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

For a field of definition of an abelian variety and prime ideal of which is of a good reduction for , the structure of as abelian group is: where , , and for . We are interested in finding an asymptotic formula for the number of prime ideals with , has a good reduction at , . 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. [ABSTRACT FROM AUTHOR]