Elliptic associators

We construct a genus one analogue of the theory of associators and the Grothendieck–Teichmüller (GT) 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 $\widehat{\mathrm{GT }}_{ell}$ , which admits an interpretation in terms of decorations of braided monoidal categories. This group acts on the tower of profinite braid groups in genus one and has the structure of a semidirect product of the profinite GT group $\widehat{\mathrm{GT }}$ by an explicit radical. We relate $\widehat{\mathrm{GT }}_{ell}$ to its prounipotent group scheme version $\mathrm{GT }_{ell}(-)$ , which also has a semidirect product structure. We construct a torsor over this group, the scheme of elliptic associators. An explicit family of elliptic associators is constructed, based on earlier joint work with Calaque and Etingof on the universal KZB connexion. The existence of elliptic associators enables one to show that the Lie algebra of $\mathrm{GT }_{ell}(-)$ is isomorphic to a graded Lie algebra, on which we obtain several results: it is a semidirect product of the graded GT Lie algebra $\mathfrak grt $ by an explicit radical; we exhibit an explicit Lie subalgebra. Elliptic associators also allow 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.