Mumford-Tate Groups and Domains: Their Geometry and Arithmetic

Introduction 1
I Mumford-Tate Groups 28
I.A Hodge structures 28
I.B Mumford-Tate groups 32
I.C Mixed Hodge structures and their Mumford-Tate groups 38
II Period Domains and Mumford-Tate Domains 45
II.A Period domains and their compact duals 45
II.B Mumford-Tate domains and their compact duals 55
II.C Noether-Lefschetz loci in period domains 61
III The Mumford-Tate Group of a Variation of Hodge Structure 67
III.A The structure theorem for variations of Hodge structures 69
III.B An application of Mumford-Tate groups 78
III.C Noether-Lefschetz loci and variations of Hodge structure .81
IV Hodge Representations and Hodge Domains 85
IV.A Part I: Hodge representations 86
IV.B The adjoint representation and characterization of which weights give faithful Hodge representations 109
IV.C Examples: The classical groups 117
IV.D Examples: The exceptional groups 126
IV.E Characterization of Mumford-Tate groups 132
IV.F Hodge domains 149
IV.G Mumford-Tate domains as particular homogeneous complex manifolds 168
Appendix: Notation from the structure theory of semisimple Lie algebras 179
V Hodge Structures with Complex Multiplication 187
V.A Oriented number fields 189
V.B Hodge structures with special endomorphisms 193
V.C A categorical equivalence 196
V.D Polarization and Mumford-Tate groups . 198
V.E An extended example 202
V.F Proofs of Propositions V.D.4 and V.D.5 in the Galois case 209
VI Arithmetic Aspects of Mumford-Tate Domains 213
VI.A Groups stabilizing subsets of D 215
VI.B Decomposition of Noether-Lefschetz into Hodge orientations 219
VI.C Weyl groups and permutations of Hodge orientations 231
VI.D Galois groups and fields of definition 234
Appendix: CM points in unitary Mumford-Tate domains 239
VII Classification of Mumford-Tate Subdomains 240
VII.A A general algorithm 240
VII.B Classification of some CM-Hodge structures 243
VII.C Determination of sub-Hodge-Lie-algebras 246
VII.D Existence of domains of type IV(f) 251
VII.E Characterization of domains of type IV(a) and IV(f) 253
VII.F Completion of the classification for weight 3 256
VII.G The weight 1 case 260
VII.H Algebro-geometric examples for the Noether-Lefschetzlocus types 265
VIII Arithmetic of Period Maps of Geometric Origin 269
VIII.A Behavior of fields of definition under the period
Map -- image and preimage 270
VIII.B Existence and density of CM points in motivic VHS 275
Bibliography 277
Index 287