On the acyclicity of reductions of elliptic curves modulo primes in arithmetic progressions

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and, for a prime $p$ of good reduction for $E$, let ${\tilde{E}}_p$ denote the reduction of $E$ modulo $p$. Inspired by an elliptic curve analogue of Artin’s primitive root conjecture posed by S. Lang and H. Trotter in 1977, J.-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes $p\le x$ for which the group ${\tilde{E}}_p({\mathbb{F}}_p)$ is cyclic. An illuminating proof of this asymptotic formula appeared in a 1983 paper of M. R. Murty, which also established the same unconditionally in the case where $E$ has complex multiplication. More recently, Akbal and Güloğlu considered the question of cyclicity of ${\tilde{E}}_p({\mathbb{F}}_p)$ under the additional restriction that $p$ lie in an arithmetic progression. In this paper, we study the question of which arithmetic progressions $a\mathrm{mod}n$ have the property that, for all but finitely many primes $p\equiv a\mathrm{mod}n$, the group ${\tilde{E}}_p({\mathbb{F}}_p)$ is not cyclic, answering a question of Akbal and Güloğlu on this issue.