A class of non-holomorphic modular forms

Francis BrownAll Souls College, Oxford(IHES, Bures-Sur-Yvette)

Modular forms are everywhereMPIM

22nd May 2017

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Two motivations

1 Do there exist modular forms which correspond to mixedmotives? Today, mixed Tate motives over Z mainly.

2 String theory. Genus one closed superstring amplitudes.

Graph G −→ Modular-invariant function

Unknown class of functions. A few known examples.(. . . , M. Green, Vanhove, . . . , Zagier, Zerbini, . . . ).

Goal: define a class of non-holomorphic modular forms.

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I. General framework

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Definitions I

LetH = {z : Im z > 0}

z = x + iy , q = e2iπz .

For simplicity, letΓ = SL2(Z) .

Definition

A real analytic function f : H→ C is modular of weights (r , s) if

f (γz) = (cz + d)r (cz + d)s f (z)

for all

γ =

(a bc d

)∈ Γ .

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Definitions II

Definition

Let Mr ,s be the C-vector space of functions f : H→ C which arereal analytic modular of weights (r , s), such that

f (q) ∈ C[[q, q]][L±]

whereL := log |q| = iπ(z − z) = −2πy .

There is an N ∈ N

f (q) =N∑

k=−N

∑m,n≥0

a(k)m,n Lkqmqn

where a(k)m,n ∈ C.

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Definitions III

LetM =

⊕r ,s

Mr ,s

Then M is a bigraded algebra. Let

w = r + s and h = r − s .

Call w the total weight. Take w , h even.

The constant part of f is

f 0 :=∑k

a(k)0,0 Lk ∈ C[L±]

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Trivial examples

L ∈M−1,−1

For any holomorphic modular form

f ∈ M2n

then f ∈M2n,0 and f ∈M0,2n, e.g.,

G2k = −b2k

4k+∑n≥1

σ2k−1(n)qn k ≥ 2

The function G2 is not modular, but

G∗2 = G2 − 1

4L∈ M2,0

is an ‘almost holomorphic’ modular form.

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Differential operators

Define

∂r = (z − z)∂

∂z+ r , ∂s = (z − z)

∂

∂z+ s .

They preserve modularity

∂r :Mr ,s −→Mr+1,s−1

∂s :Mr ,s −→Mr−1,s+1

Write∂ =

∑r

∂r , ∂ =∑

s

∂s .

sl2-action

These operators generate an sl2:

[h, ∂] = 2∂ , [h, ∂] = −2∂ , [∂, ∂] = h

where h :Mr ,s →Mr ,s is multiplication by (r − s).

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Bigraded Laplace operator

Define an operator

∆r ,s :Mr ,s −→Mr ,s

by

∆r ,s = −∂s−1∂r + r(s − 1)

= −∂r−1∂s + s(r − 1)

Then

∆0,0 = −y2( ∂2

∂x2+

∂2

∂y2

)is the Laplace-Beltrami operator. Write

∆ =∑r ,s

∆r ,s

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Petersson inner product

Subspace of cuspidal functions (no constant part)

Sr ,s ⊂Mr ,s .

Let D be a fundamental domain for Γ.

Mr ,s × Sn−s,n−r −→ C

〈f , g〉 =

∫D

f (z)g(z) yn dxdy

y2

Special cases:〈f , g〉 :Mr ,s × Sr−s −→ C

〈f , g〉 :Mr ,s × S s−r −→ C

where S2n = holomorphic cusp forms.

holomorphic (Sturm)/antiholomorphic projection

p = ph + pa : Mr ,s −→ Sr−s ⊕ S s−r

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Picture of M

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II. Primitives and obstructions

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Iterated primitives

Goal

Construct new elements from old by solving

∂rF = f for F ∈Mr ,s

This equation can’t always be solved. Obstructions:

1 Combinatorial. Not all f admit a primitive F ∈ C[[q, q]][L±].For example, f cannot contain any terms: L−rqn.

2 Modularity. Suppose ∂F = f has a solution F ∈ C[[q, q]][L±].Then F is not necessarily modular.

3 (Arithmetic. F will have transcendental coefficients.)

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Orthogonality to cusp forms

Theorem

Suppose that ∂F = f admits a solution F ∈Mr ,s . Then

〈f , g〉 = 0 for all g ∈ Sr−s+2 .

Equivalently, the holomorphic projection vanishes

ph(f ) = 0 .

Idea of proof: Stokes’ theorem∫D

Lr+1f (z)g(z)dxdy

y2= 4π2

∫∂D

F (z)g(z)dz

The right-hand side vanishes by modularity.

