S4A1 - Seminar
Local Langlands Correspondence for GL(2) - part II
von Prof. Dr. M. Rapoport und Dr. J. Stix, WS 2008/2009.

The seminar is the continuation of
Local Langlands Correspondence for GL(2)

 Time: Thursday 14-16 Place: Seminarraum D, Meckenheimer Allee 160

List of talks with details and references as .pdf.

A local field F is either a finite extension of the p-adic numbers Qp or k((t)) with k a finite field. The local Langlands correspondence relates objects from two apparently different areas of mathematics which use F.

The first has to do with symmetries of extensions of the local field F. The Weil group WF is a dense subgroup of the Galois group GalF=Gal(Falg/F) of the algebraic closure Falg over F. Representations of WF enlarge the category of representations of GalF with the advantage of some analytic structure on the set of isomorphy clases. Adding some linear algebra one arrives at the notion of a semisimple Weil-Deligne representation and the set of isomorphy classes of such of dimension n is Gn(F).

On the other side we deal with the arithmetic within the field F. The general linear group GLn(F) is an analytic group over F. Smooth representations of GLn(F) are continuous representations in C-vector spaces that are endowed with the discrete topology. The continuity assumption simply asks for open stabilizers. We denote the set of isomorphy classes of irreducible smooth representations of GLn(F) by An(F).

The local Langlands correspondence is a bijection
&pi : Gn(F) &rarr An(F)
subject to the following conditions (without which we would simply claim that two infinite sets have the same cardinality):
1. For n=1 we are dealing with characters, hence essentially the maximal abelian quotient WFab of the Weil group and GL1(F) = F&lowast. The Langlands correspondence for n=1 is merely a reformulation of local class field theory and given by &pi(&chi) &omicron artF = &chi, where artF is the Artin reciprocity map artF : WFab &rarr F&lowast of local class field theory which is an isomorphism.
2. The correspondence for arbitrary n is compatible with twist by characters that match in the sense above.
3. The correspondence is compatible with taking the contragredient representation.
4. To pairs on either side are associated complicated analytic invariants, the local L-function and the local &epsilon-factor, which have to be preserved by &pi. Moreover, these conditions 1.-4. uniquely determine the Langlands correspondence.

The Langlands correspondence presents a vast generalization of class field theory to a non abelian setup. It is part of a bigger program of current active research: the Langlands Program (1970). The local Langlands correspondence was constructed by Drinfel'd in 1974 for GL2 in the function field case and Jacquet and Langlands (1970), completed by Tunnell (1978) and Kutzko (1980) in the p-adic case. The general case of GLn for all n was proved by Laumon, Rapoport and Stuhler in 1993 in the function field case, whereas Harris and Taylor (2000) finally solved the p-adic case, with a simplified proof given by Henniart (2001) shortly afterwards.

In the seminar we will focus on the case n=2. After having constructed and studied &pi(&rho) for irreducible 2-dimensional representations &rho of the Weil group and cuspidal representations of GL2(F) using the theory of strata and cuspidal types in part 1 of the seminar (SoSe 2008), the second part (WS 2008/2009) will be devoted to the more analytical aspects of L-functions and local &epsilon-factors. We will learn the dichotomy of principal series representations versus cuspidal representations closing a gap from part 2. Then we study L-functions and local &epsilon-factors until we can finally proof the converse theorem which in particular implies the uniqueness of the Langlands correspondence. As time permits we are then prepared to proof the existence of the local Langlands correspondence for GL2(F) in the tame case leading the seminar to its climax.

We will follow closely the recent book of Bushnell and Henniart [BH]. By and large, Part 2 of the seminar is independent from the theory presented in part 1, so a fresh start is possible.

Prerequisits:
some linear representation theory of finite groups, some local class field theory, some local fields.

References:

[BH06]   Bushnell, C. J., Henniart, G., The local Langlands Conjecture for GL(2), Grundlehren der Math. Wissenschaften 335, Springer, 2006.

[Ha02] Harris, M., On the Local Langlands Correspondence, in ICM 2002, vol III, arXiv:math.NT/0304324v1, April 2003.

[Se77]   Serre, J.-P., Linear representations of finite groups, GTM 42, Springer, 1977.

[Se79]   Serre, J.-P., Local fields, GTM 67, Springer, 1979.

[We00] Wedhorn, T., The local Langlands correpondence for GL(n) over p-adic fields, arXiv:math.AG/0011210v2, 25 Nov 2000.

To be updated during the term:

 day # title speaker 16.10.2008 1 Analysis on locally pro-finite groups. Dominik Klein 23.10.2008 2 The Hecke algebra. Lennart Meier 30.10.2008 3 The mirabolic group. Martin Kreidl 06.11.2008 4 The Jacquet module. Ulrich Terstiege 13.11.2008 5 Matrix coefficients. Philipp Hartwig 20.11.2008 6 The Exhaustion Theorem. Peter Scholze 27.11.2008 7 Analysis on GL1(F). Timo Richarz. 04.12.2008 8 Analysis on GL2(F). Eugen Hellmann 11.12.2008 9 Gauß sums and stability theorem. Alexander Ivanov 18.12.2008 10 Proof of the functional equation. Oliver Lorscheid 08.01.2009 11 The Converse Theorem. Eva Viehmann 15.01.2009 12 Local ε-factors for Galois representations. nn 22.01.2009 13 Weil-Deligne representations. nn 29.01.2009 14 The local Langlands Correspondence - comparison of L-functions and ε-factors. nn 05.02.2009 15

Prof. Manfred Lehn has written a HowTo on the topic "Wie halte ich einen  guten (Pro-)Seminarvortrag ?".

Letzte Änderung: 15.03.2010, Sekretariat Prof. Dr. M. Rapoport