On the Lang-Trotter conjecture for two elliptic curves

TL;DR

This paper develops a probabilistic model for the distribution of primes related to two elliptic curves, proposes explicit formulas for constants involved, and proves the conjecture on average over families of elliptic curves.

Contribution

It introduces a new probabilistic model for two elliptic curves, including explicit Euler product formulas and analysis of the universal constant, extending previous single-curve results.

Findings

Proposes a probabilistic model predicting prime distributions for two elliptic curves.

Provides explicit Euler product representations for the asymptotic constant.

Proves the conjecture on average over families of elliptic curves.

Abstract

Following Lang and Trotter we describe a probabilistic model that predicts the distribution of primes $p$ with given Frobenius traces at $p$ for two fixed elliptic curves over $\mathbb{Q}$. In addition, we propose explicit Euler product representations for the constant in the predicted asymptotic formula and describe in detail the universal component of this constant. A new feature is that in some cases the $\ell$-adic limits determining the $\ell$-factors of the universal constant, unlike the Lang-Trotter conjecture for a single elliptic curve, do not stabilize. We also prove the conjecture on average over a family of elliptic curves following the work of David, Koukoulopoulos, and Smith.