Tristan Freiberg: On the average exponent of elliptic curves modulo $p$
Tid: Ti 2012-05-15 kl 15.00
Plats: Room 3721, Lindstedtsvägen 25, 7th floor, Department of Mathematics, KTH
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 $e_p$ denote the
exponent of this group. Assuming a certain form of the
Generalized Riemann Hypothesis, we study the average of
$e_p$ as $p \le X$ ranges over primes of good reduction, and
find that the average exponent essentially equals
$p\cdot c_E$, where the constant $c_E \in (0,1)$ depends on
$E$.