The average exponent of elliptic curves modulo $p$DOUBLON DE HAL-01094035

Jie Wu (1)

1 Analyse

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 $$ \frac{1}{\pi(x)} \sum_{p\le x} e_p = \frac{1}{2} C_E x + O_E\big(x^{5/6} (\log x)^{4/3}\big) $$ 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.

Mots clés: density theorems; Elliptic curves over global fields

Domaines: Théorie des nombres [math.NT]

Jie Wu. The average exponent of elliptic curves modulo $p$DOUBLON DE HAL-01094035. Journal of Number Theory, 2014, 135, pp.28-35. ⟨hal-00711948⟩