Singular moduli for real quadratic fieldsvonk/MIT.pdf · Singular moduli for real quadratic fields...

Singular moduli for real quadratic fields


Jan VonkMIT Number Theory Seminar, 5 May 2020

Joint work with Henri Darmon, Alice Pozzi, Yingkun Li


Outline

1 Introduction: Infinite products

2 The Dedekind–Rademacher cocycle

3 Singular moduli for real quadratic fields


Introduction: Infinite products

Outline






Consider the sine function

sinπz = πz∏

n∈ Z \{0}

(1− z

n

).

For any z ∈ Q, the value sinπz is algebraic.

sin π4 =

√2

2

sin π8 =

√2−√

22

sin π16 =

√2−√

2+√

22

Theorem (Kronecker–Weber)All finite abelian extensions of Q are (essentially) generated bycombinations of

sin(πz), z ∈ Q



Consider the j-function

j(q) = q−1 + 744 + 196884q + 21493760q2 + . . . q = e2πiz

= q−1(1− q)−744(1− q2)80256(1− q3)−12288744 · · ·

=

reg∏γ ∈ SL2(Z)

ζ3 − γzζ3 − γz

For any z ∈ K imaginary quadratic, the value j(z) is algebraic.

j(√−14) = 23

(323 + 228

√2 + (231 + 161

√2)

√2√

2− 1)3

Theorem (Kronecker–Weber)All finite abelian extensions of K are (essentially) generated bycombinations of {

sin(πz) z ∈ Q,j(z) z ∈ K .



Singular moduli

Singular moduli, like Weber’s example:

j(√−14) = 23

(323 + 228

√2 + (231 + 161

√2)

√2√

2− 1)3

,

have several interesting features. In this case:

It generates the Hilbert class field over K = Q(√−14).

It has an interesting prime factorisation (Gross–Zagier)

Nm j(√−14) = 224 · 116 · 173 · 413

It is closely related to the first derivative of the diagonal restriction ofa real analytic family of Eisenstein series (Gross–Zagier).



Let τ1, τ2 be two CM points inH∞ = {z ∈ C : Im(z) > 0}.Gross and Zagier (1985) find explicit formula for

Nm (j(τ1)− j(τ2)) ∈ Z

Algebraic proof: Its q-adic valuation is given in terms of arithmeticintersection of (oriented optimal) embeddings

Q(τ1),Q(τ2) ↪→ B∞q = �at alg conductor∞q

Analytic proof: Fourier coe�icients of Hecke’s real analytic Eisensteinseries over F , a�ached to the genus character defined by Q(τ1, τ2).

Q(τ1, τ2)

F

Q

Q(τ2)Q(τ1)

χ



Real analytic Hilbert Eisenstein series Es(z1, z2) defined by Hecke:

∑[a]∈Cl(∆1∆2)

χ(a)Nm(a)1+2s′∑

(m,n)∈a2/U

y s1y

s2

(mz1 + n)(m′z2 + n′)|mz1 + n|2s|m′z2 + n′|2s

Gross–Zagier consider its diagonal restriction Es(z, z) and show

When s = 0, have Es(z, z) = 0,

The holomorphic projection of the first derivative(∂

∂sEs(z, z)

)∣∣∣∣hols=0

has Fourier coe�icients related to logNm (j(τ1)− j(τ2)).

The holomorphic projection must vanish!⇒ formula for Nm (j(τ1)− j(τ2)).



Today: Analogues for real quadratic fields.

(Gross–Zagier) Let τ1, τ2 be CM points, consider

J∞(τ1, τ2) = j(τ1)− j(τ2) ∈ Q

Related to real analytic Eisenstein family.

ordq J∞(τ1, τ2) = Intersection multiplicities

Q(τ1),Q(τ2) ↪→ B∞q.

(With Darmon) Let τ1, τ2 be RM points, construct

Jp(τ1, τ2)?∈ Q

Related to p-adic analytic families.

ordq Jp(τ1, τ2)?= Intersection multiplicities

Q(τ1),Q(τ2) ↪→ Bpq.


The Dedekind–Rademacher cocycle

Outline






The work of Duke–Imamoglu–TothInspiration comes from work of Duke–Imamoglu–Toth on linking numbersof modular geodesics.

If γ ∈ SL2(Z) is hyperbolic, get associated knot

Kn(γ) ↪→ SL2(Z)\ SL2(R)

t 7→ SL2(Z)g(

et

e−t

), where g−1γg = diagonal

Linking Kn(γ) and trefoil↔ Dedekind–Rademacher cocycle (Ghys)

Linking Kn(γ1) and Kn(γ2)↔ Knopp cocycle (DIT)



The map γ 7→ 12c/(cz + d) is a weight 2 cocycle, trivialised by theEisenstein series E2(z). It li�s uniquely to a weight 0 cocycle:

0−→H1(SL2(Z),O)d−→ H1(SL2(Z),O2)−→ 0

The Dedekind–Rademacher symbol Φ(γ) ∈ Z is defined by

log ∆(γz)− log ∆(z) = 6 log(−(cz + d)2) + 2πiΦ(γ).

so that the unique li� is given by the right hand side. When applied to ahyperbolic matrix γ with fixed point τ , we get

6 log(−(cz + d)2) + 2πiΦ(γ)

z = τ

y yz = i∞ (+ homog.)

