  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.
