Sato-Tate groups of some weight 3 motives

2012/12/31 by Francesc Fité, Kiran S. Kedlaya, Kiran Kedlaya +1 · 16 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Cohomology #Geometry and complex manifolds #Group (periodic table) #Mathematics #Pencil (optics) #Physics #Pure mathematics #Quintic function #Symplectic geometry #Unitary state #math.AG #math.NT #msc:11G09 #msc:11M50 #msc:14J32 #msc:14K15

paper · pdf · doi:10.1090/conm/663/13350

published in Contemporary mathematics - American Mathematical Society, 57-101 (American Mathematical Society) · Minor edits to correct typos and address LMFDB modular form label changes

openalex publication_date 2016/01/01 · arxiv created 2016/02/16 · arxiv updated 2017/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

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

Citations

• Modularity of the Consani-Scholten quintic
• A Family of Calabi–Yau Varieties and Potential Automorphy II
• A family of Calabi-Yau varieties and potential automorphy