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 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(1)E(Fp)≃Z/dpZ⊕Z/epZ 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 was set as O(x2/log1/8⁡x) by [12]. 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.