Instructional Conference on Representation Theory and Arithmetic

to be held at Northwestern University, May 5-9, 2008.



The aim of this instructional conference is to provide an introduction to some aspects of representation theory that are relevant to the study of arithmetic properties of automorphic forms.

Organizers: Frank Calegari, Matthew Emerton

Instructors: The instructors for the courses will include:

  Frank Calegari (Northwestern)
  Matthew Emerton (Northwestern)
  Florian Herzig (Northwestern)
  Jacob Lurie (MIT)
  David Nadler (Northwestern)

Structure of the Conference: The conference will consist of a number of mini-courses, of between one and three lectures each, targeted at graduate students interested in the arithmetic of modular and automorphic forms.

Topics to be covered include:  

  • Modular forms: from the classical to the representation-theoretic formulation  
  • Semisimple Lie algebras and their enveloping algebras  
  • Algebraic representations of reductive groups  
  • Infinite-dimensional representations of real reductive groups  
  • Smooth representations of p-adic reductive groups  
  • Local and global theory of Hecke algebras  
  • Automorphic forms and automorphic representations  
  • An introduction to Shimura varieties  
  • An introduction to the local Langlands correspondence  
  • Galois representations associated to automorphic forms and local-global compatibility  
  • Serre's conjecture from the representation theoretic view-point

    It would be impossible to cover all the details of all these topics even with a much longer series of courses. Some of the topics will be covered more thoroughly than others, and we will focus on those aspects of the various topics that seem to be of the most importance for working number theorists. It is our hope that participants in the conference will become equipped with a significant part of the basic background knowledge that is necessary to successfully navigate the research literature on the arithmetic of automorphic forms.

    Schedule: A schedule is now available.

    Attending the conference: Due to limited space, the conference is open only to invited attendees.

    APPLICATIONS ARE NOW CLOSED

    Participants are invited to stay on after the workshop to attend the conference Current developments and directions in the Langlands program, which will be held from May 10-14 at Northwestern. If you would like to stay on for the Langlands conference, please indicate this on your application form.

    Lodging and Travel: All participants will be reimbursed for the cost of their travel (within North America) to the conference. Lodging for all participants will be provided (at no charge to the participants) at the Garrett Theological Seminary (located on the Northwestern Campus). Participants who choose to stay on for the Langlands conference will have their accommodation at Garrett extended for the duration of the conference, at no charge.

    Expected background: Participants will be presumed to have some familiarity with the classical theory of modular forms and its role in modern number theory. In particular, we expect that the participants will have some familiarity with (if not total mastery of) the following topics:

    The basic theory of modular forms and Hecke operators, as explained in (for example) the following texts:

  • A.O.L. Atkin and J. Lehner, Hecke operators on Gamma-zero m, Math. Ann. 185 (1970)
  • D. Rohrlich, Modular curves, Hecke correspondences, and L-functions, in Modular Forms and Fermat's Last Theorem
  • J.-P. Serre, A course in arithmetic
  • J.-P. Serre, Modular forms of weight one and Galois representations, in Algebraic number fields: L-functions and Galois properties (Durham, 1975)

    The statement of (although not necessarily the proof of) the relationship between elliptic curves and weight 2 modular forms expressed by the Shimura-Taniyama conjecture (now the modularity theorem of Breuil-Conrad-Diamond-Taylor-Wiles), as explained in (for example) the following surveys:

  • D. Rohrlich, Modular curves, Hecke correspondences, and L-functions, in Modular Forms and Fermat's Last Theorem
  • G. Stevens, An overview of the proof of Fermat's last theorem, in Modular Forms and Fermat's Last Theorem

    The statement of (although not necessarily the proof of) the theorem of Deligne-Serre-Shimura associating Galois representations to modular forms, as explained in (for example) the following surveys:

  • J.-P. Serre, Modular forms of weight one and Galois representations, in Algebraic number fields: L-functions and Galois properties (Durham, 1975)
  • G. Stevens, An overview of the proof of Fermat's last theorem, in Modular Forms and Fermat's Last Theorem
  • H.P.F. Swinnerton-Dyer, On l-adic representations and congruences for coefficients of modular forms, Lecture Notes in Math. 350

    The statement of (although not necessarily the proof of) Serre's conjecture (now a theorem of Khare, Kisin, and Wintenberger), as explained in (for example) the following articles:

  • K. Ribet and William Stein, Lectures on Serre's conjectures
  • J.-P. Serre, Sur les representations modulaires de degre 2 de Gal(Qbar/Q), Duke Math. J. 54 (1987)
  • G. Stevens, An overview of the proof of Fermat's last theorem, in Modular Forms and Fermat's Last Theorem

    Some references: Here are some suggested references for the material that will be covered in the various courses of the conference:

  •   W. Fulton and J. Harris, Representation theory: A first course
  •   R. Taylor, Galois representations

    This list will be expanded in the coming weeks.