Bounds for the Lang-Trotter conjectures

TL;DR

This paper improves upper bounds related to the Lang-Trotter conjectures for non-CM elliptic curves over the rationals by applying a refined Chebotarev density theorem under GRH.

Contribution

It introduces an enhanced method using a smoothed Chebotarev density theorem to tighten bounds on prime counts in the Lang-Trotter conjectures under GRH.

Findings

Improved upper bounds for prime counts in Lang-Trotter conjectures.

Application of a smoothed Chebotarev density theorem under GRH.

Enhanced understanding of Frobenius fields for non-CM elliptic curves.

Abstract

For a non-CM elliptic curve $E$ defined over the rationals, Lang and Trotter made very deep conjectures concerning the number of primes $p\leq x$ for which $a_p(E)$ is a fixed integer (and for which the Frobenius field at $p$ is a fixed imaginary quadratic field). Under GRH, we use a smoothed version of the Chebotarev density theorem to improve the best known Lang-Trotter upper bounds of Murty, Murty and Saradha, and Cojocaru and David.