# Moments of the error term in the Sato-Tate law for elliptic curves

## TL;DR

This paper establishes improved bounds for the moments of the error term in the Sato-Tate law for elliptic curves, leading to new almost-all results and a conditional CLT, using novel methods involving identities by Birch and Melzak.

## Contribution

It introduces a new approach to bounding moments of the Sato-Tate error, surpassing previous results for the first two moments and deriving new distributional theorems.

## Key findings

- Stronger bounds for first and second moments of the error term.
- New almost-all results for the error distribution.
- Conditional Central Limit Theorem for the errors.

## Abstract

We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. Our estimates are stronger than those obtained by W.D. Banks and I.E. Shparlinski (arXiv:math/0609144) and L. Zhao and the fist-named author in (arXiv:math/0608318) for the first and second moments, but this comes at the cost of larger ranges of averaging. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method is different from those used in the above-mentioned papers and builds on recent work by the second-named author and K. Sinha (arXiv:1705.04115) who derived a Central Limit Theorem on the distribution of the errors in the Sato-Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch's identities connect moments of coefficients of Hasse-Weil $L$-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak's identity is combinatorial in nature.