In his seminal work on quasi-Hopf algebras and the absolute Galois group of the rationals, Drinfeld defined the KZ (Knizhnik--Zamolodchikov) associator, a certain complex power series in two non-commuting variables which describes the monodromy of the KZ connection on the projective line minus three points. Among its coefficients are the special values of the Riemann zeta function at positive integers, a fact which hints at the deep arithmetic properties the KZ associator encodes.

More recently, Enriquez introduced an elliptic generalization of Drinfeld's theory and defined an elliptic analog of the KZ associator, the elliptic Knizhnik--Zamolodchikov--Bernard (KZB) associator. In this talk, we study some of its coefficients and their relationship to the arithmetic of (once-punctured) elliptic curves.