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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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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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
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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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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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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