Title: 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.