Mumford-Tate groups and domains. Their geometry and arithmetic (Q2882521)

The monograph under review deals with Mumford-Tate domains and Mumford-Tate groups. In contrast to large parts of the literature the viewpoint of this book is not the usage of Mumford-Tate groups as a tool to study abelian varieties or other geometric objects, but in the setting of period domains and their duals.NEWLINENEWLINEIn the first two chapters, Mumford-Tate groups and Mumford-Tate domains are introduced. Shimura domains are introduced as a subclass of Mumford-Tate domains and period domains are a subclass of Shimura domains. Mumford-Tate domains can be decomposed into a direct product of homogeneous spaces corresponding to simple subgroups of a group. Since this group may be used to be the Mumford-Tate group of the Mumford-Tate domain, an essential tool of Mumford-Tate groups is the classification of simple groups. Thus it is studied which representation of a simple group can be used to be a Mumford-Tate group of a polarized variation of Hodge structures. A main result is that the possible representations of simple groups which can be Mumford-Tate groups of a variation of Hodge structures are determined in terms of the root lattice and the Cartan decomposition. This allows a classification of simple Lie algebras of Mumford-Tate groups for Hodge structures with odd weight. Moreover different realizations of the same group as Mumford-Tate group are studied.NEWLINENEWLINEHodge structures with complex multiplication are also expounded and discussed. Such Hodge structures are used for an algorithm to determine the Mumford-Tate subdomains of a given period domain. This algorithm is executed for pure Hodge structures of weight 3 with the Hodge numbers \((1,1,1,1)\) and Hodge structures of weight \(1\) with the Hodge numbers \((2,2)\).