**Topics in automorphic forms**

This
year's topic is *Automorphy lifting
theorems*. The Langlands
reciprocity conjectures are a non-abelian generalization of the reciprocity
laws of class field theory. Let F
be a global field. Starting with
an automorphic representation π of GL(n)_{F} satisfying certain
conditions, it is often possible to use the Galois action on the l-adic
cohomology of Shimura varieties to construct a compatible system of n-dimensional l-adic representations of
the Galois group of F whose L-function coincides with that of π. The reciprocity problem is to determine
when a given l-adic Galois representation **r**
can be obtained in this way. An
automorphy lifting theorem is the statement that **r** is obtained from automorphic forms provided its reduction modulo
l is automorphic in this sense.

About
30 years ago, Mazur introduced deformation-theoretic techniques to reinterpret
the automorphy lifting problem in terms of Galois cohomology. The *Taylor-Wiles-Kisin*
method, initiated in connection with Wiles's proof of Fermat's Last Theorem,
combines properties of Galois cohomology with a study of integral Hecke
algebras to prove automorphy lifting theorems in a variety of settings. It is appropriate to call this a method
rather than a theorem, because its scope continues to be expanded. The first half of present course will
review the basic framework of the method, starting with Mazur's original
construction. The second half
presents some recent developments, including but not limited to

the
papers of Calegari-Geraghty and Hansen on patching complexes;

the
"six-author paper" on models for the p-adic Langlands correspondence;

the
work of Emerton-Gee on the Breuil-Mézard conjecture;

new
ideas of Venkatesh and his collaborators on a versions of the Taylor-Wiles-Kisin
method in the setting
of derived algebraic geometry.

** **

**Return
to Michael Harris's home page**