12 log(ετ ) Link(Kn(γ), trefoil)



Now upgrade the above to the se�ing

Γ = SL2(Z[1/p])O = Analytic functions on Hp = P1(Cp) \P1(Qp)

Want a multiplicative li� instead

0 H1(Γ,O×) H1(Γ,O×/C×p )

H1(Γ,O)

H1(Γ0(p),C×p )

H1(Γ,O2)

log dlog

d

From Euler system of Siegel units, construct ΘDR ∈ C1(Γ,O×) such that

ΘDR(γ1γ2) = ΘDR(γ1)ΘDR(γ2)γ1 × pΦp(γ1,γ2)

where Φp : Γ0(p)→ Z is Dedekind–Rademacher morphism(viewed as an element in H2(Γ,Z) ⊃ H1(Γ0(p),Z)).



Let τ be RM point, then the value

ΘDR[τ ] := ΘDR(γτ )(τ)

can be computed, e.g. p = 7 and τ = −17+√

3214 seems to give a root of

74x6−20976x5−270624x4+526859689x3−649768224x2−120922465776x+716

Conjecture (Darmon–Dasgupta)Let H/Q(τ) be the ring class field a�ached to τ , then

ΘDR[τ ] ∈ OH[1/p]×

With Henri Darmon and Alice Pozzi, we prove this conjecture. Use p-adicEisenstein family over Q(τ) with an odd class character ψ

E(p)k,ψ := Lp(ψ, 1− k) + 4

∑ν∈d−1

+

∑p - I | (ν) d

ψ(I)Nm(I)k−1

e2πi(ν1z1+ν2z2),

Remark. Dasgupta–Kakde (forthcoming) give a di�erent proof.



We consider its diagonal restriction E(p)k,ψ(z, z), and show that

When k = 1, we get

E(p)1,ψ(z, z) = Lp(ψ, 0)− 4

∞∑n=1

〈{0→∞}, Tngψ〉 qn if p split in Q(τ),

E(p)1,ψ(z, z) = 0 if p inert in Q(τ).

When p is inert in Q(τ), the ordinary projection limn→∞

Un!p of the first

derivative equals(∂

∂kE(p)k,ψ(z, z)

)∣∣∣∣ordk=1

= log(NmΘDR[ψ]) E(p)2 +

∑f cusp

log(Nm P−f (ψ)) f

where P−f (ψ) = Stark–Heegner point.

Mirrors Gross–Zagier very closely! Can even be bootstrapped to a fullproof of the conjecture on ΘDR, via the (exclusively p-adic) connectionwith deformation theory of Galois representations.



With Henri Darmon and Alice Pozzi consider the eigenvariety around E(p)1,ψ ,

which was shown to be etale over weight space by Betina–Dimitrov–Shih.

The first derivatives of the three parallel weight deformations of the Artinrepresentation 1⊕ ψ are related, exhibiting uψ ∈ OH[1/p]× such that

log(NmΘDR[ψ]) = L′p(ψ, 0) (Diagonal restrictions)= log(Nm uψ) (Galois deformations)

Remark 1. Working instead with the anti-parallel family F− through theweight (1, 1) form E(p)

1,ψ , we prove the full conjecture, without the norm.

Remark 2. Charollois–Darmon define archimedean version of ΘDR[τ ],refining the Stark conjecture. (No Galois deformation, no proof!)



Outline






The work of Duke–Imamoglu–Toth (Part II)Let us return to modular knots Kn(γ) ↪→ SL2(Z) \ SL2(R):

Linking Kn(γ) and trefoil↔ Dedekind–Rademacher cocycle (Ghys)

Linking Kn(γ1) and Kn(γ2)↔ Knopp cocycle (DIT)

The Knopp cocycle in Z 1(SL2(Z),O2) a�ached to an RM point τ is

γ 7−→∑

w ∈ SL2(Z)τ

{∞ → γ∞} ∩ {w → w ′}z − w

where the exponent is the intersection number of the geodesic from w ′ tow with the geodesic from∞ to γ∞, and hence ±1 or 0.

Choie–Zagier: Those account for all elements in Z 1par(SL2(Z),O2).



Joint with Henri Darmon, we upgrade the above to the se�ing

Γ = SL2(Z[1/p])M = Meromorphic functions on Hp = P1(Cp) \P1(Qp)

Construct multiplicative li�

0 H1(Γ,M×) H1(Γ,M×/C×p )

H1(Γ,M)

H1(Γ0(p),C×p )

H1(Γ,M2)

log dlog

d

Concretely, Θτ ∈ C1(Γ,M×) is explicit infinite product over Γ:

Θτ (γ) =∏

w ∈ Γτ

∗ (z − w){w′→w}∩{∞→γ∞}

Its li�ing obstruction is rich! When X0(p) has genus zero, we have

Θτ (γ1γ2) = Θτ (γ1)Θτ (γ2)γ1 × εΦp(γ1,γ2)τ

where Φp : Γ0(p)→ Z is the Dedekind–Rademacher morphism.



