Titlebook: Elliptic Curves and Arithmetic Invariants; Haruzo Hida Book 2013 Springer Science+Business Media New York 2013 Hecke algebra.Shimura varie

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Nontriviality of Arithmetic Invariants,tions and motives. Number theorists all agree that .-functions and .-values are useful invariants for studying the object with number-theoretic goals in mind. From .-functions, number theorists have created more invariants, for example, .- and .-invariant from .-adic .-functions and the .-invariant from exceptional zeros of .-adic .-functions.

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Review of Scheme Theory, necessary to describe elliptic curves and modular forms. Group schemes are best understood from the functorial viewpoint of scheme theory, regarding a scheme as a functor from the category of algebras to sets taking each algebra . to the set of .-rational points .(.) of the scheme ..

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Book 2013urce 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.

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Invariants, Shimura Variety, and Hecke Algebra,ptic curves in an elementary manner in Chap.2, at least we could illustrate our main objectives in this book with a rough outline of their proofs. Detailed proofs (for some of them) will be given after we become equipped with a scheme-theoretic description of elliptic curves and their moduli as the

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Geometry of Variety, is to make the book logically complete, and another is to give the foundation of the theory of towers of varieties in the language of proschemes, since the Shimura variety is a tower of varieties fundamental to the number-theoretic study of automorphic forms. If the reader is familiar with the subj