Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 2

Special Issue: In Honor of Eduard Looijenga, Part 3 of 3

Notes on the universal elliptic KZB connection

Pages: 229 – 312

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n2.a2

Author

Richard Hain (Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Abstract

In this paper, we give an exposition of the elliptic KZB connection over the universal elliptic curve and use it to compute the limit mixed Hodge structure on the unipotent fundamental group of the first order Tate curve. We also give an explicit algebraic formula for the restriction of the elliptic KZB connection to the moduli space of non-zero abelian differentials on an elliptic curve.

Keywords

Eisenstein series, moduli of elliptic curves, mixed elliptic motives, Hodge theory, algebraic de Rham theory of fundamental groups, elliptic KZB connection

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The author was supported in part by the NSF through grants DMS-0706955 and DMS-1005675.

Received 14 June 2016

Accepted 12 March 2017

Published 20 March 2020