Function Fields and Elementary Equivalence

Continuing work of Duret, we treat the relation between isomorphismand elementary equivalence of function fields over algebraicallyclosed fields. For function fields of curves, these are ‘usually’the same, but in characteristic zero, for elliptic curves withcomplex multiplication, a weak variant of elementary equivalenceof their function fields corresponds to isomorphism of the endomorphismrings of the curves, not to isomorphism of the curves themselves.1991 Mathematics Subject Classification 14H52, 11U09, 03C52.