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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 rootconjecture posed by S. Lang and H. Trotter in 1977, J-P. Serre adapted methodsof C. Hooley to prove a GRH-conditional asymptotic formula for the number ofprimes $p \leq x$ 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 questionof cyclicity of $\tilde{E}_p(\mathbb{F}_p)$ under the additional restrictionthat $p$ lie in an arithmetic progression. In this note, we study the issue ofwhich arithmetic progressions $a \bmod n$ have the property that, for all butfinitely many primes $p \equiv a \bmod n$, the group$\tilde{E}_p(\mathbb{F}_p)$ is not cyclic, answering a question of Akbal andG\"{u}lo$\breve{\text{g}}$lu on this issue.

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