On Selmer groups of geometric Galois representations

Let X be an algebraic variety defined over a global field F. The étale cohomology groups of X yield interesting representations of the absolute Galois group of F. The Selmer group of such a representation is a sub-group of its Galois cohomology, defined by local conditions, which should be related to the arithmetic of X. In the first part of this thesis we define Selmer groups and recall the use of Euler systems to construct annihilators of these Selmer groups, focusing on applications to the deformation theory of Galois representations. We also show how the existence of certain Euler systems yields an explicit reciprocity law in deformation theory. The second part of the thesis is concerned with using the geometry of a variety to construct an Euler system for its étale cohomology groups. Our main results are generalizations of methods of Flach and Mazur to higher dimensional situations. In the third part we apply these results to the case of modular curves and Kuga-Sato varieties with applications to the deformation theory of modular Galois representations of weight at least 2.