automorphic functions for moduli of polarized hodge...

Automorphic functions for moduli of polarizedHodge structures

Hossein Movasati

IMPA, Instituto de Matemática Pura e Aplicada, Brazilwww.impa.br/∼hossein/

ICTP, 2010

Outline

Motivation

Griffiths domain, another point of view

The case of elliptic curves

A family of Calabi-Yau varieties

I Upper half plane: H := {x + iy ∈ C | y > 0}.I The discrete group:

SL(2,Z) = {(

a bc d

)| a,b, c,d ∈ Z,ad − bc = 1}.

I The SL(2,Z) action on H:

Az :=az + bcz + d

, A ∈ SL(2,Z), z ∈ H.

I SL(2,Z)\H is the moduli of polarized Hodge structureswith Hodge numbers h10 = h01 = 1 and polarization of

type(

0 1−1 0

).

I One gets in a natural way the theory of modular forms onH for SL(2,Z): It is the graded C-algebra freely generatedby the Eisenstein series g2 and g3, where

gk (z) = 1 + (−1)k 4kBk

∑n≥1

σ2k−1(n)qn, k = 2,3, z ∈ H,

where q := e2πiz , Bk is the k -th Bernoulli number

B1 =16, B2 =

130, B3 =

142, . . .

andσi(n) :=

∑d |n

d i .

I Similar theories for subgroups of finite index of SL(2,Z),and in particular for Γ0(N), Γ(N), Γ1(N).

I Modular forms are good generating functions. They countunexpected objects.

I The j-function

j =g3

2

g32 − g2

3=

q−1 + 744 + 196884q + 21493760q2 + 864299970q3 + · · · .

and the monstrous moonshine conjecture, Borcherdstheorem.

I Shimura-Taniyama conjecture, Modularity theorem,counting the number of points of elliptic curves over finitefields.

I Griffiths 1970: What about other Hodge structures?I Griffiths domain D, discrete group Γ acting from left on DI Γ\D is the moduli of polarized Hodge structures of a fixed

type.

I Compactification problem: ..., E. Cattani, A. Kaplan, W.Schmid, S. Usui, K. Kato, ...

I Griffiths: Automorphic cohomology

I Griffiths 1970: What about other Hodge structures?I Griffiths domain D, discrete group Γ acting from left on DI Γ\D is the moduli of polarized Hodge structures of a fixed

type.I Compactification problem: ..., E. Cattani, A. Kaplan, W.

Schmid, S. Usui, K. Kato, ...I Griffiths: Automorphic cohomology

I For few cases of Hodge number D is a symmetricHermitian domain.

I Baily-Borel theorem (1969) on the unique algebraicstructure of quotients of symmetric Hermitian domains bydiscrete arithmetic groups.

I Shimura varieties.

Griffiths domain, another point of view

Fix a C-vector space V0, of dimension h, a natural numberm ∈ N and a h × h integer valued matrix Ψ0 such that theassociated bilinear form

Zh × Zh → Z, (a,b)→ aΨ0bt

is non-degenerate, symmetric if m is even and skew if m is odd.A lattice VZ in V0 is a Z-module generated by a basis of V0.A polarized lattice (VZ, ψZ) of type Ψ0 is a lattice VZ togetherwith a bilinear map ψZ : VZ × VZ → Z such that in a Z-basis ofVZ, ψZ has the form Ψ0.

We fix Hodge numbers

hi,m−i ∈ N ∪ {0}, hi :=m∑

j=i

hj,m−j , i = 0,1, . . . ,m, h0 = h

and a filtration

F •0 : {0} = F m+10 ⊂ F m

0 ⊂ · · · ⊂ F 10 ⊂ F 0

0 = V0, dim(F i0) = hi

on V0. We fix also a bilinear map ψ0 in V0 which is of the formΨ0 in some basis of V0.

Let P be the set of polarized lattices (VZ, ψZ) with theproperties:

1. ψC(x)(F i0,F

j0) = 0, ∀i , j , i + j > m;

2. V0 = ⊕mi=0H i,m−i(x), where H i,m−i(x) := F i

0 ∩ F m−i0

VZ;

3. (−1)i+m2 ψC(x)(v , vx ) > 0, ∀v ∈ H i,m−i(x), v 6= 0.

4. ψC = ψ0

We also call P the (generalized) period domain.

I The algebraic group G0 = Aut(V0,F •0 , ψ0) acts from theright on P and

Γ\D ∼= P/G0.

I Question: Does P enjoy a structure of an algebraic varietysuch that the action of G0 becomes algebraic?

I In the case of elliptic curves the answer is yes! Novelty?Weneed something more than automorphic functions/modularforms. We need quasi/differential automorphic functions.


Γ\D ∼= P/G0.


I In the case of elliptic curves the answer is yes! Novelty?

Weneed something more than automorphic functions/modularforms. We need quasi/differential automorphic functions.


Γ\D ∼= P/G0.


I In the case of elliptic curves the answer is yes! Novelty?Weneed something more than automorphic functions/modularforms. We need quasi/differential automorphic functions.

