Abstract
We introduce a new family of real-analytic modular forms on the upper-half plane. They are arguably the simplest class of ‘mixed’ versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in $q,\overline{q}$ and $\log |q|$ involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.
Highlights
Let H = {z : Im z > 0} be the upper-half plane with its action of SL2(Z):γ z = az + b, cz + d where γ = ab cd ∈ SL2(Z). (1.1)We shall say that a real-analytic function f : H −→ C is modular of weights (r, s) with r, s ∈ Z if for all γ ∈ SL2(Z), it satisfies f (γ z) =rs f (z). (1.2)c The Author 2020
Every f ∈ MI E is modular of weights (r, s) for integers r, s 0
Zsv where Zsv is the weight-filtered Q-algebra of single-valued multiple zeta values, viewed as constant functions of modular weights (0, 0)
Summary
We construct a space MI E of real-analytic modular functions on H, which are modular analogues of the single-valued polylogarithms. They are obtained by taking real and imaginary parts of iterated primitives of the holomorphic Eisenstein series, which are defined for all even k 4 by. Every f ∈ MI E is modular of weights (r, s) for integers r, s 0 It admits a unique expansion, for some N 0, of the form. Let lw(MI E ) ⊂ MI E denote the subspace of functions with modular weights (n, 0), that is, satisfying the classical modular transformation property with no antiholomorphic automorphy factor (the notation lw stands for ‘lowest weight’ in the sense of sl2representations). We show that this L-function is a single-variable restriction of the multiple-variable L-functions recently defined in [1]
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