The error term in the Sato-Tate theorem of Birch

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 $I$.

References in corpus (1)

• Effective equidistribution and the Sato-Tate law for families of elliptic curves