The notes in this volume correspond to advanced courses held at
the Centre de Recerca Matematica as part of the research program in
Arithmetic Geometry in the 2009-2010 academic year.
The notes by Laurent Berger provide an introduction to "p"-adic
Galois representations and Fontaine rings, which are especially
useful for describing many local deformation rings at "p" that
arise naturally in Galois deformation theory.
The notes by Gebhard Bockle offer a comprehensive course on
Galois deformation theory, starting from the foundational results
of Mazur and discussing in detail the theory of
pseudo-representations and their deformations, local deformations
at places l p and local deformations at "p" which are flat. In the
last section, the results of Bockle and Kisin on presentations of
global deformation rings over local ones are discussed.
The notes by Mladen Dimitrov present the basics of the
arithmetic theory of Hilbert modular forms and varieties, with an
emphasis on the study of the images of the attached Galois
representations, on modularity lifting theorems over totally real
number fields, and on the cohomology of Hilbert modular varieties
with integral coefficients.
The notes by Lassina Dembele and John Voight describe methods
for performing explicit computations in spaces of Hilbert modular
forms. These methods depend on the Jacquet-Langlands correspondence
and on computations in spaces of quaternionic modular forms, both
for the case of definite and indefinite quaternion algebras.
Several examples are given, and applications to modularity of
Galois representations are discussed.
The notes by Tim Dokchitser describe the proof, obtained by the
author in a joint project with Vladimir Dokchitser, of the parity
conjecture for elliptic curves over number fields under the
assumption of finiteness of the Tate-Shafarevich group. The
statement of the Birch and Swinnerton-Dyer conjecture is included,
as well as a detailed study of local and global root numbers of
elliptic curves and their classification."
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