The distribution and density of cyclic groups of the reductions of an elliptic curve over a function field

TL;DR

This paper provides an asymptotic formula for counting places where the reduction of a non-isotrivial elliptic curve over a function field forms a cyclic group, and characterizes when the density of such places is zero.

Contribution

It introduces an asymptotic formula for the distribution of cyclic reductions of elliptic curves over function fields and identifies conditions for zero density.

Findings

Asymptotic count of places with cyclic reduction

Conditions for zero Dirichlet density

Characterization of cyclic group reductions

Abstract

Let $K$ be a global field of finite characteristic $p\geq2$, and let $E/K$ be a non-isotrivial elliptic curve. We give an asympotoic formula of the number of places $\nu$ for which the reduction of $E$ at $\nu$ is a cyclic group. Moreover we determine when the Dirichlet density of those places is 0.