On the density of a set of primes associated to an Elliptic Curve

TL;DR

This paper investigates the density of primes related to elliptic curves where the number of points on the curve over finite fields is coprime to p-1, providing conjectural estimates supported by heuristics.

Contribution

It introduces a conjectural formula for the density of such primes associated with elliptic curves, backed by heuristic evidence for most cases.

Findings

Proposes a conjectural density value for primes linked to elliptic curves.

Provides heuristic evidence supporting the conjecture for most elliptic curves.

Enhances understanding of prime distributions related to elliptic curve properties.

Abstract

For a given elliptic curve $E$ defined over the rationals, we study the density of primes $p$ satisfying $\mathrm{gcd}(\#E(\mathbb{F}_p),p-1)=1$ and give a conjectural value for this density with strong heuristic evidence for most elliptic curves, in an appropriate sense.