Abstract

Let K and K' be two elliptic fields with complex multiplication over an algebraically closed field k of characteristic 0, non k-isomorphic, and let C and C' be two curves with respectively K and K' as function fields. We prove that if the endomorphism rings of the curves are not isomorphic then K and K' are not elementarily equivalent in the language of fields expanded with a constant symbol (the modular invariant). This theorem is an analogue of a theorem from David A. Pierce in the language of k-algebras