Let K be a number field. Modern algebraic number theory aims to study the absolute Galois group G_K and its representations. Let p be a prime. In these lectures, which are closely related to the course given by G. Böckle, we will investigate mod p representations r: G_K --> GL_n(F), where F is an algebraic extension of the field F_p. These are simpler objects than the ell-adic representations appearing in Böckle's course.

We are particularly interested in mod p representations that come from geometry, arising as the reduction modulo v of a p-adic representation r_{f,v} constructed from a (Hilbert) modular form as in Böckle's lectures (where v|p is a prime of K) or, more generally, as a sub-representation of the etale cohomology of a mod p sheaf on an algebraic variety defined over K. Such representations are called modular.

If K is the field Q of rational numbers, then Serre conjectured in the 1960's that every continuous, irreducible, and odd representation r: G_Q --> GL_2(F) is modular, where F is an algebraic closure of F_p and r is called odd if the image of complex conjugation is a matrix of determinant -1. Moreover, he explicitly conjectured the weights and levels of modular forms f that would give rise to r (in the case K = Q, the modular mod p representations are just the reductions modulo p of p-adic representations of the form r_{f,p}). This conjecture was proved in 2005 by Khare and Wintenberger, relying on modularity lifting theorems of Wiles, Taylor, Kisin, and others. The implication that if r is modular, then it is modular of the predicted weights, was known earlier from the work of many mathematicians.

If K is larger than Q, then one still conjectures that every continuous, irreducible, and totally odd representation r: G_K --> GL_2(F) is modular. This time there are [K:Q] different embeddings of Q into C, and one says that r is totally odd if the complex conjugation associated to each of these embeddings is mapped to a matrix of determinant -1. It is more complicated to generalize to arbitrary K the conjecture indicating what the modular weights of r should be. While the full conjecture seems out of reach at the present time, in a number of cases it is known that if f is modular, then it is modular of the specified weights. In all cases, the modular weights of r are determined by the restriction of r to decomposition groups at primes above p.

A modern approach to Serre's conjecture is to view it in the context of a mod p version of the local Langlands correspondence. Let L be a finite extension of Q_p. One expects, roughly, a correspondence between n-dimensional mod p representations of G_L and certain smooth admissible mod p representations of the group GL_n(L). Serre's conjecture can be interpreted as a statement about the GL_2(O_L)-socle of the representation associated to (the restriction to a decomposition subgroup of) r. This picture is well-understood for GL_2(Q_p) and is still murky in other cases.

The course will use the machinery introduced in Böckle's lectures and will cover the following topics:

- Modularity of mod p Galois representations
- Statement of Serre's conjecture for Q.
- Generalizations of Serre's conjecture.
- The mod p local Langlands correspondence and relation to Serre's conjecture.
- Modularity lifting and potential modularity results and their proofs (R = T theorems).
- The structure of the proof of Serre's conjecture.

Last modification: 8 January 2012.