On the Density of Coprime Reductions of Elliptic Curves

TL;DR

This paper investigates the density of primes where two non-CM elliptic curves over rationals have coprime group orders, proposing a conjecture, proving convergence, and analyzing distribution for Serre pairs.

Contribution

It formulates a conjecture on the density of such primes, proves convergence of the associated series, and derives explicit formulas and distribution results for Serre pairs.

Findings

Series defining the density converges and has an Euler product expansion.

Explicit formula for the density constant in the case of Serre pairs.

Distribution of the constants follows a specific moments pattern.

Abstract

Given non-CM elliptic curves $E_1$ and $E_2$ over $\mathbb{Q}$, we study the natural density of primes $p$ of good reduction for which the orders of the groups $E_1(\mathbb{F}_p)$ and $E_2(\mathbb{F}_p)$ are coprime. This problem may be viewed as an elliptic curve analogue of the classical question concerning the density of coprime integer pairs. Motivated by Zywina's refinement of the Koblitz conjecture, we formulate a conjecture for the density of such primes. We prove that the series defining this constant converges and admits an almost Euler product expansion. In the case of Serre pairs, we give a closed formula for the constant and use it to prove a moments result describing the distribution of these constants as $(E_1, E_2)$ varies.