**Coherent Cohomology of Shimura Varieties**

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 and uncorrected notes**
**1.
Modular
Curves

2. Lie Groups, Deligne's Axioms, and Shimura Varieties

3. Tori and Canonical Models of Shimura Varieties

4. Automorphic Vector Bundles

5. Cohomology of Lie Algebras

6. Discrete Series

7. Torus Embeddings

8. Logarithmic Growth

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