Notes on the Universal Elliptic KZB Equation

TL;DR

This paper explores the universal elliptic KZB equation, detailing its structure, computing related Hodge structures, and demonstrating its rationality over the moduli space of elliptic curves with differentials.

Contribution

It provides a comprehensive exposition of the elliptic KZB connection, computes the limit mixed Hodge structure, and proves the connection's definition over Q.

Findings

Explicit computation of the limit mixed Hodge structure on the unipotent fundamental group of the Tate curve minus its identity.

Demonstration that the elliptic KZB connection is defined over the rational numbers.

Detailed analysis of the connection over the moduli space of elliptic curves with a non-zero abelian differential.

Abstract

The universal elliptic KZB equation is the integrable connection on the pro-vector bundle over M_{1,2} whose fiber over the point corresponding to the elliptic curve E and a non-zero point x of E is the unipotent completion of \pi_1(E-{0},x). This was written down independently by Calaque, Enriquez and Etingof (arXiv:math/0702670), and by Levin and Racinet (arXiv:math/0703237). It generalizes the KZ-equation in genus 0. These notes are in four parts. The first two parts provide a detailed exposition of this connection (following Levin-Racinet); the third is a leisurely exploration of the connection in which, for example, we compute the limit mixed Hodge structure on the unipotent fundamental group of the Tate curve minus its identity. In the fourth part we elaborate on ideas of Levin and Racinet and explicitly compute the connection over the moduli space of elliptic curves with a non-zero abelian differential, showing that it is defined over Q.