Sato-Tate distributions of twists of the Fermat and the Klein quartics

TL;DR

This paper classifies the Sato-Tate distributions of Jacobians of twists of Fermat and Klein quartics, revealing new phenomena in the distribution of normalized Euler factors for certain abelian threefolds.

Contribution

It provides a complete classification of Sato-Tate distributions for these specific algebraic curves, introducing new insights into their limiting distributions.

Findings

Classified 54 distributions for Fermat quartic twists

Classified 23 distributions for Klein quartic twists

Discovered a new phenomenon where distribution is not determined by coefficients

Abstract

We determine the limiting distribution of the normalized Euler factors of an abelian threefold A defined over a number field k when A is geometrically isogenous to the cube of a CM elliptic curve defined over k. As an application, we classify the Sato-Tate distributions of the Jacobians of twists of the Fermat and Klein quartics, obtaining 54 and 23, respectively, and 60 in total. We encounter a new phenomenon not visible in dimensions 1 or 2: the limiting distribution of the normalized Euler factors is not determined by the limiting distributions of their coefficients.