Arithmetic Aspects of Mumford-Tate Domains

This chapter describes the arithmetic aspects of Mumford-Tate domains and Noether-Lefschetz loci. It first clarifies a few points concerning the structure and construction of Mumford-Tate domains before presenting a computationally effective procedure to determine the components in terms of Lie algebra representations and Weyl groups. It then shows that the normalizers of M in G are the groups stabilizing the Noether-Lefschetz locus. It also discusses the decomposition of Noether-Lefschetz loci into Hodge orientations, Weyl groups and permutations of Hodge orientations, and Galois groups and fields of definition. The results demonstrate that Mumford-Tate groups built up from well-understood real² factors are one source of easily described examples of Mumford-Tate domains.