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 $I$ .

Locations

• Bulletin of the Australian Mathematical Society - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View