Elliptic associators

TL;DR

This paper develops a genus one analogue of associators and the Grothendieck-Teichmueller group, linking elliptic curves, braid groups, and multiple zeta values through explicit constructions and algebraic structures.

Contribution

It introduces elliptic associators, constructs the elliptic Grothendieck-Teichmueller group, and relates these to braid groups and multiple zeta values in genus one.

Findings

Construction of elliptic associators based on KZB connection.

Explicit description of the elliptic GT group and its Lie algebra.

Relations established between MZVs and Eisenstein series.

Abstract

We construct a genus one analogue of the theory of associators and the Grothendieck-Teichmueller group. The analogue of the Galois action on the profinite braid groups is an action of the arithmetic fundamental group of a moduli space of elliptic curves on the profinite braid groups in genus one. This action factors through an explicit profinite group hat GT_ell, which admits an interpretation in terms of decorations of braided monoidal categories. We relate this group to its prounipotent group scheme version GT_ell(-). We construct a torsor over the latter group, the scheme of elliptic associators. An explicit family of elliptic associators is constructed, based on our earlier work with Calaque and Etingof on the universal KZB connexion. The existence of elliptic associators enables one to show that the Lie algebra of GT_ell(-) is isomorphic to a graded Lie algebra, on which we obtain several results: semidirect product structure; explicit generators. This existence also allows one to compute the Zariski closure of the mapping class group in genus one (isomorphic to the braid group B_3) in the automorphism groups of the prounipotent completions of braid groups in genus one. The analytic study of the family of elliptic associators produces relations between MZVs and iterated integrals of Eisenstein series.