Resumen de Products of CM elliptic curves

Ernst Kani

• Let K be an arbitrary field, and let E_1, · · · ,E_n be isogenous elliptic curves with complex multiplication (CM) over K. The basic problem considered in this paper is to find suitable criteria for determining whether or not a given abelian variety A′ is isomorphic to the product variety A = E_1 × ··· × E_n . A special case of this problem is to determine this in the case that {A' = E_1'\times\cdots\times E'_n} is also a product variety. The solution of this problem is based on an extension of the methods of Deuring, Shimura/Taniyama and Waterhouse of constructing isogenies of abelian varieties via ideals of their endomorphisms rings.