Towards multiple elliptic polylogarithms
Andrey Levin, Georges Racinet
TL;DR
The paper develops an explicit De Rham framework for multiple elliptic polylogarithms by constructing and analysing the De Rham fundamental torsor of punctured elliptic curves, both fiberwise and relatively over moduli spaces. It introduces a universal flat connection over the upper half-plane built from a Kronecker-type two-variable Jacobi form and shows an equivalent relative theory over smooth elliptic families via relative tannakian methods. A central achievement is an explicit description of the relative fundamental torsor on ${\mathscr M}_{1,2}$ in terms of a relative Hopf algebra ${\mathcal{R}}$ and a flat connection, together with a natural ${\mathbb{Q}}$-structure obtained by algebraic reexpression of the analytic data. The work provides concrete differential equations for parallel transport in families, clarifies the role of modularity, and establishes how these elliptic polylogarithmic objects descend to algebraic models, enabling arithmetic applications. Overall, the paper lays foundational tools for a theory of multiple elliptic polylogarithms in a De Rham, tannakian, and arithmetic setting, with explicit formulas and moduli-stack interpretations.
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.