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

TL;DR

This paper investigates the conditions under which the reduced elliptic curve groups modulo primes in specific arithmetic progressions are non-cyclic, extending prior work on the distribution of cyclic groups in elliptic curve reductions.

Contribution

It characterizes arithmetic progressions for which the group of points on elliptic curves modulo primes is non-cyclic for all but finitely many primes, answering a question posed by Akbal and Gülöğlu.

Findings

Identifies arithmetic progressions with finitely many exceptions to non-cyclicity

Provides criteria for when elliptic curve reductions are non-cyclic in progressions

Extends understanding of elliptic curve group structures in number theory

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 $p \leq x$ for which the group $\tilde{E}_p(\mathbb{F}_p)$ is cyclic. More recently, Akbal and G\"{u}lo$\breve{\text{g}}$lu 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 note, we study the issue of which arithmetic progressions $a \bmod n$ have the property that, for all but finitely many primes $p \equiv a \bmod n$, the group $\tilde{E}_p(\mathbb{F}_p)$ is not cyclic, answering a question of Akbal and G\"{u}lo$\breve{\text{g}}$lu on this issue.