The 1995 work of Wiles and Taylor-Wiles opened up a whole new technique in algebraic number theory and, a decade on, the waves caused by this incredibly important work are still being felt. This book, authored by a leading researcher, describes the striking applications that have been found for this technique. In the book, the deformation theoretic techniques of Wiles-Taylor are first generalized to Hilbert modular forms (following Fujiwara's treatment), and some applications found by the author are then discussed. With many exercises and open questions given, this text is ideal for...

This book is a comprehensive and systematic account of the theory of p-adic and classical modular forms and the theory of the special values of arithmetic L-functions and p-adic L-functions. The approach is basically algebraic, and the treatment is elementary. No deep knowledge from algebraic geometry and representation theory is required. The author's main tool in dealing with these problems is taken from cohomology theory over Riemann surfaces, which is also explained in detail in the book. He also gives a concise but thorough treatment of analytic continuation and functional equation....

This book is a comprehensive and systematic account of the theory of p-adic and classical modular forms and the theory of the special values of arithmetic L-functions and p-adic L-functions. The approach is basically algebraic, and the treatment is elementary. No deep knowledge from algebraic geometry and representation theory is required. The author's main tool in dealing with these problems is taken from cohomology theory over Riemann surfaces, which is also explained in detail in the book. He also gives a concise but thorough treatment of analytic continuation and functional equation....

This book provides a comprehensive account of a key, perhaps the most important, theory that forms the basis of Taylor-Wiles proof of Fermat's last theorem. Hida begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and recent results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond. He offers a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology...

In the early years of the 1980s, while I was visiting the Institute for Ad- vanced Study (lAS) at Princeton as a postdoctoral member, I got a fascinating view, studying congruence modulo a prime among elliptic modular forms, that an automorphic L-function of a given algebraic group G should have a canon- ical p-adic counterpart of several variables. I immediately decided to find out the reason behind this phenomenon and to develop the theory of ordinary p-adic automorphic forms, allocating 10 to 15 years from that point, putting off the intended arithmetic study of Shimura varieties via...

This book provides a comprehensive account of a key, perhaps the most important, theory that forms the basis of Taylor-Wiles proof of Fermat's last theorem. Hida begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and recent results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond. He offers a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology...

In the early years of the 1980s, while I was visiting the Institute for Ad- vanced Study (lAS) at Princeton as a postdoctoral member, I got a fascinating view, studying congruence modulo a prime among elliptic modular forms, that an automorphic L-function of a given algebraic group G should have a canon- ical p-adic counterpart of several variables. I immediately decided to find out the reason behind this phenomenon and to develop the theory of ordinary p-adic automorphic forms, allocating 10 to 15 years from that point, putting off the intended arithmetic study of Shimura varieties via...

A comprehensive account of the theory of moduli spaces of elliptic curves (over integer rings) and its application to modular forms. The construction of Galois representations, which play a fundamental role in Wiles' proof of the Shimura-Taniyama conjecture, is given.

This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including " "-invariant, "L"-invariant, and similar topics. This book can be regarded as an introductory text to the author's previous book "p-Adic Automorphic Forms on Shimura Varieties." Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader. Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great...

This introduction to Shimura varieties covers key topics including non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; elliptic and modular curves over rings and more.