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

2022/06/02 by Jones, Nathan, Lee, Sung Min
#11F80 (Primary) #11G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2206.00872

Abstract

Let E be an elliptic curve defined over ℚ and, for a prime p of good reduction for E let 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 ≤ x for which the group E_p(\mathbbF_p) is cyclic. More recently, Akbal and Gülo\breveglu considered the question of cyclicity of E_p(\mathbbF_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 ≡ a \bmod n, the group E_p(\mathbbF_p) is not cyclic, answering a question of Akbal and Gülo\breveglu on this issue.

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