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.
Highlights
This paper studies examples of real analytic functions on the upper half plane satisfying a modular transformation property of the form f az + b cz + d
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 Mf over Q
We can construct nonholomorphic modular forms which are associated with iterated extensions of the pure motives Mf
Summary
It is known that the functions Im(z)r Er,r(z) all occur as modular graph functions (2) Their relation with motives (1) comes about by expressing the Er,s as integrals. The function IG depends neither on the edge numbering, nor on the choice of orientation of G It defines a function IG on the upper half plane which is real analytic and invariant under the action of SL2(Z) (Fig. 1). (1) Zerbini [36] has shown that in all known examples, the ‘zeroth modes’ of modular graph functions involve a certain class of multiple zeta values ζ
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