"p-adic fields" by Vincent Sécherre.

This first course will be an introduction to p-adic numbers and p-adic fields.
We will study the structure of these fields and their finite extensions, and will see how they appear naturally in number theory.

"Local class field theory" by Vincent Sécherre.

Local class field theory gives a way of classifying the abelian extensions of a given p-adic field contained in a given algebraic closure. It can be considered
as the achievement of the local Langlands program for the group GL(1), for it gives a relationship between the group GL(1,F), for F a p-adic field, and the
absolute Galois group of F. We will explain how this relationship can be described, and will construct algebraic elements over F which generate the
various abelian extensions of F.

"Introduction to p-adic reductive groups and their representations (I and II)" by Shaun Stevens.

These talks will introduce the basic notions of reductive groups over a p-adic field (root datum, parabolic subgroup, complex dual group etc.), concentrating on examples to illustrate the notions. This will be followed by an introduction to the theory of smooth representations of these groups (parabolic induction, cuspidality, etc.).

PART I

PART II

"Unramified representations of a p-adic reductive group" by Vincent Sécherre.

An irreducible smooth representation of a p-adic reductive group G is said to be K-unramified if it has non-zero vectors which are invariants by a given maximal compact open subgroup K of G. We will see how the K-unramified representations of G can be classified, up to isomorphism, by unordered finite families of non-zero complex numbers and how this provides an example of the Langlands functoriality.

"An introduction to adeles and global class field theory" by Jose María Tornero.

"Automorphic forms on GL(2) (I, II and III)" by Pierre-Henri Chaudouard.

This course is an introduction to automorphic forms on GL(2). First we will review classical modular forms on Poincaré half-plane. Then we will introduce
automorphic forms and we will sketch proofs of many basic facts about them. We will make the link with local representation theory.

"Introduction to L-functions (I and II)" by Wee Teck Gan.

An L-function is a meromorphic function which is attached to a representation of a p-adic or real reductive Lie group in the local setting, and a modular form or
automorphic representation in the global setting. In the first lecture, I will describe the theory of L-functions for GL(1), in the adelic framework  given to it by Tate. In the second lecture, I will discuss certain higher dimensional extensions, i.e. to GL(n), such as the theory of Godemont-Jacquet,  the theory of Rankin-Selberg integrals, due to Jacquet-Piatetski-Shapiro and Shalika and the Langlands-Shahidi theory.

PART I

PART II

"An Introduction to Functoriality (I and II)" by Jim Cogdell.

In these two talks we will give an introduction to Langlands' Principle of Functoriality. The first talk will be devoted to GL(n). We will describe the formulation of the Langlands Correspondence for GL(n) and some of the ideas in the proofs of the local and global correspondences. The second lecture will be devoted to the more general Principle of Functoriality. We will describe Langlands' definition of the L-group, how it is used to formulate the Functoriality Principle, both local and global, and  then describe the cases where we understand Functoriality and some of the techniques used.

"The mod. l Langlands correspondence for p-adic GL(n), p not l (I and II)", by Guy Henniart.

Let p and l be distinct prime numbers. Let F be a finite extension of Q_p, F ^{\bar} an algebraic closure of F, G_F the Galois group of F ^{\bar} over F. The Langlands correspondence for complex smooth irreducible representations of GL(n,F), which relates them to complex representations of G_F of dimension n, also makes sense for representations in Q_l ^{\bar} vector spaces. With a suitable and natural normalization, it relates Q_l ^{\bar}-representations of L(n,F) with an invariant Z_l ^{\bar} lattice to Galois representations with the same properties. The goal of the lectures will be to explain the result of Vignéras, which says that two cuspidal representations of GL(n,F), with an invariant lattice, which have isomorphic reductions mod.l, correspond to Galois representations with isomorphic reductions mod.l. In this way we get a natural Langlands correpondence mod.l.

"Number fields with prescribed ramification and selfdual automorphic representations of GL(2n)" by Gaëtan Chenevier.

Abstract : "Let S be a finite set of primes, Q_S a maximal algebraic extension of Q unramified outside S (and infinity), and p a prime in S.
If S contains at least two primes, I will show that the field embeddings Q_S ---> Q_p^{bar} have a dense image, or which is the same, that the natural maps (decomposition groups at p)

       Gal(Q_p^{bar}/Q_p) ---> Gal(Q_S/Q)

are injective. Some ingredients of the proof : l-adic cohomology of unitary Shimura varieties, local and global Langlands correspondence, construction of selfdual automorphic representations of GL(2n) with prescribed properties. This is a joint work with Laurent Clozel."

Some geometrico-cohomological methods in the local Langlands program by Jean-François Dat.

We will essentially be concerned with two topics. The first one could be named "buildings and their applications to representation theory". The second one, more advanced, "moduli spaces, etale cohomology, and Langlands correspondences". The aim will be rather to try and explain the ideas underlying these approaches than to give a formal treatment. Time permitting, we may also evoke the complex-geometric approach to the classification of simple modules
of affine Hecke algebras.

