Elliptic modular graph forms I: Identities and generating series

TL;DR

This paper introduces elliptic modular graph forms (eMGFs), explores their identities and generating series, and develops reduction formulas and algebraic relations, advancing the understanding of modular forms related to elliptic curves.

Contribution

It defines eMGFs as generalizations of modular graph functions incorporating Abelian group characters and derives new identities and reduction formulas for these functions.

Findings

Derived holomorphic subgraph reduction formulas.

Established algebraic identities between eMGFs.

Developed generating series for eMGFs.

Abstract

Elliptic modular graph functions and forms (eMGFs) are defined for arbitrary graphs as natural generalizations of modular graph functions and forms obtained by including the character of an Abelian group in their Kronecker--Eisenstein series. The simplest examples of eMGFs are given by the Green function for a massless scalar field on the torus and the Zagier single-valued elliptic polylogarithms. More complicated eMGFs are produced by the non-separating degeneration of a higher genus surface to a genus one surface with punctures. eMGFs may equivalently be represented by multiple integrals over the torus of combinations of coefficients of the Kronecker--Eisenstein series, and may be assembled into generating series. These relations are exploited to derive holomorphic subgraph reduction formulas, as well as algebraic and differential identities between eMGFs and their generating series.