On the average congruence class bias for cyclicity and divisibility of the groups of $\mathbb{F}_p$-points of elliptic curves

Abstract

In 2009, W. D. Banks and I. E. Shparlinski studied the average densities of primes $p \leq x$ for which the reductions of elliptic curves of small height modulo $p$ satisfy certain arithmetic properties, namely cyclicity and divisibility of the number of points by a fixed integer $m$. In this paper, we refine their results, restricting the primes $p$ under consideration to lie in an arithmetic progression $k \bmod{n}$. Furthermore, for a fixed modulus $n$, we investigate statistical biases among the different congruence classes $k \bmod{n}$ of primes satisfying the aforementioned properties.