SPEAKER: David Pierce (UIUC)

TITLE: Elementary equivalence and elliptic curves

ABSTRACT: We consider the classification up to elementary equivalence of function fields over an algebraically closed field. (We use the language of fields with constants from the ground field.) Duret (JSL, '86, '92) showed that in the case of function fields of curves, elementary equivalence implies isomorphism, except for most elliptic curves with complex multiplication. We generalize his technique and characterize how far it goes. In particular, we show that elliptic function fields with complex multiplication agree on certain AE-sentences if and only if their endomorphism rings are isomorphic.