Assume that X0(p) has genus 0, i.e. p = 2, 3, 5, 7, 13 (for simplicity)

If τ1, τ2 are coprime RM points inHp, the quantity

Jp(τ1, τ2) := Θτ1 (γτ2 )(τ2)

is arithmetically very rich, and should be thought of as analogous to thequantity J∞(τ1, τ2) := j(τ1)− j(τ2) considered by Gross–Zagier:

Conjecture (Darmon–V.)Let Hi be the ring class field a�ached to τi , then

Jp(τ1, τ2) ∈ H1H2

and is acted on in the obvious way by Gal(H1H2/Q).

Remark 1. This implies complex conjugation always acts by inversion.Remark 2. For general primes, li�ing obstructions are killed by principalparts of weakly holomorphic forms of weight 1/2, giving a map

M!!1/2(Γ0(4p))−→H1(Γ,M×).



Let ∆1 = 13, then for below choices of p and τ consider the quantity

Jp

(1 +√

132

, τ

)

τ p = 11 p = 19 p = 59

2√

2 3−4√−1

53−4√−1

5 1

3√

2 11+21√−3

2·195−4√−6

11 1

4√

2 57−176√−1

5·375−12√−1

133+4√−1

5

7√

2 118393−8328√−14

52·59·8393+95

√−7

22·6737+9√−7

22·11

8√

2 1312−1425√−1

13·14943+924

√−1

52·373+4√−1

5

9√

2 11387+12320√−3

192·6743+4100

√−6

112·83 1

Observe that for any pair of primes p, q there seems to be some relation

“ordp”(q-adic invariant) = “ordq”(p-adic invariant).



Let p = 2, then the RM values of

J2

(1 +√

132

,−)

at the six RM points of discriminant 621 are all roots of

53266281197421626898704636823062295969007036119297599934916 x6

−27836752624445107255550537796183532261306810430217742390746 x5

−29297701627429700833818885363891546270240998098759334148135 x4

+87958269550388100260309855891207245711288562805656560629805 x3

−29297701627429700833818885363891546270240998098759334148135 x2

−27836752624445107255550537796183532261306810430217742390746 x+53266281197421626898704636823062295969007036119297599934916 = 0

This generates ring class field of conductor 621 over K = Q(√

13).



The q-adic valuation of Jp(τ1, τ2) is conjecturally given by arithmeticintersection numbers of geodesics on Shimura curve Xpq a�ached toquaternion algebra Bpq . In this example, the constant term is

22 · 77 · 194 · 375 · 472 · 592 · 673 · 972 · 1092 · 229 · 241 · 379 · 631 · 709 · 733 · 1009

James Rickards developed algorithms to compute these intersectionnumbers. He finds that the only q for which some geodesic of discriminant13 intersects some geodesic of discriminant 621 on X2,q are:

q 7 19 37 47 59 67 97 109〈−,−〉q,∞ 7 4 5 2 2 3 2 2

q 229 241 379 631 709 733 1009〈−,−〉q,∞ 1 1 1 1 1 1 1



Towards a proof?

The connection between di�erences of singular moduli and derivatives ofreal analytic families of modular forms place singular moduli in thecontext of the Kudla programme, which seeks connections betweenalgebraic cycles and Fourier coe�icients of families of automorphic forms.

For real quadratic singular moduli, it is reasonable to try to establishsimilar connections in the context of an emerging p-adic Kudla programme.It has the additional connection

p-Adic families of modular forms

lDeformations of Galois representations



Ongoing work with Henri Darmon and Yingkun Li adapts this strategy tomeromorphic cocycles. Replaces Eisenstein family by a family Fk+1/2

obtained from a Hida family of theta series.

Again there is vanishing at k = 1, and(∂

∂kFk+1/2

)∣∣∣∣ordk=1

=∑D

log(Nm Jp(χ,D))qD + (Alg.terms)

where the algebraic terms are determined by the deformation theory of theHida family around weight k = 1 worked out by Darmon–Lauder–Rotger.

When X0(p) has genus zero, this ordinary projection must vanish! Flow ofinformation is reversed from the CM se�ing of Gross–Zagier.



Conclusion

We hope that this may lead to a tentative ‘RM theory’ which provides richarithmetic invariants that play a role in an emerging p-adic Kudlaprogramme similar to that of CM singular moduli.

The lack of a geometric substitute for ‘RM elliptic curve’ is o�set by manyadvantages of the p-adic se�ing, especially the connection with thearithmetic of number fields via the theory of Galois deformations.

Thanks for having me!

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