The error term in the Sato-Tate theorem of Birch (1906.03534v1)

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 $I$.