Character sums for elliptic curve densities

Abstract

If $E$ is an elliptic curve over $\mathbb{Q}$, then it follows from work of Serre and Hooley that, under the assumption of the Generalized Riemann Hypothesis, the density of primes $p$ such that the group of $\mathbb{F}_p$-rational points of the reduced curve $\tilde{E}(\mathbb{F}_p)$ is cyclic can be written as an infinite product $\prod \delta_\ell$ of local factors $\delta_\ell$ reflecting the degree of the $\ell$-torsion fields, multiplied by a factor that corrects for the entanglements between the various torsion fields. We show that this correction factor can be interpreted as a character sum, and the resulting description allows us to easily determine non-vanishing criteria for it. We apply this method in a variety of other settings. Among these, we consider the aforementioned problem with the additional condition that the primes $p$ lie in a given arithmetic progression. We also study the conjectural constants appearing in Koblitz's conjecture, a conjecture which relates to the density of primes $p$ for which the cardinality of the group of $\mathbb{F}_p$-points of $E$ is prime.