An elliptic modular form of weight k can be interpreted as a global section of a certain natural vector bundle (depending on k) over the modular curve. More generally, the classical holomorphic Hilbert or Siegel modular forms are sections of vector bundles on the corresponding modular varieties, which are special examples of Shimura varieties. Holomorphic modular forms can be defined on any Shimura variety, and, as in the classical examples, these forms can be identified with global sections of vector bundles, called automorphic vector bundles.
Automorphic vector bundles admit several equivalent natural constructions; they form a tensor
category that can be identified (after making choices) with the category of finite-dimensional
representations of a certain algebraic group. Most of the objects of this category do not correspond to holomorphic modular forms, because they do not admit holomorphic global sections. Instead, the higher (coherent) cohomology of the associated locally free sheaves can be interpreted in terms of non-holomorphic automorphic forms that are harmonic with respect to the natural invariant hermitian metric. In order to do this correctly, one usually has to replace the Shimura variety by one of its toroidal compactifications. The coherent cohomology of these compactifications then links automorphic forms to the Hodge theory of Shimura varieties. This theory was developed in the 1980s in order to generalize results of Shimura and others on the relations between special values of L-functions and periods of integrals, in the spirit of an important conjecture of Deligne. More recently interest in this theory has been revived, with applications to p-adic families of modular forms, the construction of p-adic Galois representations, and the study of deformations of Galois representations (extending the Taylor-Wiles method). At the same time, the rapid development of a relative theory of automorphic forms (Jacquet's relative trace formula, the Gan-Gross-Prasad and Ichino-Ikeda conjectures, as well as recent work of Wei Zhang and Sakellaridis-Venkatesh) has defined a new class of automorphic periods that can be given arithmetic normalizations by means of coherent cohomology.
The course will present the basis of this theory. After a rapid introduction to the analytic and
geometric theory of Shimura varieties, we will define automorphic vector bundles and the coherent
cohomology of their canonical extensions to toroidal compactifications. The Hodge theory of these
bundles will be interpreted in terms of relative Lie algebra cohomology, which will require a (rapid) review of basic representation theory. The end of the course will describe several applications of these methods to problems in number theory.
Some informal notes
- Modular curves
- Lie Groups, Deligne's Axioms, and Shimura Varieties
- Tori and Canonical Models of Shimura Varieties
- Automorphic Vector Bundles
- Cohomology of Lie Algebras
- Discrete Series
- Torus Embeddings
- Logarithmic Growth