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This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.
Book Synopsis An Introduction to Non-Abelian Class Field Theory by : Toyokazu Hiramatsu
Download or read book An Introduction to Non-Abelian Class Field Theory written by Toyokazu Hiramatsu and published by World Scientific. This book was released on 2016-09-13 with total page 188 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.
"This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1."--Publisher's website.
Book Synopsis An Introduction to Non-Abelian Class Field Theory by : Toyokazu Hiramatsu
Download or read book An Introduction to Non-Abelian Class Field Theory written by Toyokazu Hiramatsu and published by . This book was released on 2016 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1."--Publisher's website.
This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.
Book Synopsis An Introduction to Non-Abelian Class Field Theory by : Toyokazu Hiramatsu
Download or read book An Introduction to Non-Abelian Class Field Theory written by Toyokazu Hiramatsu and published by World Scientific Publishing Company. This book was released on 2017 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph provides a brief exposition of automorphic forms of weight 1 and their applications to arithmetic, especially to Galois representations. One of the outstanding problems in arithmetic is a generalization of class field theory to non-abelian Galois extension of number fields. In this volume, we discuss some relations between this problem and cusp forms of weight 1.
Class field theory brings together the quadratic and higher reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. This book provides an accessible introduction to class field theory. It takes a traditional approach in that it attempts to present the material using the original techniques of proof, but in a fashion which is cleaner and more streamlined than most other books on this topic. It could be used for a graduate course on algebraic number theory, as well as for students who are interested in self-study. The book has been class-tested, and the author has included lots of challenging exercises throughout the text.
Book Synopsis Class Field Theory by : Nancy Childress
Download or read book Class Field Theory written by Nancy Childress and published by Springer Science & Business Media. This book was released on 2008-10-28 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: Class field theory brings together the quadratic and higher reciprocity laws of Gauss, Legendre, and others, and vastly generalizes them. This book provides an accessible introduction to class field theory. It takes a traditional approach in that it attempts to present the material using the original techniques of proof, but in a fashion which is cleaner and more streamlined than most other books on this topic. It could be used for a graduate course on algebraic number theory, as well as for students who are interested in self-study. The book has been class-tested, and the author has included lots of challenging exercises throughout the text.
Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. This book explores the consequences and the practical use of these results in detailed studies and illustrations of classical subjects. In the corrected second printing 2005, the author improves many details all through the book.
Book Synopsis Class Field Theory by : Georges Gras
Download or read book Class Field Theory written by Georges Gras and published by Springer Science & Business Media. This book was released on 2005-02-16 with total page 513 pages. Available in PDF, EPUB and Kindle. Book excerpt: Global class field theory is a major achievement of algebraic number theory based on the functorial properties of the reciprocity map and the existence theorem. This book explores the consequences and the practical use of these results in detailed studies and illustrations of classical subjects. In the corrected second printing 2005, the author improves many details all through the book.
Book Synopsis Knots and Primes by : Masanori Morishita
Download or read book Knots and Primes written by Masanori Morishita and published by Springer Nature. This book was released on with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group theoretical principle, admitting an entirely elementary description. This de scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here.
Book Synopsis Class Field Theory by : J. Neukirch
Download or read book Class Field Theory written by J. Neukirch and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: Class field theory, which is so immediately compelling in its main assertions, has, ever since its invention, suffered from the fact that its proofs have required a complicated and, by comparison with the results, rather imper spicuous system of arguments which have tended to jump around all over the place. My earlier presentation of the theory [41] has strengthened me in the belief that a highly elaborate mechanism, such as, for example, cohomol ogy, might not be adequate for a number-theoretical law admitting a very direct formulation, and that the truth of such a law must be susceptible to a far more immediate insight. I was determined to write the present, new account of class field theory by the discovery that, in fact, both the local and the global reciprocity laws may be subsumed under a purely group theoretical principle, admitting an entirely elementary description. This de scription makes possible a new foundation for the entire theory. The rapid advance to the main theorems of class field theory which results from this approach has made it possible to include in this volume the most important consequences and elaborations, and further related theories, with the excep tion of the cohomology version which I have this time excluded. This remains a significant variant, rich in application, but its principal results should be directly obtained from the material treated here.
Research Paper (postgraduate) from the year 2011 in the subject Mathematics - Number Theory, grade: Postgraduate, University of Sheffield, language: English, abstract: This document is a continuation of my Semester 1 project on class field theory. In the previous work, we made a rounded exposition of the fundamentals of class field theory but in order to preserve the document length the main proofs had to be skipped. We concentrate on filling in the gaps in this second installment. Due to the need to complete the arguments left open last semester and the need for applications this part of the project is a little longer than it should have been. It was not mentioned in the previous project but the class field theory we are studying here is global class field theory. There is such a thing as local class field theory in which we study the Abelian extensions of local fields (essentially fields that arise as completions of a number field with respect to places). Actually we touch on these ideas slightly in this project but never quite get to de_ning a local Artin map and looking at the local analogues of the main theorems of global class field theory. For those wanting to continue on to study local class field theory, consider Chapter 7 of [2] To start off this project we shall first restate the main de_nitions and theorems. This will be brief and those wanting to remind themselves of the details should consult my Semester 1 project. There will be very little motivation or technical results here since this was the purpose of the work done previously. We then set out to prove the main theorems of class field theory. With our present knowledge this would not be a simple task and we soon find that we first have to invent or discover new concepts such as the idele group and the corresponding idele class group. These are topological devices that take stock of all completions of a number eld at once. Such constructions will make the theory much easier to understand and formulate, whilst at the same time generalising the theory to all Abelian extensions. The cohomology of nite Abelian groups will be introduced and used alongside the idele theory to establish an important inequality. We use L-series in conjunction with the ideal theory to establish another important inequality. Combining the two inequalities will give a nice result that allows us to prove Artin reciprocity. In order to prove the existence theorem we resort to using Kummer n-extensions and the notion of a class eld. This middle chunk of the project will be quite technical but hopefully enjoyable and illuminating. [...]
