On the Asymptotics for Invariants of Elliptic Curves Modulo p

Let E be an elliptic curve defined over Q. Let E(F_p) denote the elliptic curve modulo p. It is known that there exist integers i _p and f _p such that E(F_p) ∼= Z/i _pZ × Z/i _p f_p Z. We study questions related to i _p and f p. In particular, for any α > 0 and k ∈ N, we prove there exist positive constants c_α and c_k such that for any A > 0

Σ(log i_p)^α = c_&#945 li(x) + O(x/(logx)^A)

and

Σ Tk(i_p) = c_k li(x) + O(x/(log x)^A)

unconditionally for CMelliptic curves, where τk (n) is the number of ways of writing n as a product of k positive integers. For a CM curve E and 0 < α < 1, we prove that there exists a constant c'_α > 0 such that Σ i^α_p = c'_α li(x) + O(x^{3+α/4 (log x)1-α/2)}