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arXiv · 1906.03534

The error term in the Sato-Tate theorem of Birch

M. Ram Murty·Neha Prabhu
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Abstract

We establish an error term in the Sato-Tate theorem of Birch. That is, for $p$ prime, $q=p^r$ we show that $\#\{ (a,b) \in \mathbb{F}_q^2 : θ_{a,b}\in I\} =μ_{ST}(I)q^2 + O_r(q^{7/4})$ for any interval $I\subseteq[0,π]$ where for an elliptic curve $E: y^2= x^3 +ax +b$, the quantity $θ_{a,b}$ is defined by $2\sqrt{q}\cosθ_{a,b} = q+1-E(\mathbb{F}_q)$ and $μ_{ST}(I)$ denotes the Sato-Tate measure of the interval $I$.

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M. Ram Murty, Neha Prabhu. 2019-06-08. The error term in the Sato-Tate theorem of Birch. https://doi.org/10.1017/s0004972718001648

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