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THE ERROR TERM IN THE SATO–TATE THEOREM OF BIRCH

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Arithmetic algebraic geometry

Published online by Cambridge University Press:  08 February 2019

M. RAM MURTY  and
NEHA PRABHU Open the ORCID record for NEHA PRABHU [Opens in a new window]
M. RAM MURTY
Affiliation:
Queen’s University, Kingston, Ontario K7L 3N6, Canada email [email protected]
NEHA PRABHU*
Affiliation:
Queen’s University, Kingston, Ontario K7L 3N6, Canada email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

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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

$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$
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 $I$.

Keywords

Sato–Tate theoremelliptic curves

MSC classification

Primary: 11G05: Elliptic curves over global fields
Secondary: 11K38: Irregularities of distribution, discrepancy
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 1 , August 2019 , pp. 27 - 33
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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