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

Abstract

Let E be an elliptic curve defined over Q and, for a prime p of good reduction for E let Ẽp 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 ≤ x for which the group Ẽp(Fp) is cyclic. More recently, Akbal and Güloglu considered the question of cyclicity of Ẽp(Fp) under the additional restriction that p lie in an arithmetic progression. In this note, we study the issue of which arithmetic progressions a mod n have the property that, for all but finitely many primes p ≡ a mod n, the group Ẽp(Fp) is not cyclic, answering a question of Akbal and Güloglu on this issue.