Note on the Distribution of the Traces of Frobenius

TL;DR

This paper corrects a previous claim about the distribution of Frobenius traces for elliptic curves over finite fields, showing they are dense but not equidistributed on [-1,1], and provides an elementary proof.

Contribution

It corrects the existing literature by clarifying the distribution of Frobenius traces and offers a simple proof demonstrating their density without equidistribution.

Findings

The traces are dense in [-1,1].

The traces are not equidistributed on [-1,1].

Provides an elementary proof of the density result.

Abstract

Let $E$ be a nonsingular elliptic curve over the rational numbers, and let $\tau^n=p^n+1-\#E(\mathbb{F}_{p^n})$. A result in the current literature claims that the normalized traces of Frobenius $\alpha_n=(\tau^n+\overline{\tau}^n)/2p^{n/2}$ are equidistributed on the interval $[-1,1]$ as $n\to \infty$. This short note has two goals: 1. Proposes a correction to this result on the distribution of the traces of Frobenius. 2. Provides a simple elementary proof of this result. More precisely, it is shown that the sequence $\{\alpha_n:n\geq1\}$ is dense but not equidistributed on the interval $[-1,1]$.