THEMATIC PROGRAMS

December 20, 2014

January-June 2012 (Winter/Spring )
Thematic Program on Galois Representation
s

March 12-16, 2012
Galois Representations, Shimura Varieties, and Automorphic Forms;
A Workshop for Graduate Students and Postdoctoral Fellows


Supported by the National Science Foundation
Award #1101503

Lectures to be held in the morning each day.

Recent developments in the Langlands programme have relied on an impressive amount of machinery -- the Kisin-Taylor-Wiles method, cyclic base change, the Fundamental Lemma, the geometry of unitary Shimura varieties, and the labyrinthian theory of automorphic forms. One consequence is that non-experts can become needlessly lost in the early stages of a paper (or during the first five minutes of a talk) by technical details, even when the key ideas are actually quite intuitive. The goal of this workshop is to demystify much of the technical jargon in the subject with an emphasis on examples, with a special emphasis on comparing the similarities and differences of the general theory with the special case of modular forms and modular curves.

Lecturers include Frank Calegari, Ana Caraiani, Florian Herzig, Keerthi Madapusi Pera, Yiannis Sakellaridis and Chung Pang Mok.

Monday, March 12
9:30 - 10: 00 Coffee and Welcome
10:00 - 11:00 F. Calegari (Northwestern University) (notes)
Classical Modular Forms
11:00 - 11: 30 Break
11:30 - 12:30 F. Herzig (University of Toronto) (notes)
Hecke Operators
12:30 - 2:15 Lunch
2:15 - 3:15 K. Madapusi Pera (Harvard University) (notes)
Modular Curves over Q
3:15 - 3:45 Break
3:45 - 4:45 C. P. Mok (McMaster University) (notes)
Galois Representations for Modular Forms
Tuesday, March 13
9:30 - 10: 00 Coffee
10:00 - 11:00 Y. Sakellaridis (Rutgers University) (notes)
Representations of GL2(R)
11:00 - 11: 30 Break
11:30 - 12:30 F. Calegari (Northwestern University) (notes)
Galois Deformations
12:30 - 2:15 Lunch
2:15 - 3:15 F. Calegari (Northwestern University) (notes)
Galois Deformations II
3:15 - 3:45 Break
3:45 - 4:45 F. Calegari (Northwestern University) (notes)
Taylor-Wiles
Wednesday, March 14
9:30 - 10: 00 Coffee
10:00 - 11:00 C. P. Mok (McMaster University) (notes)
GL(2)
11:00 - 11: 30 Break
11:30 - 12:30 Y. Sakellaridis (Rutgers University) (notes)
Automorphic Forms
12:30 - 2:15 Lunch
2:15 - 3:15 A. Caraiani (Harvard University) (notes)
Simple Shimura Varieties
3:15 - 4:10 Break
4:10 - 5:00 P. Scholze (Universität Bonn) (notes)
Colloquium
Thursday, March 15
9:30 - 10: 00 Coffee
10:00 - 11:00 A. Caraiani (Harvard University) (notes)
Local-Langlands and Local-Global
11:00 - 11: 30 Break
11:30 - 12:30 K. Madapusi Pera (Harvard University) (notes)
Local Models
12:30 - 2:15 Lunch
2:15 - 3:15 F. Calegari (Northwestern University) (notes)
GSp(4)
3:15 - 3:45 Break
3:45 - 4:45 K. Madapusi Pera (Harvard University) (notes)
Local Models II
Friday, March 16
9:30 - 10: 00 Coffee
10:00 - 11:00 K. Madapusi Pera (Harvard University) (notes)
p-adic Hodge Theory
11:00 - 11: 30 Break
11:30 - 12:30 F. Calegari (Northwestern University) (notes)
Kisin-Taylor-Wiles

Topics to be covered include:

Part I: Classical modular forms:
Modular Forms as functions on lattices, as holomorphic functions, as sections of coherent sheaves, as cohomology of local systems Hecke operators, admissible representations, Local Langlands for GL_2 Modular curves over number fields and over rings of integers. The special fibre of X_0(p), The Eichler-Shimura relation Representations of GL_2(R)

Part II: Galois Representations for GL(2):
Galois Representations for classical modular forms
Galois Representations locally at p (introduction)
Deformations of Galois Represenations
The idea behind the Taylor-Wiles argument

Part III: Generalizations to other groups
Reductive algebraic groups, the Langlands dual groups, algebraic automorphic forms Special Cases: Hilbert modular forms, Siegel modular forms, Inner Forms of GL(2), Jacquet-Langlands Simple Shimura varieties Local Models The Local Langlands conjecture + global reciprocity (statements) Kisin's modification of Taylor-Wiles

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