Average behaviors of invariant factors in Mordell-Weil groups of CM elliptic curves modulo p

Let E be an elliptic curve defined over Q and with complex multiplication by O"K, 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(F"p) has a structure(1)E(F"p)~Z/d"pZ@?Z/e"pZ with uniquely determined d"p|e"p. We give an asymptotic formula for the average order of e"p over primes p@?x of good reduction, with improved error term O(x^2/log^A@?x) for any positive number A, which previously was set as O(x^2/log^1^/^8@?x) by [12]. Further, we obtain an upper bound estimate for the average of d"p, and a lower bound estimate conditionally on nonexistence of Siegel-zeros for Hecke L-functions.