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.