For a field of definition $k$ of an abelian variety $\Av$ and prime ideal $\ip$ of $k$ which is of a good reduction for $\Av$, the structure of $\Av(\F_{\ip})$ as abelian group is:

    \Av(\F_{\ip})\simeq \Z/d_1(\ip)\Z\oplus\cdots\oplus\Z/d_g(\ip)\Z\oplus\Z/e_1(\ip)\Z\oplus\cdots\oplus\Z/e_g(\ip)\Z,

    where $d_i(\ip)|d_{i+1}(\ip)$, $d_g(\ip)|e_1(\ip)$, and $e_i(\ip)|e_{i+1}(\ip)$ for $1\leq i<g$.

    We are interested in finding an asymptotic formula for the number of prime ideals $\ip$ with $N\ip<x$, $\Av$ has a good reduction at $\ip$, $d_1(\ip)=1$. We succeed in 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 CM field.