Average of the first invariant factor of the reductions of abelian varieties of CM type

Abstract

Abstract. For a field of definition k of an abelian variety A and prime ideal p of k which is of a good reduction for A, the structure of A(Fp) as abelian group is:We are interested in finding an asymptotic formula for the number of prime ideals p with N p < x, A has a good reduction at p, d1(p) = 1. We succeed in proving this under the assumption of the Generalized Riemann Hypothesis (GRH). Unconditionally, we achieve a short range asymptotic for abelian varieties of CM type, and the full cyclicity theorem for elliptic curves over a number field containing the CM field.