Course Description: Galois Representations

Lecturer: Gebhard Böckle

Much of modern number theory is concerned with the study of the absolute Galois group of number fields K and its ramification properties. Class field theory achieves this goal for abelian extensions of K. The Langlands program is the main conjectural framework for arbitrary extensions of K. It relates (largely conjecturally) automorphic forms to Galois representations and vice versa.

A small but very well understood and already very powerful part of the Langlands program is the interplay between classical (and Hilbert) modular forms and certain p-adic Galois representations. Concretely, one can attach to any classical Hecke eigenform f and every place v of its coefficient field a 2-dimensional v-adic Galois representation r_{f,v} of the absolute Galois group of the rationals Q.

Except for the prime p below v, the ramification properties of r_{f,v} correspond to ramification properties of the automorphic representation associated with f and are described in parts by the conductor. The r_{f,v} for varying v form a strictly compatible system of Galois representations. A uniform description of the local ramification properties are provided by the Weil-Deligne representation attached to the r_{f,v}. Modular forms thus provide a vast set of examples of interesting Galois representations whose ramification properties are understood rather completely. So they play an important role in understanding the absolute Galois group of Q.

Conversely if we have a 2-dimensional p-adic Galois representation of the absolute Galois group of Q one would like to know when it arises from a modular form. If this is the case one calls the representations modular. Here Fontaine and Mazur have given a precise conjectural characterization of modular representations the conjecture is close to being a theorem by Emerton and Kisin, independently. A related result on mod p Galois representation is the theorem of Khare and Wintenberger aka the conjecture of Serre. It will be central in the talks by M. Schein.

A main tool in proving modularity conjectures are deformation rings as defined by Mazur and refined by Wiles et al. These rings R are constructed by Galois theoretic properties which in turn are motivated by the conjecture of Fontaine and Mazur. From their construction it follows that R admits a surjective homomorphism to a p-adic Hecke algebra T, depending on R. The ring R parameterizes Galois representations of a certain type, the ring T can be interpreted as a ring which parameterizes modular Galois representations of a certain type. An R=T theorem thus asserts, in a strong sense, that certain p-adic Galois representations are of the form r_{f,v} for some v above p.

In the course on Galois representations, I intend to cover the following topics:

The aim of the latter part of the lecture series is simply to define and motivate deformation rings satisfying geometric properties as described by the Fontaine-Mazur conjectures. R=T theorems will be discussed in the talks by M. Schein.

Lecture notes (dated 16/03/2012).

Last modification: 20 March 2012.