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THE ERROR TERM IN THE SATO–TATE THEOREM OF BIRCH
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 1 / August 2019
- Published online by Cambridge University Press:
- 08 February 2019, pp. 27-33
- Print publication:
- August 2019
-
- Article
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- You have access
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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 for any interval
$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$
$I\subseteq [0,\unicode[STIX]{x1D70B}]$, 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 
$I$.
