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THE ERROR TERM IN THE SATO–TATE THEOREM OF BIRCH
Published online by Cambridge University Press: 08 February 2019
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
$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$
$I\subseteq [0,\unicode[STIX]{x1D70B}]$, 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
$I$.
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- Research Article
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- © 2019 Australian Mathematical Publishing Association Inc.
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