Elliptic curves and big Galois representations

by Daniel Delbourgo

"The mysterious properties of modular forms lie at the heart of modern number theory. This book develops a generalisation of the method of Euler systems to a two-variable deformation ring. The resulting theory is then used to study the arithmetic of elliptic curves, in particular the Birch and Swinnerton-Dyer (BSD) formula." "Three main steps are outlined. The first is to parametrise 'big' cohomology groups using (deformations of) modular symbols. One can then establish finiteness results for big Selmer groups. Finally, at weight two, the arithmetic invariants of these Selmer groups allow the control of data from the BSD conjecture." "This is the first book on the subject, and the material is introduced from scratch; both graduate students and professional number theorists will find this an ideal introduction to the subject. Material at the very forefront of current research is included, and numerical examples encourage the reader to interpret abstract theorems in concrete cases."--Jac