Elliptic KZB connections via universal vector extensions

Fonseca, Tiago J.; Matthes, Nils

doi:10.2140/ant.2025.19.1369

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Elliptic KZB connections via universal vector extensions

Tiago J. Fonseca and Nils Matthes

Vol. 19 (2025), No. 7, 1369–1425

DOI: 10.2140/ant.2025.19.1369

Abstract

Using the formalism of bar complexes and their relative versions, we give a new, purely algebraic, construction of the so-called universal elliptic KZB connection in arbitrary level. We compute explicit analytic formulae, and we compare our results with previous approaches to elliptic KZB equations and multiple elliptic polylogarithms in the literature.

Our approach is based on a number of results concerning logarithmic differential forms on universal vector extensions of elliptic curves. Let $S$ be a scheme of characteristic 0, $E \to S$ be an elliptic curve, $f : E^{♮} \to S$ be its universal vector extension, and $π : E^{♮} \to E$ be the natural projection. Given a finite subset of torsion sections $Z \subset E (S)$ , we study the dg-algebra over $𝒪_{S}$ of relative logarithmic differentials $𝒜 = f_{*} Ω_{E^{♮} ∕ S}^{∙} (\log π^{- 1} Z)$ . In particular, we prove that the residue exact sequence in degree 1 splits canonically, and we derive the formality of $𝒜$ . When $S$ is smooth over a field $k$ of characteristic 0, we also prove that sections of $𝒜^{1}$ admit canonical lifts to absolute logarithmic differentials in $f_{*} Ω_{E^{♮} ∕ k}^{1} (\log π^{- 1} Z)$ , which extends a well-known property for regular differentials given by the “crystalline nature” of universal vector extensions.

Keywords

KZB equations, unipotent connections, universal vector extension, multiple elliptic polylogarithms, unipotent fundamental group

Mathematical Subject Classification

Primary: 11G05, 14F40, 32G34

Milestones

Received: 1 September 2023

Revised: 28 June 2024

Accepted: 3 September 2024

Published: 3 June 2025

Authors

Tiago J. Fonseca
Departamento de Matemática Instituto de Matemática, Estatística e Computação Científica Universidade Estadual de Campinas Campinas Brazil

Nils Matthes
Department of Mathematical Sciences University of Copenhagen Copenhagen Denmark

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Vol. 19, No. 7, 2025