S4A1 - Seminar
Local Langlands Correspondence for GL(2)
von Prof. Dr. M. Rapoport und Dr. J. Stix, SoSe 2008.

The seminar will continue next term as
Local Langlands Correspondence for GL(2) - part II

 Time: Tuesday 14-16 Place: Seminarraum B Beringstr. 4

List of talks with details and references as ps or 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 and on the construction of &pi(&rho) for irreducible 2-dimensional representations &rho of the Weil group. Due to these restrictions we avoid the complications of Weil-Deligne representations and essentially all the analysis. The corresponding class of representations of GL2(F) are the cuspidal representations, the class of new representations which cannot be understood via representations of GLn(F) with smaller n.

We will follow closely the recent book of Bushnell and Henniart [BH]. In particular we will focus on the theory of cuspidal types.

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 08.04.2006 1 Smooth representations. Michael Mueller 15.04.2006 2 Induction and Frobenius reciprocity. Benjamin Mocnik 22.04.2006 3 The Weil group of a local field and its representations. Gero Mayr-Gollnitzer 29.04.2006 4 Weil group representations of dimension 2. Timo Richarz 06.05.2006 5 Representation theory of GL2(Fq). Stefan Kraemer 13.05.2006 - --- break due to Pentecost --- - 20.05.2006 6 Chain orders in M2(F) and characters. Martin Kreidl 27.05.2006 7 Automorphic induction I: case of level 0. Alexander Ivanov 03.06.2006 8 Fundamental strata. Robert Kucharczyk 10.06.2006 9 Simple strata: minimal elements and intertwining properties. Eugen Hellmann 17.06.2006 10 Cuspidal types. Peter Scholze 24.06.2006 11 Automorphic induction II: case of level > 0. Ulrich Terstiege 01.07.2006 12 Tame parametrization. Brian Smithling 08.07.2006 13 15.07.2006 14

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

Letzte Ă„nderung: 15.03.2010, Sekretariat Prof. Dr. M. Rapoport