The Universal Elliptic KZB Connection in Higher Level

TL;DR

This paper studies the level N elliptic KZB connection, showing it forms an admissible variation of mixed Hodge structure and degenerates to the cyclotomic KZ connection, advancing understanding of Galois actions on fundamental groups.

Contribution

It demonstrates that the elliptic KZB connection underlies an admissible variation of mixed Hodge structure and describes its degeneration to the cyclotomic KZ connection.

Findings

Connection underlies an admissible variation of mixed Hodge structure.

Degenerates to the cyclotomic KZ connection over singular fibers.

Progress towards understanding Galois actions on unipotent fundamental groups.

Abstract

The level $N$ elliptic KZB connection is a flat connection over the universal elliptic curve in level $N$ with its $N$-torsion sections removed. Its fiber over the point $(E,x)$ is the unipotent completion of $\pi_1(E - E[N],x)$. It was constructed by Calaque and Gonzalez. In this paper, we show that the connection underlies an admissible variation of mixed Hodge structure and that it degenerates to the cyclotomic KZ connection over the singular fibers of the compactified universal elliptic curve. These are the first steps in a larger project to compute the action of the Galois group of mixed Tate motives unramified over $\mathbb{Z}[\mathbf{\mu}_N,1/N]$ on the unipotent fundamental group of $\mathbb{P}^1 - \{0,\mathbf{\mu}_N,\infty\}$ and to better understand Goncharov's higher cyclotomy.