Character theory approach to Sato–Tate groups

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.