Geometric Modular Forms And Elliptic Curves

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Geometric Modular Forms And Elliptic Curves

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Author : Haruzo Hida
language : en
Publisher: World Scientific
Release Date : 2012

Geometric Modular Forms And Elliptic Curves written by Haruzo Hida and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.

1. An algebro-geometric tool box. 1.1. Sheaves. 1.2. Schemes. 1.3. Projective schemes. 1.4. Categories and functors. 1.5. Applications of the key-lemma. 1.6. Group schemes. 1.7. Cartier duality. 1.8. Quotients by a group scheme. 1.9. Morphisms. 1.10. Cohomology of coherent sheaves. 1.11. Descent. 1.12. Barsotti-Tate groups. 1.13. Formal scheme -- 2. Elliptic curves. 2.1. Curves and divisors. 2.2. Elliptic curves. 2.3. Geometric modular forms of level 1. 2.4. Elliptic curves over C. 2.5. Elliptic curves over p-adic fields. 2.6. Level structures. 2.7. L-functions of elliptic curves. 2.8. Regularity. 2.9. p-ordinary moduli problems. 2.10. Deformation of elliptic curves -- 3. Geometric modular forms. 3.1. Integrality. 3.2. Vertical control theorem. 3.3. Action of GL(2) on modular forms -- 4. Jacobians and Galois representations. 4.1. Jacobians of stable curves. 4.2. Modular Galois representations. 4.3. Fullness of big Galois representations -- 5. Modularity problems. 5.1. Induced and extended Galois representations. 5.2. Some other solutions. 5.3. Modularity of Abelian Q-varieties

Geometric Modular Forms And Elliptic Curves

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Author : Haruzo Hida
language : en
Publisher: World Scientific
Release Date : 2000-09-27

Geometric Modular Forms And Elliptic Curves written by Haruzo Hida and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-09-27 with Mathematics categories.

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.

Geometric Modular Forms And Elliptic Curves 2nd Edition

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Author : Haruzo Hida
language : en
Publisher: World Scientific
Release Date : 2011-12-28

Geometric Modular Forms And Elliptic Curves 2nd Edition written by Haruzo Hida and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-12-28 with Mathematics categories.

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 Lie groups) is added to Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated at the end of Chapter 2 (in order to make the proof of regularity of the moduli of elliptic curve more conceptual), and in Chapter 4, though limited to ordinary cases, newly incorporated are Ribet's theorem of full image of modular p-adic Galois representation and its generalization to ‘big’ Λ-adic Galois representations under mild assumptions (a new result of the author). Though some of the striking developments described above is out of the scope of this introductory book, the author gives a taste of present day research in the area of Number Theory at the very end of the book (giving a good account of modularity theory of abelian ℚ-varieties and ℚ-curves).

The 1 2 3 Of Modular Forms

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Author : Jan Hendrik Bruinier
language : en
Publisher: Springer
Release Date : 2009-09-02

The 1 2 3 Of Modular Forms written by Jan Hendrik Bruinier and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-09-02 with Mathematics categories.

This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.

Elliptic Curves Second Edition

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Author : James S Milne
language : en
Publisher: World Scientific
Release Date : 2020-08-20

Elliptic Curves Second Edition written by James S Milne and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-08-20 with Mathematics categories.

This book uses the beautiful theory of elliptic curves to introduce the reader to some of the deeper aspects of number theory. It assumes only a knowledge of the basic algebra, complex analysis, and topology usually taught in first-year graduate courses.An elliptic curve is a plane curve defined by a cubic polynomial. Although the problem of finding the rational points on an elliptic curve has fascinated mathematicians since ancient times, it was not until 1922 that Mordell proved that the points form a finitely generated group. There is still no proven algorithm for finding the rank of the group, but in one of the earliest important applications of computers to mathematics, Birch and Swinnerton-Dyer discovered a relation between the rank and the numbers of points on the curve computed modulo a prime. Chapter IV of the book proves Mordell's theorem and explains the conjecture of Birch and Swinnerton-Dyer.Every elliptic curve over the rational numbers has an L-series attached to it.Hasse conjectured that this L-series satisfies a functional equation, and in 1955 Taniyama suggested that Hasse's conjecture could be proved by showing that the L-series arises from a modular form. This was shown to be correct by Wiles (and others) in the 1990s, and, as a consequence, one obtains a proof of Fermat's Last Theorem. Chapter V of the book is devoted to explaining this work.The first three chapters develop the basic theory of elliptic curves.For this edition, the text has been completely revised and updated.

Rational Points On Elliptic Curves

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Author : Joseph H. Silverman
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

Rational Points On Elliptic Curves written by Joseph H. Silverman and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.

In 1961 the second author deliv1lred a series of lectures at Haverford Col lege on the subject of "Rational Points on Cubic Curves. " These lectures, intended for junior and senior mathematics majors, were recorded, tran scribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and por tions have appeared in various textbooks (Husemoller [1], Chahal [1]), but they have never appeared in their entirety. In view of the recent inter est in the theory of elliptic curves for subjects ranging from cryptogra phy (Lenstra [1], Koblitz [2]) to physics (Luck-Moussa-Waldschmidt [1]), as well as the tremendous purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience. We have attempted to maintain much of the informality of the orig inal Haverford lectures. Our main goal in doing this has been to write a textbook in a technically difficult field which is "readable" by the average undergraduate mathematics major. We hope we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rigorous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than to rigorously prove.

Heads In Grammatical Theory

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Author : Greville G. Corbett
language : en
Publisher: Cambridge University Press
Release Date : 1993-06-24

Heads In Grammatical Theory written by Greville G. Corbett and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1993-06-24 with Language Arts & Disciplines categories.

A study of the idea of the 'head' or dominating element of a phrase.

Modular Forms A Classical And Computational Introduction 2nd Edition

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Author : Lloyd James Peter Kilford
language : en
Publisher: World Scientific Publishing Company
Release Date : 2015-03-12

Modular Forms A Classical And Computational Introduction 2nd Edition written by Lloyd James Peter Kilford and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-03-12 with Mathematics categories.

Modular Forms is a graduate student-level introduction to the classical theory of modular forms and computations involving modular forms, including modular functions and the theory of Hecke operators. It also includes applications of modular forms to various subjects, such as the theory of quadratic forms, the proof of Fermat's Last Theorem and the approximation of π. The text gives a balanced overview of both the theoretical and computational sides of its subject, allowing a variety of courses to be taught from it.This second edition has been revised and updated. New material on the future of modular forms as well as a chapter about longer-form projects for students has also been added.

P Adic Aspects Of Modular Forms

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Author : Baskar Balasubramanyam
language : en
Publisher: World Scientific
Release Date : 2016-06-14

P Adic Aspects Of Modular Forms written by Baskar Balasubramanyam and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-06-14 with Mathematics categories.

The aim of this book is to give a systematic exposition of results in some important cases where p-adic families and p-adic L-functions are studied. We first look at p-adic families in the following cases: general linear groups, symplectic groups and definite unitary groups. We also look at applications of this theory to modularity lifting problems. We finally consider p-adic L-functions for GL(2), the p-adic adjoint L-functions and some cases of higher GL(n).

Elliptic Curves And Arithmetic Invariants

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Author : Haruzo Hida
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-13

Elliptic Curves And Arithmetic Invariants written by Haruzo Hida and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-13 with Mathematics categories.

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.