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This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including μ-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 resource for experts in the field, but it is also accessible to advanced graduate students studying number theory.  Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.

Curves, Elliptic. --- Invariants. --- Number theory. --- Curves, Elliptic --- Invariants --- Number theory --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Elliptic functions. --- Elliptic integrals --- Functions, Elliptic --- Integrals, Elliptic --- Number study --- Numbers, Theory of --- Mathematics. --- Algebraic geometry. --- Number Theory. --- Algebraic Geometry. --- Transcendental functions --- Functions of complex variables --- Integrals, Hyperelliptic --- Geometry, algebraic. --- Algebraic geometry --- Geometry

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This book provides 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. In addition, the book presents an outline of the proof of diverse modularity results of two-dimensional Galois representations (including that of Wiles), as well as some of the author's new results in that direction. In this new second edition, a detailed description of Barsotti-Tate groups (including formal Li

Curves, Elliptic. --- Forms, Modular. --- Modular forms --- Forms (Mathematics) --- Elliptic curves --- Curves, Algebraic