Part I: Galois Deformations -- On p-adic Galois Representations -- Deformations of Galois Representations -- Part II: Hilbert Modular Forms -- Arithmetic Aspects of Hilbert Modular Forms and Varieties -- Explicit Methods for Hilbert Modular Forms -- Part III: Elliptic Curves -- Notes on the Parity Conjecture.

The notes in this volume correspond to advanced courses given at the Centre de Recerca Matemàtica (Bellaterra, Barcelona, Spain) as part of the Research Programme in Arithmetic Geometry in the 2009-2010 academic year. They are now available in printed form due to the many requests received by the organizers to make the content of the courses publicly available. The material covers the theory of p-adic Galois representations and Fontaine rings, Galois deformation theory, arithmetic and computational aspects of Hilbert modular forms, and the parity conjecture for elliptic curves.