Applications of the square sieve to a conjecture of Lang and Trotter for a pair of elliptic curves over the rationals

TL;DR

This paper investigates the distribution of primes for which two non-isogenous elliptic curves over rationals share the same Frobenius field, employing Heath-Brown's square sieve to derive bounds under various hypotheses.

Contribution

It applies Heath-Brown's square sieve to establish new upper bounds for the prime counts related to Frobenius fields of two elliptic curves, both conditionally and unconditionally.

Findings

Conditional upper bounds assuming GRH

Unconditional upper bounds

Extension of the Lang-Trotter conjecture context

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$. Let $p$ be a prime of good reduction for $E$. Then, for a prime $p \neq \ell$, the Frobenius automorphism associated to $p$ (unique up to conjugation) acts on the $\ell$-adic Tate module of $E$. The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of $\ell$. Its splitting field is called the Frobenius field of $E$ at $p$. Let $E_1$ and $E_2$ be two elliptic curves defined over $\mathbb{Q}$ that are non-isogenous over $\overline{\mathbb{Q}}$ and also without complex multiplication over $\overline{\mathbb{Q}}$. In analogy with the well-known Lang-Trotter conjecture for a single elliptic curve, it is natural to consider the asymptotic behaviour of the function that counts the number of primes $p \leq x$ such that the Frobenius fields of $E_1$ and $E_2$ at $p$ coincide. In this short note, using Heath-Brown's square sieve, we provide both conditional (upon the Generalized Riemann Hypothesis) and unconditional upper bounds.