A Class Number Associated with a Product of Two Elliptic Curves

In this paper we shall consider the product. E×E' of two mutually isogenous elliptic curves E, E' whose rings of endomorphisms are the ring Z of rational integers. We ask whether E×E' can be a Jacobian variety of some curve ; and further in how many essentially different ways. In other words we try to obtain a formula for the number H of isomorphism classes of canonically polarized Jacobian varieties (E×E', Y), Y being a theta divisor. The number H proves to be closely connected with the number of ideal classes and the number of ambiguous ideal classes of a certain imaginary quadratic field Q √<-m> [§8]. The method of this paper is basically the same as that of a study [2], in which the rings of endomorphisms of E, E' are the principal order of an imaginary quadratic field. I wish to express here my hearty thanks to my friend M. Nishi for his suggestions and encouragement