On Certain Type of Jacobian Varieties of Dimension 2
摘要
The purpose of this paper is to show a method of constructing a Jacobian variety isogenous to a product of two non-isogenous elliptic curves. We first treat in § 1 the classical case making use of the theory of complex tori ; a normal form of Riemann matrix will be given for each Jacobian variety obtained. In § 2 and the subsequent sections we treat the abstract case, where we assume that the rings of endomorphisms of the two elliptic curves are both isomorphic to the ring Z of rational integers. Any Jacobian variety of such type can be obtained by our method. Also we can determine the structure of the rings of endomorphisms of Jacobian varieties of such type ; and prove, as a simple application, that the ring of endomorphisms of the Jacobian variety of a generic curve of genus 2 is isomorphic to Z.