Divisibility of orders of reductions of elliptic curves

TL;DR

This paper investigates the divisibility properties of the number of points on elliptic curves over finite fields, providing explicit families and density estimates related to torsion subgroups and reductions modulo primes.

Contribution

It introduces explicit families of elliptic curves with computable divisibility properties of their reductions and estimates the density of primes with specific congruence classes.

Findings

Explicit families of elliptic curves with known divisibility properties

Density estimates for primes with specific reduction behaviors

Elliptic curves over function fields with controlled divisibility of reductions

Abstract

Let $E$ be an elliptic curve defined over $\mathbb Q$ and $\widetilde{E}_p$ denote the reduction of $E$ modulo a prime $p$ of good reduction for $E$. The divisibility of $|\widetilde{E}_{p}(\mathbb{F}_p)|$ by an integer $m\ge 2$ for a set of primes $p$ of density $1$ is determined by the torsion subgroups of elliptic curves that are $\mathbb Q$-isogenous to $E$. In this work, we give explicit families of elliptic curves $E$ over $\mathbb Q$ together with integers $m_E$ such that the congruence class of $|\widetilde{E}_p(\mathbb{F}_p)|$ modulo $m_E$ can be computed explicitly. In addition, we can estimate the density of primes $p$ for which each congruence class occurs. These include elliptic curves over $\mathbb Q$ whose torsion grows over a quadratic field $K$ where $m_E$ is determined by the $K$-torsion subgroups in the $\mathbb Q$-isogeny class of $E$. We also exhibit elliptic curves over $\mathbb Q(t)$ for which the orders of the reductions of every smooth fiber modulo primes of positive density strictly less than $1$ are divisible by given small integers.