Universiteit Leiden Mathematisch Instituut
Geometry Seminar

Spring semester 2006:

This Semester Fabio Mainardi will give a series of lectures on the Langlands correspondence for GL_r over function fields. Besides this, there will be talks by invited speakers on various subjects.

The aim of Mainardi's lectures is to explain what Lafforgue proved (and not at all how he proved it). At the end of this page you can find some relevant literature. A tentative programme is:

1. Admissible representations of GL(N,F) where F is a non-archimedean local field:
   1. basic definitions and properties,
   2. L and epsilon factors (following Godement and Jacquet),
   3. examples: N=1, L-factors of spherical representations,
   4. L and epsilon factors of pairs: equivalence criterion in terms of epsilon factors.
2. Automorphic representations of GL(N,F) where F is a function field:
   1. definition,
   2. L-functions,
   3. multiplicity one (Shalika),
   4. on functional equations of automorphic L-functions (Shahidi),
   5. converse theorems (Cogdell, Piatetski-Shapiro).
3. Statement of Langlands conjectures, local and global.

Schedule:

• Wednesday, 15 February 2006, 16:30-17:30+, lecture room 401 of the Snellius building
  Fabio Mainardi: Smooth and admissible representations of GL(N); induced representations
• Wednesday, 1 March 2006, 16:30-18:00+, lecture room 401 of the Snellius building
  Fabio Mainardi: Contragredient representations; Hecke algebras
• Wednesday, 15 March 2006, 16:30-18:00+, lecture room 401 of the Snellius building
  Fabio Mainardi: Reducibility of induced representations; unramified representations; Satake isomorphism
• Wednesday, 29 March 2006, 11:30-13:45, lecture room 402 of the Snellius building
  Gerard Helminck: Jacquet's theorem and supercuspidal representations
• Wednesday, 5 April 2006, 16:30-~17:30, lecture room 401 of the Snellius building
  Lars Halvard Halle: Stable reduction of curves.

  I will discuss the problem of computing the stable reduction of a family of curves. In positive characteristic there is no general algorithm for doing this, contrary to the characteristic zero case. There is however a geometrical characterization, due to T. Saito, of those families that only need a tamely ramified baseextension to acheive stable reduction. For such families, I will explain how to compute the stable reduction, generalizing the characteristic zero algorithm.

• Wednesday, 12 April 2006, 15:45-17:30, lecture room 401 of the Snellius building
  Fabio Mainardi: Unramified representations; Whittaker models; Rankin-Selberg convolutions
• Tuesday, 25 April 2006, 13:45-15:30, lecture room 402 of the Snellius building
  Fabio Mainardi: Unramified representations; Whittaker models; Rankin-Selberg convolutions. Click here for the course notes.
• Tuesday, 9 May, 14:00-16:00, room 611A of the mathematical institute in Utrecht (!)
  Fabio Mainardi: Functional equation and epsilon factors for the Rankin-Selberg convolutions. Click here for the course notes.
• Friday, 2 June, 11:00-13:00, room 401 of the Snellius building
  Fabio Mainardi: Weil-Deligne representations, statement of the local Langlands correspondence, examples. Click here for the course notes.
• Thursday, 8 June, 14:30, room 401 of the Snellius building
  Fabio Mainardi: Last talk of the series.

References for Mainardi's lectures:

• Bernstein, Zelevinsky: Representations of GL(n,F), where F is a local non-archimedean field, Russian Mathematical Surveys 31
• Borel: Automorphic L-functions, Proc.Symp.Pure Math. 33
• Bump: Automorphic forms and representations, Cambridge Press
• P. Cartier: Representations of p-adic groups: a survey, Proceedings of Symposia in Pure Mathematics XXXIII, vol.1
• Casselman: Introduction to the theory of admissible representations of p-adic reductive groups
• Gelbart, Shahidi: Analytic theory of automorphic L-functions, Academic Press
• R. Godement, H. Jacquet: Zeta functions of simple algebras, Lecture Notes in Mathematics 260
• Henniart: Representations des groupes reductif p-adiques, Seminaire Bourbaki 736
• Jacquet: Sur les representations des groups reductifs p-adiques, C.R.A.S Paris 280, page 1271-1272
• H. Jacquet, Piatetski-Shapiro, Shalika: Rankin-Selberg convolutions, Amer.J.Math.105
• Kudla: The local Langlands correspondence: the non-archimedean case Proc.Symp.Pure Math. 55
• Jacquet, R. Langlands: Automorphic forms on GL(2)
• L. Lafforgue: Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147
• G. Laumon: La correspondance de Langlands sur les corps de fonctions, Seminaire Bourbaki 873
• Tate: Number theoretic background, Proc.Symp.Pure Math. 33