Average of the First Invariant Factor of the Reductions of Abelian Varieties of CM Type

Let E be an elliptic curve defined over Q and with complex multiplication by OK, the ring of integers in an imaginary quadratic field K. Let p be a prime of good reduction for E. It is known that E(Fp) has a structure.;E(Fp) ≃ Z/dpZ ⊕ Z/ epZ (0.1).;with uniquely determined dp∥ep. We give an asymptotic formula for the average order of ep over primes p ≤ x of good reduction, with improved error term O(x2= logA x) for any positive number A, which previously O(x2= log1/8 x) by [Wu]. Further, we obtain an upper bound estimate for the average of dp, and a lower bound estimate conditionally on nonexistence of Siegel-zeros for Hecke L-functions.;Then we extend the methods to abelian varieties of CM type. 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:;A(Fp) ≃ Z/d1(p) Z ⊕ · · · ⊕ Z/ dg(p)Z ⊕ Z/e1(p) Z ⊕ · · · ⊕ Z= eg(p)Z, (0.2).;where di(p)| di+1(p), dg(p)|e 1(p), and ei( p)|ei+1(p ) for 1 ≤ i