Mathematical Institute of the University of Bonn

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

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

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

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List of talks with details and references as ps or pdf.

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A local field F is either a finite extension of the p-adic numbers Q_p 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 W_F is a dense subgroup of the Galois group Gal_F=Gal(F^alg/F) of the algebraic closure F^alg over F. Representations of W_F enlarge the category of representations of Gal_F 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 G_n(F).

On the other side we deal with the arithmetic within the field F. The general linear group GL_n(F) is an analytic group over F. Smooth representations of GL_n(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 GL_n(F) by A_n(F).

The local Langlands correspondence is a bijection &pi : G_n(F) &rarr A_n(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 W_F^ab 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 art_F = &chi, where art_F is the Artin reciprocity map art_F : W_F^ab &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 GL_n 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 GL_n(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.

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

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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.

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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

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

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