Elliptic Curves and Big Galois Representations
There are many deep conjectures relating special values of L-functions with arithmetic invariants. This book is concerned with special values of L-functions attached to modular forms. After a brief review of elliptic curves and modular forms, the Perrin-Riou and Kato work on the theory of Euler systems for modular forms is introduced. Then corresponding p-adic L-functions are constructed via modular symbols attached to Euler systems. The technical heart of the book contains the theory of 2-variable Euler systems realized in terms of the lambda-deformation of the space of modular symbols. The construction is compatible with the analytic theory of Greenberg-Stevens. The rest of the book is devoted to a study of the arithmetic of p-ordinary families of modular forms. For example, the association of Selmer groups over a one-variable deformation ring is discussed and the p-part of the Tate-Shafarevich group of an elliptic curve is computed.