Hodge classes on abelian varieties of low dimension

Abstract

In this paper we study Hodge classes on complex abelian varieties X If dimX then it is wellknown that every Hodge class on X is a linear combination of products of divisor classes In the authors showed that if X is simple of dimension then every Hodge class is a linear combination of products of divisor classes and Weil classesif there are any The notion of a Weil class shall be briey reviewed in The aim of this note is to extend this to arbitrary abelian varieties of dimension In order to state our main results let us describe some special cases a The abelian variety X is isogenous to a product X X where X is an elliptic curve with complex multiplication by an imaginary quadratic eld k and where X is a simple abelian threefold such that there exists an embedding k EndX b The abelian variety X is simple of dimension such that EndX is a eld containing an imaginary quadratic eld k which acts on the tangent space TX with multiplicities See x for further explanation c The abelian variety X is simple of dimension with D EndX a denite quaternion algebra over Q Type III in the Albert classication Note that for every D n Q the subalgebra Q D is an imaginary quadratic eld d The abelian variety X is simple of dimension with EndX Q