The elliptic KZB connection and algebraic de Rham theory for unipotent fundamental groups of elliptic curves

TL;DR

This paper develops an algebraic de Rham framework for unipotent fundamental groups of punctured elliptic curves, utilizing the elliptic KZB connection to provide explicit Tannaka duality results.

Contribution

It introduces an algebraic de Rham theory for unipotent fundamental groups of elliptic curves using the elliptic KZB connection, enabling explicit Tannaka duality.

Findings

Explicit algebraic de Rham theory for unipotent fundamental groups

Application of elliptic KZB connection in this context

Explicit Tannaka duality for unipotent connections

Abstract

In this paper, we develop an algebraic de Rham theory for unipotent fundamental groups of once punctured elliptic curves over a field of characteristic zero using the universal elliptic KZB connection of Calaque-Enriquez-Etingof and Levin-Racinet. We use it to give an explicit version of Tannaka duality for unipotent connections over an elliptic curve with a regular singular point at the identity.