Sato-Tate groups of some weight 3 motives

TL;DR

This paper classifies the possible Sato-Tate groups for certain weight 3 motives with specific Hodge numbers, describes families realizing these groups, and provides numerical evidence supporting their equidistribution, including a family from Calabi-Yau threefolds.

Contribution

It establishes a group-theoretic classification of Sato-Tate groups for weight 3 motives with specified Hodge numbers and describes families realizing these groups.

Findings

Classified Sato-Tate groups for the motives considered.

Identified families of motives that realize these groups.

Numerical evidence supports equidistribution and the full USp(4) group in a Calabi-Yau family.

Abstract

We establish the group-theoretic classification of Sato-Tate groups of self-dual motives of weight 3 with rational coefficients and Hodge numbers h^{3,0} = h^{2,1} = h^{1,2} = h^{0,3} = 1. We then describe families of motives that realize some of these Sato-Tate groups, and provide numerical evidence supporting equidistribution. One of these families arises in the middle cohomology of certain Calabi-Yau threefolds appearing in the Dwork quintic pencil; for motives in this family, our evidence suggests that the Sato-Tate group is always equal to the full unitary symplectic group USp(4).