On the Average Exponent of Elliptic Curves Modulo p

Given an elliptic curve E defined over $\mathbb Q$ and a prime p of good reduction, let $\tilde E(\mathbb F_p)$ denote the group of $\mathbb F_p$-points of the reduction of E modulo p, and let ep denote the exponent of this group. Assuming a certain form of the generalized Riemann hypothesis (GRH), we study the average of ep as $p\leqslant X$ ranges over primes of good reduction, and find that the average exponent essentially equals p⋅cE, where the constant cE>0 depends on E. For E without complex multiplication (CM), cE can be written as a rational number (depending on E) times a universal constant, $c:=\Pi_q\big(1-\frac {q^3}{(q^2-1)(q^5-1)}\big)$, the product being over all primes q. Without assuming GRH, we can determine the average exponent when E has CM, as well as give an upper bound on the average in the non-CM case.