Square-free discriminants of Frobenius rings

TL;DR

This paper studies the distribution of primes for which the discriminant of Frobenius endomorphisms of elliptic curves is square-free, providing asymptotic formulas and exploring connections to the Lang-Trotter conjecture.

Contribution

It establishes the asymptotic behavior of primes with square-free Frobenius discriminants for average elliptic curves over $\\mathbb{Q}$, linking to deep conjectures in number theory.

Findings

Derived the asymptotic formula for the count of such primes.

Connected the distribution to the Lang-Trotter conjecture.

Discussed implications for the endomorphism rings of elliptic curves.

Abstract

Let $E$ be an elliptic curve over $\Q$. It is well known that the ring of endomorphisms of $E_p$, the reduction of $E$ modulo a prime $p$ of ordinary reduction, is an order of the quadratic imaginary field $Q(\pi_p)$ generated by the Frobenius element $\pi_p$. When the curve has complex multiplication (CM), this is always a fixed field as the prime varies. However, when the curve has no CM, very little is known, not only about the order, but about the fields that might appear as algebra of endomorphisms varying the prime. The ring of endomorphisms is obviously related with the arithmetic of $a_p^2-4p$, the discriminant of the characteristic polynomial of the Frobenius element. In this paper, we are interested in the function $\pi_{E,r,h}(x)$ counting the number of primes $p$ up to $x$ such that $a_p^2-4p$ is square-free and in the congruence class $r$ modulo $h$. We give in this paper the precise asymptotic for $\pi_{E,r,h}(x)$ when averaging over elliptic curves defined over the rationals, and we discuss the relation of this result with the Lang-Trotter conjecture, and with some other problems related to the curve modulo $p$.