The Lang-Trotter Conjecture for products of non-CM elliptic curves

TL;DR

This paper investigates the distribution of primes with fixed Frobenius traces for products of non-CM elliptic curves over , formulates conjectural asymptotics, and provides computational evidence supporting the Lang-Trotter type conjecture.

Contribution

It extends the Lang-Trotter conjecture framework to abelian surfaces isogenous to products of non-CM elliptic curves, including explicit constants and computational verification.

Findings

Formulated conjectural asymptotics for primes with fixed Frobenius traces.

Computed explicit constants for the asymptotic formulas.

Provided computational evidence supporting the conjecture.

Abstract

Inspired by the work of Lang-Trotter on the densities of primes with fixed Frobenius traces for elliptic curves defined over $\mathbb{Q}$ and by the subsequent generalization of Cojocaru-Davis-Silverberg-Stange to generic abelian varieties, we study the analogous question for abelian surfaces isogenous to products of non-CM elliptic curves over $\mathbb{Q}$. We formulate the corresponding conjectural asymptotic, provide upper bounds, and explicitly compute (when the elliptic curves lie outside a thin set) the arithmetically significant constants appearing in the asymptotic. This allows us to provide computational evidence for the conjecture.