Formes modulaires p-adiques

In this course we introduce the main notions relative to the classical theory of modular forms. A complete treatise in a similar style can be found in the author's book joint with F. Str{\"o}mberg [1].

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$.

These are expanded notes of some lectures given by the author for a workshop held at the Indian Statistical Institute, Bangalore in June, 2010, giving an exposition on the modular representations of finite groups of Lie type and $p$-adic groups, and the modular Langlands correspondence. These notes give an overview of the subject with several examples.

The purpose of this informal article is to introduce the reader to some of the objects and methods of the theory of p-adic representations. My hope is that students and mathematicians who are new to the subject will find it useful as a starting point. It consists mostly of an expanded version of the notes for my two lectures at the "Dwork trimester" in June 2001.

In this article, I discuss material which is related to the recent proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama--Shimura.

These are the lecture notes from a five-hour mini-course given at the Winter School on Galois Theory held at the University of Luxembourg in February 2012. Their aim is to give an overview of Serre's modularity conjecture and of its proof by Khare, Wintenberger, and Kisin, as well as of the results of other mathematicians that played an important role in the proof. Along the way we remark on some recent (as of 2012) work concerning generalizations of the conjecture.

The present notes contain the material of the lectures given by the author at the summer school on ``Modular Forms and their Applications'' at the Sophus Lie Conference Center in the summer of 2004.

In this expository article, we present a brief introduction to the theory of Hilbert modular forms and Galois representations, and describe what it means to attach a compatible system of Galois representations to a Hilbert modular form.

Let $p$ be a prime. We discuss $p$-adic properties of various arithmetical functions related to the coefficients of modular form and generating functions. Modular forms are considered as a tool of solving arithmetical problems. Examples of congruences between modular forms modulo $p$ and modulo $p^n$ are given, and the use of a computer for the study of modular forms is illistrated.

These are the notes for an eponymous course given by the authors at the summer school on p-adic arithmetic geometry in Hangzhou.