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Number theory --- Forms, Modular. --- Galois theory. --- Homology theory.
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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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L-functions. --- Eisenstein series. --- 511.3 --- 511.3 Analytical, additive and other number-theory problems. Diophantine approximations --- Analytical, additive and other number-theory problems. Diophantine approximations --- Series, Eisenstein --- Automorphic functions --- Functions, L --- -Number theory --- -Analytical, additive and other number-theory problems. Diophantine approximations --- Eisenstein series --- L-functions
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Algebraic geometry --- Curves, Elliptic. --- Forms, Modular.
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The theory of p-adic and classic modular forms, and the study of arithmetic and p-adic L-functions has proved to be a fruitful area of mathematics over the last decade. Professor Hida has given courses on these topics in the USA, Japan, and in France, and in this book provides the reader with an elementary but detailed insight into the theory of L-functions. The presentation is self contained and concise, and the subject is approached using only basic tools from complex analysis and cohomology theory. Graduate students wishing to know more about L-functions will find that this book offers a unique introduction to this fascinating branch of mathematics.
L-functions. --- Eisenstein series. --- Series, Eisenstein --- Automorphic functions --- Functions, L --- -Number theory
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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
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This book provides a comprehensive account of a key (and perhaps the most important) theory upon which the Taylor-Wiles proof of Fermat's last theorem is based. The book begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and results on elliptic modular forms, including a substantial simplification of the Taylor-Wiles proof by Fujiwara and Diamond. It contains a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology groups. The final chapter presents a proof of a non-abelian class number formula and includes several new results from the author. The book will be of interest to graduate students and researchers in number theory (including algebraic and analytic number theorists) and arithmetic algebraic geometry.
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Automorphic forms --- Number theory. --- p-adic numbers --- Formes automorphes --- Nombres p-adiques --- Nombres, Théories des --- Galois, Théorie de
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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.
Mathematics --- Number theory --- Algebraic geometry --- Geometry --- landmeetkunde --- wiskunde --- getallenleer --- geometrie
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