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Corollary

Let f ∈ S be a non-zero holomorphic cusp form. Then

∂F = Lk f

has no modular solutions F ∈M.

‘Cusp forms have no modular primitives’

But! this leaves open the possibility that

∂F = LkG2n+2

might have a solution. Indeed, it does.

Out of this crack of light, a vast landscape will unfold!

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Example: real analytic Eisenstein series

Definition

For r , s ≥ 0 and r + s = w > 0

Er ,s(z) =w !

(2iπ)w+2

1

2

∑m,n 6=0,0

L(mz + n)r+1(mz + n)s+1

.

These are the unique solutions to

∂Ew ,0 = LGw+2

∂Er ,s = (r + 1)Er+1,s−1 s ≥ 1

and

∂E0,w = LGw+2

∂Er ,s = (s + 1)Er−1,s+1 r ≥ 1

such that coefficient of L−w/2 in Ew/2,w/2 vanishes.16 / 35

Picture

Dotted arrows are action of ∂, ∂.17 / 35

Properties

Eigenfunctions of the Laplacian:

∆Er ,s = −w Er ,s

Orthogonal to cusp forms

p(Er ,s) = 0

Constant part involves odd zeta values:

E0r ,s = a L + a′ζ(2n + 1) L−2n

where a, a′ ∈ Q.

( Corresponds to 0→ Q(2n)→ E → Q(−1)→ 0 )

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Relation with Maass waveforms

Theorem

If f ∈M is an eigenfunction of the Laplacian, it is a linearcombination with coeffs. in C[L±] of eigenfunctions of the form:

real analytic Eisenstein series Er ,s ,

an almost holomorphic modular form,

an almost antiholomorphic modular form.

Er ,sand

classical

M Maass

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III. Modular iterated integrals

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Overview

Now solve equations like

∂F = Ls+1f Er ,s + Lg

where f and g holomorphic modular forms. Use the solutions F togenerate new primitives, and so on, recursively.

This generates a huge space MI ⊂M of modular iteratedintegrals. It is filtered by the length: MIk .

Recall orthogonality condition ph(∂F ) = 0. Implies

g =∑h

−〈f Er ,s , h〉h ,

sum over basis of Hecke cusp eigenforms. By Rankin-Selberg, thelength 2 elements of MI2 have coefficients involving L(f ⊗ h, n).

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Vector bundles and equivariance

DefineV2n =

⊕r+s=2n

X rY sQ

It is equipped with a right-action of SL2:

(X ,Y )∣∣γ

= (aX + bY , cX + dY ) .

Definition

A function f : H→ V2n ⊗ C is equivariant if

f (γz)∣∣γ

= f (z) for all γ ∈ Γ

Key point: Equivariant f can be uniquely written

f (z) =∑

r+s=2n

fr ,s(z) (X − zY )r (X − zY )s

where fr ,s : H→ C modular of weights (r , s)22 / 35

Single-valued periods

We want to construct equivariant sections of V2n.

Idea: apply the single-valued machine.

Example:

log z =

∫ z

1

dt

t

It is a multi-valued function on C×. Analytic continuation around0 gives a discontinuity

log z 7→ log z + 2πi .

Since 2πi is imaginary, the function

log |z | = Re log z

is single-valued. It is the single-valued version of log z .

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Example

F (z) = 2πi

∫ →1∞

zG2n+2(τ)(X − τY )2ndτ

Is not quite equivariant:

F (γz)∣∣γ− F (z) = Cγ(X ,Y ) for all γ ∈ Γ

Cγ is the Eisenstein cocycle:

Cγ = a ζ(2n + 1) Y 2n∣∣∣γ−id

+ (2πi) e02n(γ)

where a ∈ π−2nQ, and e02n(γ) ∈ Q[X ,Y ]. Set

E(z) = 2 Re F (z)− 2 a ζ(2n + 1)Y 2n .

It is equivariant, and its coefficients are the Er ,s .

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Double Eisenstein integrals

For any f ∈ M2n+2, define

f (z) = 2πi f (z)(X − zY )2ndz .

It is equivariant.

