# The error term in the Sato-Tate theorem of Birch

## TL;DR

This paper derives an explicit error term for the distribution of Frobenius angles in the Sato-Tate theorem for elliptic curves over finite fields, refining the understanding of their statistical behavior.

## Contribution

It provides a new explicit error bound in the Sato-Tate distribution for elliptic curves over finite fields, improving previous asymptotic results.

## Key findings

- Error term of O_r(q^{7/4}) established
- Quantitative distribution of Frobenius angles confirmed
- Enhanced precision in Sato-Tate theorem for elliptic curves

## Abstract

We establish an error term in the Sato-Tate theorem of Birch. That is, for $p$ prime, $q=p^r$ we show that $\#\{ (a,b) \in \mathbb{F}_q^2 : \theta_{a,b}\in I\} =\mu_{ST}(I)q^2 + O_r(q^{7/4})$ for any interval $I\subseteq[0,\pi]$ where for an elliptic curve $E: y^2= x^3 +ax +b$, the quantity $\theta_{a,b}$ is defined by $2\sqrt{q}\cos\theta_{a,b} = q+1-E(\mathbb{F}_q)$ and $\mu_{ST}(I)$ denotes the Sato-Tate measure of the interval $I$.