An outline of the course, including references, can be found here.

I will upload hand-written lecture notes on this page.

Registration on Müsli is encouraged.

Time and location of the lectures: Wednesdays 2-4 pm, INF 205 / Seminarraum 11

Office Hours: Thursdays 15:30-16:30, INF 205 / Room 3/225

The lectures are in English. Passing the examination at the end of the course gives right to 4CP.

The examination will consist in the written solution of a homework problem (in German or English as the student prefers) and of the presentation of part of the solution. The oral part of the examination will take place on August 3rd in Room 3/225. The final grade will come from the evaluation of the written and oral presentations.

Short outline of the course:

A way to study the Galois group of an extension of number fields is to investigate its actions on finite dimensional vector spaces over finite or p-adic fields. Mazur introduced some tools to understand how a Galois representation over a finite field can be "deformed" into a Galois representation over a p-adic field. Many p-adic representations encoding arithmetic information can be constructed from geometric objects such as elliptic curves, modular forms and their generalizations. The study of deformations gives significant insights on such representations. Mazur's ideas were largely developed in the last few decades and played a role in important recent developments in number theory, such as the proofs of Fermat's Last Theorem and Fontaine-Mazur's conjecture.

The aim of this course is to introduce Galois representations and the theory of their p-adic deformation, and to study the first properties of the deformation spaces.

Main references:

F. Goûvea, Lecture notes for the course "Deformations of Galois Representations".

A. Mézard, Lecture notes for the course "Déformations de représentations galoisiennes", in French.

B. Mazur, Deforming Galois representations, in Galois groups over Q, M.S.R.I. Publications, Springer-Verlag, 1989.

N. Boston, Explicit deformation of Galois representations, Inventiones Mathematicae 103, 1991.

Other references:

G. Böckle, Lecture notes for the course "Deformations of Galois Representations".

M. Kisin, Lecture notes on Deformations of Galois Representations.

J. Tilouine, Deformations of Galois representations and Hecke algebras, Narosa Publ. House, 1996.

H. Hida, Modular forms and Galois cohomology, Cambridge studies in advanced mathematics 69, Cambridge University Press, 2000.

Some further reading (with material not included in the course):

M. Emerton, Deformation of Galois Representations and Applications, talk at the IHP Conference on the occasion of the Galois bicentennial.

T. Gee, Modularity lifting theorems, notes for the Arizona Winter School 2013 on Modular forms and modular curves.

G. Chenevier, The infinite fern and families of quaternionic modular forms, notes for a Master's Course for the Galois trimester at Institut Henri Poincaré, Paris.