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

TL;DR

This paper refines previous results on the average distribution of primes related to elliptic curves, focusing on primes in arithmetic progressions and revealing biases among different residue classes for properties like cyclicity and divisibility.

Contribution

It extends earlier work by analyzing primes in specific arithmetic progressions and uncovers biases among residue classes concerning elliptic curve properties.

Findings

Identification of biases among residue classes for primes related to elliptic curves.

Refinement of average density results for primes in arithmetic progressions.

Enhanced understanding of the distribution of primes with elliptic curve properties.

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.