Modular forms and multiple zeta values
Henrik Bachmann
Fachbereich Mathematik, Bereich Algebra und Zahlentheorie
Advisor: Prof. Dr. Ulf KühnNotation: s1, . . . , sl ∈ N, τ ∈ H = {x + yi ∈ C | y > 0} and q = e

2πiτ

.

Shuffle regularized multiple Eisenstein seriesMultiple zeta values and multiple Eisenstein series
Definition: The multiple zeta value (MZV) of weight s1 + · · · + sl andlength l is defined byX1
ζ(s1, . . . , sl) :=( for s1 ≥ 2 and s2, . . . , sl ≥ 1) .
s1sl ∈ Rn.
.
.
n1l
n >n >···>n >012l

These numbers fulfil a lot of Q-linear relations, which conjecturally all come fromthe so called extended double shuffle realtions, for example
5197
ζ(3) = ζ(2, 1) ,ζ(12) = 168ζ(5, 7) + 150ζ(7, 5) + 28ζ(9, 3) . (1)691
MZV are of particular interest in quantum field theory and number theory [4]. We
study their connection to modular forms by using multiple Eisenstein series:

There exists an algebra I 1 of so called formal iterated integrals which was introduced by Goncharov in [5]. It is spanned by formal symbols I(s1, . . . , sl), with
s1, . . . , sl ≥ 1, which satisfy relations coming from real iterated integrals. The
product is given by the shuffle product. This space has the structure of a Hopf1

11

algebra, i.e. in particular it has a coproduct ∆ : I −→ I ⊕ I which can becomputed in a nice combinatorial way by summing over all dissections of a halfcircle.
Example: ∆(I(2, 3)) = I(2, 3) ⊗ 1 + 3I(3) ⊗ I(2) + 2I(2) ⊗ I(3) + 1 ⊗ I(2, 3).
I(2) ⊗ I(3)Definition: The multiple Eisenstein series are defined byX
1Gs1,...,sl (τ ) =( for s1≥ 3 and s2, . . . , sl≥ 2) .
n1sl
λ 1 · · · λlObservation ([4, 3]): The coefficients in the coproduct are thesame as in the Fourier expansion of multiple Eisenstein series.

0≻λ1≻···≻λlλi∈Zτ +Z∃Proposition: [3] There exist algebra homomorphisms Z : I 1 → R
1and g : I → C[[q]] such that Z (I(s1, . . . , sl)) = ζ(sl, . . . , s1) andg (I(s1, . . . , sl)) = [sl, . . . , s1] for s1, . . . , sl ≥ 2.
∃∃

∃

∃

∃

Theorem: [3] For all s1, . . . , sl ≥ 1 the shuffle regularized multiple
Eisenstein series Gs1,...,sl have the following properties:• They fulfill the shuffle product, i.e. we have an algebra homormorphism
1
I → C[[q]] by sending the generators I(s1, . . . , sl) to Gs1,...,sl (q).
• For integers s1, . . . , sl ≥ 2 they equal the multiple Eisenstein series∃Theorem: [1] The multiple Eisenstein series have a Fourier expansion
given by an explicit but complicated Q[−2πi]-linear combinations of products of multiple zeta value and generating functions of multiple
divisor sums [s1, . . . , sl] ∈ Q[[q]].

Definition: For integers s1, . . . , sl ≥ 1 define the shuffle regularized
multiple Eisenstein series, as the image of the generator I(s1, . . . , sl) in1
I under the algebra homomorphism (Z ⊗ g ) ◦ ∆:

Gs1,...,sl (τ ) := (Z ⊗ g ) ◦ ∆ I(s1, . . . , sl) .
∃The case l = 2 of double Eisenstein series was considered in [4], where the authors
also showed a connection of relations between MZV and cusp forms via periodpolynomials (second relation in (1)).

∃For l = 1 and even s1 > 2 these are the classical Eisenstein series which aremodular forms for SL2(Z).

∃The λi ∈ Zτ + Z are lattices points and ≻ is an order on Z + Zτ given bym1τ + n1 ≻ m2τ + n2 :⇔ (m1 > m2 ∨ (m1 = m2 ∧ n1 > n2)) .

∃The brackets [s1, . . . , sl] in the Fourierexpansion are kind of a combinatorial
object which are for s1, . . . , sl > 0 given byX1
s1−1
sl −1 u1v1+···+ul vl[s1, . . . , sl] :=v1 . . . vl q
∈ Q[[q]] .
(s1 − 1)! . . . (sl − 1)! u >···>u >01lv1,...,vl >0

In [2] it is shown that the space spanned by these brackets has the structure of abi-filtered algebra with a derivation given by q dqd . It has the space of modular
forms as a subalgebra.
Idea: Extend the definition of the multiple Eisenstein series
Gs1,...,sl for s1 ≥ 2 and s2, . . . , sl ≥ 1 by studying the space of thegenerating functions of multiple divisor sums and a connection to the coproduct on the space of formal iterated integrals.

∃Examples: G3,2(τ ) = ζ(3, 2)+3ζ(3)(−2πi)2[2]+2ζ(2)(−2πi)3[3]+(−2πi)5[3, 2].

51
1654
2
3G3,2,2(τ ) =ζ(3, 2, 2) +ζ(2, 3) + ζ(3, 2) (2πi) [2] + ζ(2, 2)(2πi) [3]553457+ 3ζ(3)(2πi) [2, 2] + 4ζ(2)(2πi) [3, 2] + (2πi) [3, 2, 2] ,Gs1,...,sl (q) = Gsl,...,s1(q)and therefore they fulfill the stuffle product in these cases.
References[1] H. Bachmann, “Multiple Zeta-Werte und die Verbindung zu Modulformen
durch Multiple Eisensteinreihen”, Master thesis, University of Hamburg, 2012.
[2] H. Bachmann, U. Kühn, “The algebra of generating functions for multiple divisor sums and applications to multiple zeta values”, arXiv:1309.3920[math.NT].
[3] H. Bachmann, K. Tasaka, “The algebra of multiple Eisenstein series and theGoncharov coproduct”, in preparation.
[4] H. Gangl, M. Kaneko, D. Zagier, “Double zeta values and modular forms”, in“Automorphic forms and zeta functions” p. 71-106, World Sci. Publ., Hackensack, NJ, 2006.
[5] A.B. Goncharov, “Galois symmetries of fundamental groupoids and noncommutative geometry”, Duke Math. J., 128(2) (2005), 209–284.

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