Let $E$ be an elliptic curve defined over ${ \mathbb Q}$. For a prime $p$ of good reduction for $E$, denote by $e_p$ the exponent of the reduction of $E$ modulo $p$. Under GRH, we prove that there is a constant $C_E\in (0, 1)$ such that $$ \frac{ 1}{ \pi(x)} \sum_{ p\le x} e_p = 1/2 C_E x + O_E\big(x^{ 5/6} (\log x)^{ 4/3}\big) $$ for all $x\ge 2$, where the implied constant depends on $E$ at most. When $E$ has complex multiplication, the same asymptotic formula with a weaker error term $O_E(1/(\log x)^{ 1/14})$ is established unconditionally. These improve some recent results of Freiberg and Kurlberg.
Categories: math.NT