A class of non-holomorphic modular forms I

Abstract

This introductory paper studies a class of real analytic functions on the upper half plane satisfying a certain modular transformation property. They are not eigenfunctions of the Laplacian and are quite distinct from Maass forms. These functions are modular equivariant versions of real and imaginary parts of iterated integrals of holomorphic modular forms and are modular analogues of single-valued polylogarithms. The coefficients of these functions in a suitable power series expansion are periods. They are related both to mixed motives (iterated extensions of pure motives of classical modular forms) and to the modular graph functions arising in genus one string perturbation theory.This paper studies examples of real analytic functions on the upper half plane satisfying a modular transformation property of the formfor integers r, s. They do not satisfy a simple condition involving the Laplacian. The raison d'être for this class of functions is two-fold:(1) Holomorphic modular forms f with rational Fourier coefficients correspond to certain pure motives M f over Q. Using iterated integrals, we can construct nonholomorphic modular forms which are associated with iterated extensions of the pure motives M f . Their coefficients are periods. (2) In genus one closed string perturbation theory, one assigns a lattice sum to a graph [16], which defines a real analytic function on the upper half plane invariant under SL 2 (Z).It is an open problem to give a complete description of this class of functions and prove their conjectured properties.In this introductory paper, we describe elementary properties of a class M of modular forms. Within this class are modular iterated integrals, which are analogues of singlevalued polylogarithms, and are obtained by solving a differential equation in M. The basic prototype are real analytic Eisenstein series, defined by E r,s (z) = w! (2πi) w+1 1 2 (m,n) =(0,0)

A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

“…3 we review some properties of the space M ! of real analytic modular forms from [2], and its subspaces HM ! (Sect.…”

“…One easily shows (see [7] or version 1 of [2]) that f ∈ D n+1 M ! −n if and only if [C f ] ∈ B 1 ( ; V n ) ⊗ C, and hence the previous map descends to an isomorphism…”

“…There is a correspondence [2,Sect. 7.2], between sections of the trivial bundle V n ⊗ C on H and families of functions F r,s : H → C for r + s = n with r, s ≥ 0.…”

“…This paper is the third in a series [2,3] studying subspaces of the vector space M ! of real analytic functions f : H → C which are modular of weights (r, s) for r, s ∈ Z, i.e.…”

“…is equipped with differential operators ∂, ∂ closely related to Maass' raising and lowering operators [17], and a Laplacian . In [2], we defined a subspace MI ! ⊂ M !…”