Summer School 2009

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Clay Mathematics Institute 2009 Summer School

Galois Representations

June 15 - July 10, 2009

University of Hawaii at Manoa Honolulu, Hawaii

Schedule: Week 1, Week 2, Week 3, Week 4

Many advances on the algebraic side of number theory in the last 15 years (such as the solutions of the Shimura-Taniyama conjecture, Sato-Tate conjecture and Serre's conjecture, as well as decisive progress on the Fontaine-Mazur conjecture and Main Conjectures for modular forms) have relied in an essential way on improvements in the theory of Galois representations. For example, such improvements have enabled the local and global aspects of modularity lifting theorems to be extended far beyond the traditional 2-dimensional case over the rational numbers, and have led to generalizations of the "classical" theory of p-adic modular forms in a way that makes more effective use of representation theory and geometry to obtain results on the arithmetic of L-values.

The aim of the three main courses is to present an overview of many of these ideas and applications, aimed at advanced graduate students and postdocs with a strong background in number theory, Galois cohomology, and basic algebraic geometry. One course will focus entirely on local problems (p-adic representations of Galois groups of p-adic fields), a second course will have a more global flavor (Galois deformation theory and global applications), and a third (on L-values) will rely on the other two courses. In the final week of the program there will be three mini-courses that build on themes introduced in the foundational courses. These will address aspects of the following topics: proofs of p-adic comparison isomorphisms (Andreatta), introduction to the p-adic Langlands correspondence (Emerton), and construction of global p-adic Galois representations (Shin).

Foundational Courses

• p-padic Hodge Theory
  Olivier Brinon, Brian Conrad
  (Φ, Γ)-modules, applications to potentially semi-stable deformation rings and families of Galois representations
  Brinon and Conrad Lecture notes
  
  Reading list

  The recommended background reading for the course in p-adic Hodge theory is contained in the following two files, though the larger file (prelimnotes.pdf) contains much more (for those who have time to go further). Precise instructions on what reading is recommended from these files prior to arriving in Hawaii is provided on the first page of the larger file.

  Prelimnotes.pdf, witt.pdf
• Deformation of Galois Representations and Modular Forms
  Mark Kisin, Jacques Tilouine
  Deformation theory of Galois representations, pseudo-representations, with applications to eigenvarieties, construction of Galois representations, Taylor-Wiles method
  Kisin lecture notes
  Tilouine lecture notes
  
• Iwasawa Theory and Automorphic Applications
  Joel Bellaiche, Chris Skinner
  Iwasawa theory and Hida theory with applications to Selmer groups, automorphic forms, and the arithmetic of special values of L-functions
  Bellaiche lecture notes

Scientific Committee

• Brian Conrad (Stanford University)
• David Alexandre Ellwood (CMI)
• Mark Kisin (University of Chicago)
• Chris Skinner (Princeton University)

Lecturers

• Fabrizio Andreatta (University of Milan)
• Joel Bellaiche (Brandeis University)
• Olivier Brinon (Université Paris 13)
• Brian Conrad (Stanford University)
• Matthew Emerton (Northwestern University)
• Mark Kisin (University of Chicago)
• Sug Woo Shin (University of Chicago)
• Chris Skinner (Princeton University)
• Jacques Tilouine (Université Paris 13)

Lecture notes

• Olivier Brinon and Brian Conrad
• Mark Kisin
• Jacques Tilouine
• Joel Bellaiche
• Matthew Emerton

University of Hawaii, Mathematics Department

Group photos