A refined version of the Lang-Trotter Conjecture

TL;DR

This paper refines the Lang-Trotter conjecture on the distribution of Frobenius traces of elliptic curves, incorporating the semicircular law to provide a more accurate asymptotic formula with a uniform error term.

Contribution

It introduces a refined version of the Lang-Trotter conjecture that accounts for the semicircular law, ensuring consistency with Chebotarev and Sato-Tate, and provides numerical evidence.

Findings

Refined the Lang-Trotter conjecture to include the semicircular law.

Ensured the new conjecture is consistent with Chebotarev and Sato-Tate.

Provided numerical evidence supporting the refined conjecture.

Abstract

Let $E$ be an elliptic curve defined over the rational numbers and $r$ a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of $E$ and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to $x$ which have Frobenius trace equal to $r$, where $r$ is a {\it fixed} integer. However, as shown in this note, this asymptotic estimate cannot hold for {\it all} $r$ in the interval $|r|\le 2\sqrt{x}$ with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato-Tate conjecture. The purpose of this note is to refine the Lang-Trotter conjecture, by taking into account the "semicircular law", to an asymptotic formula that conjecturally holds for arbitrary integers $r$ in the interval $|r|\le 2\sqrt{x}$, with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev theorem for a fixed division field, and with the Sato-Tate conjecture. We also present numerical evidence for the refined conjecture.