have an action:
Cl`
pKq ñ EmbpOKq “ t conjugacy classes of optimal embeddingsu.
§ If σ P Cl`
pKq and ψ P EmbpOKq, write ψσ
for the translate.
Global class-field theory gives
rec: Cl`
pKq
»
Ñ GalpH`
K{Kq, H`
K “ narrow Hilbert class field of K.
ψ P EmbpOKq ; ΘK “ rγψ bτψs P H1pΓ, Div Hpq:
§ γψ “ ψpuKq P Γ0pMq Ă Γ, where Oˆ
K,1{tors “ xuKy.
§ τψ is fixed by γψ (one can consistently choose one of two options).
Define Jψ “ xµf , Θψy P Cp.
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7. Conjecture (Darmon, 2001)
1 For each ψ P EmbpOKq, there exists Pψ P EpH`
Kq and t P Qˆ s.t.:
Jψ “ t logEpPψq.
2 For each σ P Cl`
pKq, the points Pψ satisfy
Pψσ “ recpσq´1
pPψq.
Using “multiplicative integrals” ; Pψ satisfying Jψ “ logEpPψq.
§ Part 1 of conjecture then says that Pψ P EpH`
Kq bQ.
Darmon–Green and Darmon–Pollack: numerical evidence.
The construction has been generalized to many different settings
(Greenberg, Gartner, Trifkovic, Guitart–M.–Sengun, Longo–Vigni,
Rotger–Seveso,. . . ).
Guitart–M.: more numerical evidence supporting these conjectures.
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8. Main Theorem of [BD09]
Define: JK “
ÿ
ψPEmbpOK q
Jψ P Cp.
Theorem (Bertolini–Darmon, 2009)
Suppose that:
E has split multiplicative reduction at p (i.e. appEq “ 1).
D prime q ‰ p of multiplicative reduction.
Then:
1 There exists a point PK P EpKq and t P Qˆ such that JK “ t logEpPq.
2 The point PK is of infinite order if and only if L1pE{K, 1q ‰ 0.
In [BD09] they prove the “rationality” not only of JK, but also of
“twisted traces” of the points Jψ by genus characters.
Analogues of this result:
§ For “quaternionic Darmon points” (Longo–Vigni).
§ For “Darmon cycles” (Seveso).
Strategy of proof “inspired by” a very famous proof from 1863. . .
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9. Proof Strategy
1 Construct a p-adic L-function Lppf8{K, kq attached to f8 and K.
2 Relate JK to Lppf8{K, kq:
JK “ L1
ppf8{K, kq.
3 Factor Lppf8{K, kq2 in terms of Mazur–Kitagawa p-adic L-functions:
Lppf8{K, kq2
“ D
k´2
2 Lppf8, k, k{2qLppfK
8 , k, k{2q.
4 Deduce the theorem from previous results on the Mazur–Kitagawa
p-adic L-functions appearing in the RHS.
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10. A p-adic L-function attached to f8 and K
E ; f P Snew
2 pΓ0pNqq ; If “ I`
f P H1pΓ0pNq, Zqw8“1.
If lives in a family: there exists µµ P H1pΓ0pMq, D:
˚qord such that
1 ρ2pµµq “ If , and
2 For all k P U X Zě2, Dλpkq P Cˆ
p such that ρkpµµq “ λpkqIfk
.
For each ψ P EmbpOKq, consider γψ as before, and τψ, ¯τψ the fixed
points by ψpKˆq acting on Hp.
Very Important Fact: Jτ “
ż
pZ2
pq1
logpx ´ τψyqdµµγτ px, yq.
Define Lppf8, ψ, kq as
Lppf8, ψ, kq “
ż
pZ2
pq1
ppx ´ τψqpx ´ ¯τψqq
k´2
2 dµµγψ
px, yq.
Definition
The “square-root p-adic L-function” is:
Lppf8{K, kq “
ÿ
ψPEmbpOK q
Lppf8, ψ, kq.
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11. Lppf8, ψ, kq “
ż
pZ2
pq1
ppx ´ τψqpx ´ ¯τψqq
k´2
2 dµµγψ
px, yq,
Lppf8{K, kq “
ÿ
ψ
Lppf8, ψ, kq.
Theorem A - p-adic Gross–Zagier
We have Lppf8{K, 2q “ 0 and L1
ppf8{K, 2q “ JK.
Proof
Lppf8{K, ψ, 2q “
ş
pZpq1 dµµγτ px, yq “ µµγτ pP1pQpqq “ 0.
L1
ppf8{K, ψ, 2q “ 1
2
ş
pZ2
pq1 plogpx ´ τψyq ` logpx ´ ¯τψyqq dµµγτ px, yq
“ 1
2 pJτ ` FrobppJτ qq
“ 1
2 pJτ ´ wM Jτσ q (for some σ depending only on τ).
“ 1
2 pJτ ` Jτσ q (since wM “ ´ap “ ´1).
Summing over all embeddings we get
L1
ppf8{K, 2q “
1
2
p1 ´ wM q
ÿ
σPCl`
pKq
Jτσ “
ÿ
ψ
Jψ.
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12. Factorization
Theorem B - Factorization
For all k P U,
Lppf8{K, kq2
“ D
k´2
2 Lppf8, k, k{2qLppfK
8 , k, k{2q.
Proof
1 Interpolation property for Lppf8{K, kq2 (Popa 2006):
For all k P U X Zě2,
Lppf8{K, kq2
“ λpkq2
p1 ´ appkq´2
pk´2
q2
D
k´2
2 L˚
pf7
k{K, k{2q.
“ λpkq2p1 ´ appkq´2pk´2q2D
k´2
2 L˚pf7
k, k{2qL˚pfK,7
k , k{2q
2 Use the interpolation property of the MK p-adic L-functions (RHS) to
show that the factorization occurs for all k P U X Zě2.
3 Conclude using that U X Zě2 is dense in U.
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13. End of proof
JK “ L1
ppf8{K, 2q (A - p-adic GZ)
Lppf8{K, kq2
“ D
k´2
2 Lppf8, k, k{2qLppfK
8 , k, k{2q (B - Factorization)
Easy observations:
1 D
k´2
2 “ 1 ` Opk ´ 2q,
2 Lppf8, k, k{2q “ Oppk ´ 2q2
q (exceptional zero case)
Deduce that
J2
K “
1
2
ˆ
d2
dk2
ˇ
ˇ
ˇ
k“2
Lppf8, k, k{2q
˙
LppfK
8 , 2, 1q.
Need to understand the two terms in the RHS.
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14. End of proof (continued)
A ` B ùñ J2
K “
1
2
ˆ
d2
dk2
ˇ
ˇ
ˇ
k“2
Lppf8, k, k{2q
˙
LppfK
8 , 2, 1q.
Theorem C - Bertolini–Darmon 2007
1 There exists P P EpQq, and 1 P Qˆ, such that
d2
dk2
ˇ
ˇ
ˇ
k“2
Lppf8, k, k{2q “ 1 log2
EpPq.
2 P is nontorsion if and only if L1pE, 1q ‰ 0.
appEKq “ ´1 ùñ LppfK
8 , 2, 1q “ 2L˚pEK, 1q “ 2 2 P Qˆ.
Get log2
EpPKq “ 1 2 log2
EpPq, and one sees that 1 2 “ t2 is a
square.
Taking square roots yields the theorem:
JK “ t logEpPq, and P is nontorsion iff L1
pE{K, 1q ‰ 0.
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15. Thank you !
Bibliography, code and slides at:
http://www.warwick.ac.uk/mmasdeu/
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