On the asymptotics for invariants of elliptic curves modulo p

Let E be an elliptic curve defined over Q. Let E(Fp) denote the elliptic curve modulo p. It is known that there exist integers i p and f p such that E(Fp) ∼= Z/ i pZ × Z/ i p f pZ. 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 cα and ck such that for any A > 0 ∑ p≤x (log i p) α = cα li(x) + O ( x (log x)A ) and ∑ p≤x τk(i p) = ck li(x) + O ( x (log x)A ) unconditionally for CM elliptic 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 ∑ p≤x ip = c′ α li(x)+ O ( x 3+α 4 (log x) 1−α 2 ) ∗Research of the first author supported by an NSERC PGS-D and a MaxPlanck-Institut Stipend ∗∗Research of the second author supported by an NSERC Discovery Grant