Shimura Varieties: A Hodge-Theoretic Perspective

This chapter discusses certain cleverly constructed unions of modular varieties, called Shimura varieties, in the Hodge-theoretic perspective. The Shimura varieties can show the minimal (i.e., reflex) field of definition of a Hodge/zero locus setting, and also reveal quite a bit about the interplay between “upstairs” and “downstairs” (in Ď and Γ‎\D, respectively) fields of definition of subvarieties. Hence, the chapter defines the Hermitian symmetric domains in D as well as the locally symmetric varieties Γ‎\D. It then discusses the theory of complex multiplication, before introducing Shimura varieties ∐ᵢ(Γ‎ᵢ\D) as well as three key Adélic lemmas, before finally laying out the fields of definition.

Cohomology of Complex Torus Bundles Associated to Cocycles

Equivariant holomorphic maps of Hermitian symmetric domains into Siegel upper half spaces can be used to construct families of abelian varieties parametrized by locally symmetric spaces, which can be regarded as complex torus bundles over the parameter spaces. We extend the construction of such torus bundles using 2-cocycles of discrete subgroups of the semisimple Lie groups associated to the given symmetric domains and investigate some of their properties. In particular, we determine their cohomology along the fibers.

Theta functions on Hermitian symmetric domains and fock representations

One way of realizing representations of the Heisenberg group is by using Fock representations, whose representation spaces are Hilbert spaces of functions on complex vector space with inner products associated to points on a Siegel upper half space. We generalize such Fock representations using inner products associated to points on a Hermitian symmetric domain that is mapped into a Seigel upper half space by an equivariant holomorphic map. The representations of the Heisenberg group are then given by an automorphy factor associated to a Kuga fiber variety. We introduce theta functions associated to an equivariant holomorphic map and study connections between such generalized theta functions and Fock representations described above. Furthermore, we discuss Jacobi forms on Hermitian symmetric domains in connection with twisted torus bundles over symmetric spaces.