MATHEMATICS
G6245, Fall 2013
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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