I Hodge structures arising from elliptic curves: the perioddomain

P := SL(2,Z)\{(

x1 x2x3 x4

)∈ SL(2,C) | Im(x1x3) > 0}

G0 :=

{(k k ′

0 k−1

)| k ∈ C− {0}, k ′ ∈ C

}The (generalized) period map gives a biholomorphismbetween T (C) and P, where T = Spec(C[t1, t2, t3, 1

27t23−t3

2]).

I The inverse of the period map restricted to

{(

z −11 0

)| z ∈ H}

gives us the Eisenstein series g1,g2,g3.

g1 is not a modular form. It is the main building block of quasi ordifferential modular forms. The theory of quasi modular formsfor SL(2,Z): homogeneous polynomials of the ring

C[g1,g2,g3], deg(g1) = 2, deg(g2) = 4, deg(g3) = 6

Quasi modular forms appear as generating functions:

1. Counting holomorphic maps from curves to an ellipticcurve with a fixed ramification data, R. Dijkgraaf,M. Douglas, D. Zagier, M. Kaneko.

2. Counting rational curves on K 3 surfaces, Yau-Zaslow(1996), Beauville(1999), Göttsche(1994),Bryan-Leung(1999).

Quasi modular forms appear as generating functions:1. Counting holomorphic maps from curves to an elliptic

curve with a fixed ramification data, R. Dijkgraaf,M. Douglas, D. Zagier, M. Kaneko.

2. Counting rational curves on K 3 surfaces, Yau-Zaslow(1996), Beauville(1999), Göttsche(1994),Bryan-Leung(1999).

The corresponding universal family of elliptic curves:

Et : y2 − 4(x − t1)3 + t2(x − t1) + t3, t = (t1, t2, t3) ∈ T (C).

with xdxy ∈ H1

dR(E/T ). The period map is given by

pm : T (C)→ P,

t 7→ 1√2πi

(∫δ1

dxy

∫δ1

xdxy∫

δ2

dxy

∫δ2

xdxy

).

{δ1, δ2} is a basis of the Z-module H1(Et ,Z) such that the

intersection matrix in this basis is(

0 1−1 0

). Different choices

of δ1, δ2 will lead to the action of SL(2,Z) which is alreadyabsorbed in P.

Under the period map the action of G0 is given

t•g := (t1k−2+k ′k−1, t2k−41 , t3k−6), t ∈ T (C),g =

(k k ′

0 k−1

)∈ G0.

Using explicit calculation of the Gauss-Manin connection offamilies of elliptic curves, one can show that the inverse of the

period map restricted to {(

z −11 0

)| z ∈ H} is a solution of the

ordinary differential equation:

R :

t1 = t2

1 −1

12 t2t2 = 4t1t2 − 6t3t3 = 6t1t3 − 1

3 t22

This is called the Ramanujan relations between Eisensteinseries.

The corresponding theory in the case of

Γ(2) = {A ∈ SL(2,Z) | A ≡2 I}

gives us the differential equation:t1 = t1(t2 + t3)− t2t3t2 = t2(t1 + t3)− t1t3t3 = t3(t1 + t2)− t1t2

,

which is due to Darboux 1878 and Halphen 1881.

T is the moduli of the pairs (E , ω), where E is anelliptic curve over C and ω ∈ H1

dR(E)\F 1 andF 1 ⊂ H1

dR(E) is the one dimensional subspacegenerated by a differential form of the first kind.

Hodge structures arising from Calabi-Yauvarieties

Let Mψ, ψ5 6= 1 be the the quintic hypersurfaces in P4 given by

the polynomial:

Q := x51 + x5

2 + x53 + x5

4 + x55 − 5ψx1x2x3x4x5

The group G = 〈g1,g2,g3,g4〉 acts on Mψ, where for instance

g2 : (x1, x2, x3, x4, x5) 7→ (x1, ζ5x2, x3, x4, ζ45x5)

Let Wψ be the desingularization of Mψ/G. We have

dimC(H3(Wψ,Q)) = 4, dimC H i,3−i = 1

Candelas-de la Ossa-Green-Parkes 1991: They calculated theYukawa coupling attached to Wψ:

W = 5 + 2875q

1− q+ 609250.23 q2

1− q2 + · · ·+ nj j3qj

1− qj + · · ·

Using Mirror symmetry they predicted that nj must be thenumber of rational curves of degree j in a generic quintic in P4.The Gromov-Witten invariants Nd can be calculated using thewell-known formula

Nd =∑k |d

nd/k

k3

1. Clemens conjecture: There exits a finite number of rationalcurves of a fixed degree in a generic quintic in P4.

2. Gopakumar-Vafa conjecture: The numbers nj ’s are positiveintegers (a precise approach to nj ’s is also done byGromov-Witten invariants).

3. Conjecture(Duco Van Straten informed me): Themonodromy group of the family Wψ has an infinite index inSp(4,Z).

The periods ∫δω, δ ∈ H3(Wz ,Z)

satisfy the Picard-Fuchs((z

ddz

)4

− 5z(5zddz

+ 1)(5zddz

+ 2)(5zddz

+ 3)(5zddz

+ 4)

)= 0.

wherez = ψ−5.