F 2a,2b(z) = Im∫ →

1∞

zG2aG2b − Re

(∫ →1∞

zG2a

)×∫ →

1∞

zG2b

+

∫ →1∞

zf +

∫ →1∞

zg + c

is equivariant for suitable choices of modular forms f , g andconstant c. Lowest weight modular component solves

∂F = LG2aE2b−2,0 + Lf

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Orthogonality relations

Orthogonality to cusp forms uniquely determines f :

〈f , h〉+ 〈G2aE2b−2,0, h〉 = 0 for all h ∈ S2a+2b−2

By Rankin-Selberg, the right-hand term is proportional to

L(h, 2a− 1)L(h, 2a + 2b − 2)

if h eigenform. Manin period relations implies we can constructcombinations in which all cusp forms drop out. Example:

9(F 4,10 − F 10,4) + 14(F 6,8 − F 8,6)

is an iterated integral involving Eisenstein series only. Want togeneralise: space MIE ⊂MI of modular Eisenstein integrals.

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IV. General construction

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A Lie algebra

Let Lie(a, b) be the free Lie algebra on two generators a, b. It hasa right-action of SL2.

There exist derivations (Tsunogai) for all n ≥ −1

ε2n(b) = −ad(b)2n(a)

ε2n([a, b]) = 0

They generate a Lie algebra ugeom ⊂ Der Lie(a, b).

It corresponds to a group scheme Ugeom.

Satisfy many relations, for example (Pollack):

[ε10, ε4]− 3[ε8, ε6] = 0

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Let

ω = −ad(ε0)dq

q+∑n≥1

2

(2n)!ε2n+2G2n+2(q)

dq

q

Then the generating series of iterated integrals

J(z) = 1 +

∫ →1∞

zω +

∫ →1∞

zωω + . . .

solves the differential equation (version of KZB)

dJ = ωJ .

It satisfies for all γ ∈ Γ:

J(γz)∣∣γ

= J(z)Gγ

for some non-abelian cocycle

Gγ ∈ Z 1(Γ,Ugeom(C))

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Theorem

Exist b ∈ Ugeom(C), and φ ∈ (AutUgeom)SL2(C) such that

b∣∣−1

γφ(Gγ) b = Gγ for all γ ∈ Γ .

Definition

Jev(z) = J(z)b−1φ(J(z))−1

Notice that the holomorphic differential equation is unchanged:

∂Jev

∂z= ωJev .

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Theorem

The series Jev is modular equivariant

Jev(γz)∣∣γ

= Jev(z) .

The coefficients of Jev are equivariant real analytic functions on H.They generate a space of modular forms

MIE ⊂M

which only involve iterated integrals of Eisenstein series.

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Main theorem

We have constructed explicitly a space of modular forms

MIE ⊂M .

Theorem

MIE is an algebra, with modular weights (r , s) for r , s ≥ 0.

(Expansion).Every f ∈MIE admits an expansion

f ∈ Zsv [[q, q]][L±]

where Zsv is the ring of single-valued multiple zeta values.

(Length filtration).It admits a filtration by length MIE

k ⊂MIE . In length one,MIE

1 is generated by the functions Er ,s .

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Main theorem continued

(Differential structure).

∂MIEk ⊂ MIE

k + E [L]×MIEk−1

∂MIEk ⊂ MIE

k + E [L]×MIEk−1

where E is the space of holomorphic Eisenstein series.

(Weight grading).The space MIE has a grading:

degM L = 2 and degM Er ,s = 2 .

(Finiteness).The space grMk MI

E ∩Mr ,s of bounded modular weight andbounded M-degree is finite dimensional. An element is uniquelydetermined by finitely many terms in its q, q expansion.

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Main theorem continued

(Structure).There is a non-canonical algebra isomorphism

lw(O(Ugeom))∼−→ (MIE )•,0

(Laplace equation).Every element of MIE satisfies an inhomogeneous Laplaceequation with eigenvalue −w .

Further properties (in progress):

1 (Double shuffle). The Lie algebra ugeom embeds into a spaceof polar solutions to linearised double shuffle equations.

2 (Galois action). The ring O(Ugeom) admits an action of themotivic Galois group of mixed Tate motives over Z.

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Conclusion

1 Bigraded space M =⊕

r ,sMr ,s of real-analytic modular

forms. Differential structure ∂, ∂.

2 Large subspace MI ⊂M of iterated primitives ofholomorphic modular forms. Correspond to non-trivialextensions of pure motives. Finiteness: uniquely determinedby a finitely many coefficients in the q, q-expansion.

3 Explicit construction MIE ⊂MI of iterated primitives ofEisenstein series. Correspond to mixed Tate motives over Z.

4 Closed genus one superstring amplitudes should lie in thecomponent of MIE [L±] of modular weights (0, 0). Explainsconjectural properties, plus existence of many relations.

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