Note on the Distribution of the Traces of Frobenius

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 $\{{\alpha }_n:n\ge 1\}$ is dense but not equidistributed on the interval $[-1,1]$.

Eleven Pages. Keywords: Algebraic curve; Traces of Frobenius; Sato-Tate distribution