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 $p\le x$ 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 $k\mathrm{mod}n$. Furthermore, for a fixed modulus $n$, we investigate statistical biases among the different congruence classes $k\mathrm{mod}n$ of primes satisfying the aforementioned properties.

In the updated version, we have improved the error bound and amended the commentary on the past achievements