We can take a basis δ0, δ1, δ2, δ3 be a basis of H3(Wψ,Q) suchthat the monodromy around z = 0 is given by

1 0 0 01 1 0 012 1 1 016

12 1 1

We introduce a new coordinate around z = 0:

q = e2πi

∫δ1ω∫

δ0ω,

The expression

W :=

(52

∫δ1ω∫δ2ω −

∫δ0ω∫δ3ω

(∫δ0ω)2 − 5

6(

∫δ1ω∫

δ0ω

)3

)

is invariant under the monodromy and hence it can be written interms of the new variable q.

The theory quasi/differential of automorphic functions forHodge structures arising from the mentioned Calabi-Yauvarieties is the algebra generated by the solutions of theordinary differential equation:

t0 = 1t5

(65 t5

0 + 13125 t0t3 − 1

5 t4)

t1 = 1t5

(−125t60 + t4

0 t1 + 125t0t4 + 13125 t1t3)

t2 = 1t5

(−1875t70 −

15 t5

0 t1 + 2t40 t2 + 1875t2

0 t4 + 15 t1t4 + 2

3125 t2t3)

t3 = 1t5

(−3125t80 −

15 t5

0 t2 + 3t40 t3 + 3125t3

0 t4 + 15 t2t4 + 3

3125 t23 )

t4 = 1t5

(5t40 t4 + 1

625 t3t4)

t5 = t6t5

t6 = (−725 t8

0 −24

3125 t40 t3 − 3

5 t30 t4 − 2

1953125 t23 ) + t6

t5(12t4

0 + 2625 t3)

Taket = 5q

∂t∂q

and write each ti as a formal power series in q, ti =∑∞

n=0 ti,nqn

and substitute in the above differential equation. We see that itdetermines all the coefficients ti,n uniquely with the initialvalues:

t0,0 =15, t0,1 = 24, t4,0 = 0, t5,0 6= 0

124 t0 = 1

120 + q + 175q2 + 117625q3 + 111784375q4 +

1269581056265 + 160715581780591q6 +218874699262438350q7 + 314179164066791400375q8 +469234842365062637809375q9+722875994952367766020759550q10 + O(q11)

−1750 t1 = 1

30 + 3q + 930q2 + 566375q3 + 526770000q4 +

592132503858q5 + 745012928951258q6 +1010500474677945510q7 + 1446287695614437271000q8 +2155340222852696651995625q9+3314709711759484241245738380q10 + O(q11)

−150 t2 = 7

10 + 107q + 50390q2 + 29007975q3 +

26014527500q4 + 28743493632402q5+35790559257796542q6 + 48205845153859479030q7 +68647453506412345755300q8+101912303698877609329100625q9 +156263153250677320910779548340q10 + O(q11)

−15 t3 = 6

5 + 71q + 188330q2 + 100324275q3 +

86097977000q4 + 93009679497426q5+114266677893238146q6 + 152527823430305901510q7 +215812408812642816943200q8+318839967257572460805706125q9 +487033977592346076373921829980q10 + O(q11)

−t4 =0− 1q1 + 170q2 + 41475q3 + 32183000q4 + 32678171250q5 +38612049889554q6 + 50189141795178390q7 +69660564113425804800q8 + 101431587084669781525125q9

153189681044166218779637500q10 + O(q11)

25t5 = −1125 + 15q + 938q2 + 587805q3 + 525369650q4 +

577718296190q5 + 716515428667010q6 +962043316960737646q7 + 1366589803139580122090q8 +2024744003173189934886225q9+3099476777084481347731347688q10 + O(q11)

15625t6 = 0− 15q + 26249q2 + 3512835q3 + 2527019900q4 +2381349669050q5+2699403828169815q6 + 3414337117855753978q7 +4647615139046603293280q8+6668975996587015549602975q9 +9957519516309695103093241870q10 + O(q11)

TheoremThe expression −(t4−t5

0 )2

625t35

is the Yukawa coupling calculated by

Candelas and others:

−(t4 − t50 )2

625t35

= 5 + 2875q

1− q+ 609250 · 22 q2

1− q2 +

317206375 · 32 q3

1− q3 + · · ·+ ndd3 qd

1− qd + · · ·

TheoremThe functions ti , i = 0,1,2,3,4 are algebraically independentover C, this means that there is no polynomial P in fivevariables with coefficients in C such that P(t0, t1, t2, t3, t4) = 0.

I Differential modular forms and some analytic relationsbetween Eisenstein series, Ramanujan Journal, 2007:Contains the case of elliptic curves, Ramanujan differentialequation and ...

I Moduli of polarized Hodge structures, Bull. Soc. Bras. Mat.39(1), 81-107,2008.

I Modular foliations and periods of hypersurfaces, preprint.(algorithms for calculating Gauss-Manin connection,Modular differential equations, Darboux-Halphendifferential equations and ...)

I Eisenstein series for Calabi-Yau varieties (announcement),under preparation (the modular differential equationattached to Calabi-Yau varieties).

All these can be found in my homepage

www.impa.br/∼hossein

automorphic functions for moduli of polarized hodge...

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