Seminar in Algebraic Geometry and Number Theory

Abstract: In this talk I will discuss a family of real analytic modular forms on the upper half plane, dubbed modular graph forms (MGFs), which are constructed by assigning to a graph a lattice sum. These MGFs form an algebra of functions shown to satisfy in a variety of examples many intriguing differential and algebraic relations. In particular they admit a Fourier-like expansion which is believed to involve only rational numbers and a particular class of periods called single-valued multiple zeta values. 

It is an open problem to give a complete characterisation of this class of functions and prove their conjectural properties.

Towards this goal, I will explain how the MGFs world is related to Francis Brown's theory of single-valued iterated Eichler integrals of holomorphic Eisenstein series.

I will show that an infinite class of MGFs can be constructed from SL_2-equivariant versions of Brown's iterated integrals of two holomorphic Eisenstein series. I will also consider a broader class of modular invariant functions constructed from mixed equivariant iterated integrals of holomorphic Eisenstein series and holomorphic cusp forms.

In particular we will find that for this broader class, the associated periods now also involve completed non-critical L-values of holomorphic cusp forms.    

This talk presents joint work with Axel Kleinschmidt and Oliver Schlotterer.