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

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 [bar over Q]-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.