Cyclicity of elliptic curves modulo primes in arithmetic progressions

Abstract

We consider the reduction of an elliptic curve defined over the rational numbers modulo primes in a given arithmetic progression and investigate how often the subgroup of rational points of this reduced curve is cyclic.

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

“…These three properties, which represent cases which are excluded from the proof of (21), are not mutually exclusive. As we observed earlier, if E satisfies property (1) then C E,a,n = 0, and so (21) cannot possibly hold in this case (this was also observed in [1, Proposition 1]). What about elliptic curves satisfying properties (2) or (3) but not (1)?…”

“…Inspired by Conjecture 1.1, Y. Akbal and A. M. Güloglu [1] considered the question of cyclicity of Ẽp (F p ) for the subset of those primes p which lie in a fixed arithmetic progression (this question was also considered in the Ph.D. dissertation of J. Brau [6]). Specifically, for any fixed a, n ∈ N with gcd(a, n) = 1, let us define the counting function π E,a,n (x) by π E,a,n (x) := p ≤ x : p ∤ N E , p ≡ a mod n and Ẽp (F p ) is cyclic .…”

“…(The left-hand condition above implies that C E,a,n = 0, so the implication ( 6) is consistent with Theorem 1.2.) They then posed the following question (see [1,Question 1]).…”

“…Remark 1.10. For any elliptic curve E over Q and a, n ∈ N with gcd(a, n) = 1, Theorems 1 and 2 of [1] give, for any A ≥ 2, an unconditional lower bound x (log x) A ≪ π E,a,n (x), (21) provided gcd(a − 1, n) is a power of two and none of the following three properties is fulfilled:…”