Elliptic curves on abelian surfaces

Abstract

The purpose of this paper is to present two theorems which give an overview of the set of elliptic curves lying on an abelian surface and to discuss several applications. One of these applications is a classical theorem of Biermann (1883) and Humbert (1893) on the characterization of abelian surfaces containing elliptic curves in terms of the “singular relations” of Humbert. As a by-product one obtains a purely algebraic description of such relations and hence also ofHumbert surfaces.