Topic outline

  • General

    Welcome to Galois Representations!

    In this course we study the arithmetic properties of the representations of the absolute Galois group of a number field. Such representation can be seen as “incarnations” of the Galois automorphisms as autopmorphisms of vector spaces over different fields and of different dimensions, using methods of linear algebra and arithmetic of these fields to shed some light on the global behaviour of the Galois automorphisms.

    For us, the most relevant representations are those attached to geometric objects such as elliptic curves or, more in general, modular forms. These are particular instances of so-called “automorphic representations”, and have had a striking impact in solving old-standing problems in Number Theory like Fermat’s Last Theorem (Taylor, Wiles et al.) and more in general the Serre modularity conjecture (Dieulefait, Khare and Wintenberger).

    A key feature of the theory is a dictionary between Galois representations and representations of the reductive group GL(2,F), with F a number field. This correspondence was developed by Langlands in the 60s and is one of the deepest and active topics in modern Number Theory. It aims at answering, for instance, how automorphic representations behave under functorial operations like tensor products, extension/restriction of the number field, inversion and alike.

    The main idea we will address is the concept of deformation, another key input in Wiles´s proof and vastly generalized nowadays. In particular, we will address the existence of universal deformation rings which, under certain hypothesis parametrize all the deformations of a given residual one having prescribed properties.

    Our course will start with basic definitions and facts of general representation theory. After that we will recall how arithmetic representations can be attached to elliptic curves and Dirichlet characters and will study how to deform these (residual representations), first form a purely categorical point of view (Schlessinger conditions) and then by explicit methods.

    Further, we will give a very useful algebro-geometric insight studying the Zariski cotangent and some basics of Galois cohomology to introduce modular deformations and, time permitting a brief introduction to Hida families.

    For this course we will follow the notes of the Park City Mathematics Institute “Deformation of Galois representations” by Fernando-Gouvea and the now classical notes “An introduction to the deformation of Galois representations” by Barry Mazur, which will be upload in due time.

    There won’t be a clear division between theory and exercise sessions but, roughly speaking, we will have a problem solving session every two weeks and a lot of claims will be left to the student to work by him/herself.

    Likewise, although the attendance is not mandatory to pass the course it is extremely advisable to attend all the lectures as the course will cover a very extensive number of topics.

    Prerequisites are: Abstract Algebra, Galois Theory  and Algebraic Number Theory as well as being familiar with some basic notions of elliptic curves (the definition, the group law and basic properties on the j-invariant and the discriminant). Some basic facts on commutative algebra will be provided in the course.

    Finally, the assessment will consist in two assignments of about 10-15 exercises each plus a final essay.