Towards multiple elliptic polylogarithms

TL;DR

This paper explores elliptic analogs of polylogarithms related to the fundamental group of the projective line minus three points, deriving explicit algebraic formulas and structures for families of elliptic curves.

Contribution

It introduces a new framework for multiple elliptic polylogarithms using analytic uniformisation and explicit algebraic formulas, extending to moduli stacks.

Findings

Derived the fundamental nilpotent De Rham torsor for elliptic curves.

Extended results to families over the moduli stack $M_{1,2}$.

Provided natural $ ext{Q}$-structures for these elliptic polylogarithms.

Abstract

We investigate the elliptic analogs of multi-indexed polylogarithms that appear in the theory of the fundamental group of the projective line minus three points as sections of a universal nilpotent bundle with regular singular connection. We use an analytic uniformisation to derive the fundamental nilpotent De Rham torsor of a single elliptic curve in terms of a double Jacobi form introduced by Kronecker. We then extend this result to any smooth family, relatively to the base, i.e., to the moduli stack $M_{1,2}$ over $M_{1,1}$. Everything relies on explicit formulas that turn out to be algebraic for rational (families of) elliptic curves, and we conclude by providing the corresponding natural $\QM$ structure.