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,$q=p^{r}$and an elliptic curve$E:y^{2}=x^{3}+ax+b$, we show that$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$for any interval$I\subseteq [0,\unicode[STIX]{x1D70B}]$, where the quantity$\unicode[STIX]{x1D703}_{a,b}$is defined by$2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$and$\unicode[STIX]{x1D707}_{ST}(I)$denotes the Sato–Tate measure of the interval$I$.