Character theory approach to Sato-Tate groups

TL;DR

This paper introduces a character theory approach using orthogonality relations of compact Lie groups to study Frobenius distributions and Sato-Tate groups, offering more efficient approximations with fewer samples.

Contribution

It presents a novel method applying character theory to analyze Sato-Tate groups, reducing sample size requirements compared to traditional moment statistic approaches.

Findings

Fewer sample points needed for accurate Sato-Tate group approximation.

The approach provides strong evidence supporting the generalized Sato-Tate conjecture.

Efficient convergence achieved with 2^{10} to 2^{12} points.

Abstract

In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato-Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato-Tate groups and strong evidence of the correctness of the generalized Sato-Tate conjecture. In fact, $2^{10}$ to $2^{12}$ points provide satisfactory convergence. Even for $g = 2$, the classical approach using moment statistics requires about $2^{30}$ sample points to obtain such information.