"Introduction to the p-adic Langlands program" by Benjamin Schraën.

In this course, we will give a overview of recent developments of the mod p and p-adic correspondences, essentially for GL_2(Qp), coming form Breuil, Berger, Colmez, Emerton, Paskunas...

BASIC REFERENCES

C. Bushnell, G. Henniart, `The local Langlands Conjecture for GL(2)' . Grundlehren der mathematischen Wissenschaften 335 (2006).

J. Bernstein, S. Gelbart, (Editors) An introduction to the Langlands program', Birkhäuser Boston (2004).

Some other references:

On local and global class field theory:

S. Lang, `Algebraic number theory' Springer-Verlag, NY.

J. Tate, `Number theoretic background', in Automorphic forms, representations, and L-functions, Proc. of Symp. 
in Pure Math., vol. 33.2, Providence, RI, AMS, Amer. Math. Soc., 1979.  pp. 3–26.

A. Weil, `Basic Number Theory', Springer-Verlag, NY.

On representation theory of p-adic groups:

I.N. Bernstein and A.V. Zelevinsky, `Representations of the groups GL(n,F) where F is a local non-archimedean Field', Uspekhi Mat. Nauk., Vol 31, No. 3, 1976, 74-75.

I.N. Bernstein and A.V. Zelevinsky, `Induced representations of reductive p-adic groups I', Ann. Sci. ENS 10 (1977), pp. 441-472.

A. Borel, `Automorphic L-functions', in Proc. Symp. Pure Math 33 vol. 2 (1979), AMS, pp. 27-62.

P. Cartier, `Representations of p-adic groups: A survey', in Automorphic Forms, Representations, and L-functions, Proc. Symp. Pure Math. vol. 33 part I, pp. 111-156.

W. Casselman, Introduction to the Theory of Admissible Representations of p-adic Reductive Groups, unpublished, dated 1975.

F. Rodier, Représentations de GL(n,k) où k est un corps p-adique', Séminaire Bourbaki no 587 (1982), Asterisque 92-93, 201-218.

A.V. Zelevinsky, `Induced representations of the group GL(n) over a p-adic field, II. On irreducible representations of GL(n) ', Ann. Sci. ENS (IV) 13 (1980), pp. 165-210.

On reductive groups:

A. Borel, Linear Algebraic Groups, Benjamin, New York, 1969.

J. Humphreys, Linear algebraic groups, Springer-Verlag, Berlin and New York, 1975.

T. A. Springer, Linear Algebraic Groups, 2nd edition, Birkhauser, Boston, 1998.

On L-functions:

R. Godement and H. Jacquet, Zeta Functions of Simple Algebras, Lecture Notes in Mathematics no. 260 (1972), Springer-Verlag, New York, 1972.

J. Tate, Ph.D. thesis, Princeton University, 1950, reprinted in Algebraic Number Theory, ed. J.W.S. Cassels and A. Frohlich, Thompson Book Co., Washington, D.C., 1967.

On Automorphic forms:

D. Bump, `Automohphic forms and representations ', Cambridge Studies in Advanced Mathematics 55 (1998).

S. Gelbart, `Automorphic forms on adele groups', Annals of Math. Studies 83, Princeton University Press (1975).

H. Jacquet and R.P. Langlands, Automorphic Forms on GL(2), Lecture Notes in Mathematics no. 114, Springer-Verlag, New York, 1970.

On the Langlands program:

H. Carayol, `Preuve de la conjecture de Langlands locale pour GL(n) : travaux de Harris-Taylor et Henniart ', Sem. Bourbaki, exp n. 857, mars 1999, Asterisque 266 (2000), 11-244.

S. Kudla, Local Langlands conjecture: the non-archimedean case, in Jannsen et al. (ed.), Motives, Proc. of Symp. in Pure Math., vol. 55.2., Providence, RI, AMS, Amer.  Math. Soc., 1994. , pp. 365–391.

G. Henniart, Progrès récents en fonctorialité de Langlands `, Séminaire Bourbaki, exp n. 890, 2000-2001.

On l-modular representations: 

M.-F. Vignéras, `Représentations $\l$-modulaires d'un groupe réductif p-adique avec l \neq p '. Progress in Math 131 Birkhauser 1996.

M.-F. Vignéras, `Correspondance locale de Langlands semi-simple pour GL(n,F) modulo $l \neq p$ '. Inventiones 2001, 144 page 197-223.

On p-modular representations:

C. Breuil, `Introduction générale aux volumes d'Astérisque sur le programme de Langlands p-adique pour GL2(Qp)', to appear at Astérisque. Available at https://www.ihes.fr/~breuil/publications.html

C. Breuil, `Representations of Galois and of GL2 in characteristic p', cours à l'université de Columbia, automne 2007. Available at https://www.ihes.fr/~breuil/publications.html