Book Synopsis Class Field Theory: Proofs and Applications by : Daniel Fretwell
Download or read book Class Field Theory: Proofs and Applications written by Daniel Fretwell and published by GRIN Verlag. This book was released on 2011-07-27 with total page 51 pages. Available in PDF, EPUB and Kindle. Book excerpt: Research Paper (postgraduate) from the year 2011 in the subject Mathematics - Number Theory, grade: Postgraduate, University of Sheffield, language: English, abstract: This document is a continuation of my Semester 1 project on class field theory. In the previous work, we made a rounded exposition of the fundamentals of class field theory but in order to preserve the document length the main proofs had to be skipped. We concentrate on filling in the gaps in this second installment. Due to the need to complete the arguments left open last semester and the need for applications this part of the project is a little longer than it should have been. It was not mentioned in the previous project but the class field theory we are studying here is global class field theory. There is such a thing as local class field theory in which we study the Abelian extensions of local fields (essentially fields that arise as completions of a number field with respect to places). Actually we touch on these ideas slightly in this project but never quite get to de_ning a local Artin map and looking at the local analogues of the main theorems of global class field theory. For those wanting to continue on to study local class field theory, consider Chapter 7 of [2] To start off this project we shall first restate the main de_nitions and theorems. This will be brief and those wanting to remind themselves of the details should consult my Semester 1 project. There will be very little motivation or technical results here since this was the purpose of the work done previously. We then set out to prove the main theorems of class field theory. With our present knowledge this would not be a simple task and we soon find that we first have to invent or discover new concepts such as the idele group and the corresponding idele class group. These are topological devices that take stock of all completions of a number eld at once. Such constructions will make the theory much easier to understand and formulate, whilst at the same time generalising the theory to all Abelian extensions. The cohomology of nite Abelian groups will be introduced and used alongside the idele theory to establish an important inequality. We use L-series in conjunction with the ideal theory to establish another important inequality. Combining the two inequalities will give a nice result that allows us to prove Artin reciprocity. In order to prove the existence theorem we resort to using Kummer n-extensions and the notion of a class eld. This middle chunk of the project will be quite technical but hopefully enjoyable and illuminating. [...]
This book paves the way for a proper understanding of current and future issues on global warming, air pollution, depletion of natural resources, cyberattacks on smart grids, amongst others, by unifying various diverse disciplines of science to focus on a sustainable green society of the future.Readers will find applications of science described through the practical use of mobilities, in this case, the electric vehicles.The book could be used to teach and study on issues of global warming through the window of electric vehicles. The first three chapters can be used for teaching applications of mechanics, quantum mechanics, thermodynamics, and fluid mechanics. Chapter 5 provides rudiments of control theory in anticipation of control theory through number theory and algebraic geometry. Chapters 6 and 7 contain aspects of climatology, global warming, and electric vehicles to green grid. This is the only such comprehensive introductory book in the market that provides the readers hints, suggestions and directions to ponder for a sustainable future through renewable sources.
Book Synopsis From Vehicles To Grid To Electric Vehicles To Green Grid: Many A Little Makes A Miracle by : Li Fuhuo
Download or read book From Vehicles To Grid To Electric Vehicles To Green Grid: Many A Little Makes A Miracle written by Li Fuhuo and published by World Scientific. This book was released on 2019-07-24 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book paves the way for a proper understanding of current and future issues on global warming, air pollution, depletion of natural resources, cyberattacks on smart grids, amongst others, by unifying various diverse disciplines of science to focus on a sustainable green society of the future.Readers will find applications of science described through the practical use of mobilities, in this case, the electric vehicles.The book could be used to teach and study on issues of global warming through the window of electric vehicles. The first three chapters can be used for teaching applications of mechanics, quantum mechanics, thermodynamics, and fluid mechanics. Chapter 5 provides rudiments of control theory in anticipation of control theory through number theory and algebraic geometry. Chapters 6 and 7 contain aspects of climatology, global warming, and electric vehicles to green grid. This is the only such comprehensive introductory book in the market that provides the readers hints, suggestions and directions to ponder for a sustainable future through renewable sources.
This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author's 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry.Starting with a description of Iwasawa's classical results on his proof of the main conjecture under the Kummer-Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation.The fundamentals in the first five chapters are as follows:Many open problems are presented to stimulate young researchers pursuing their field of study.
Book Synopsis Elementary Modular Iwasawa Theory by : Haruzo Hida
Download or read book Elementary Modular Iwasawa Theory written by Haruzo Hida and published by World Scientific. This book was released on 2021-10-04 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the first to provide a comprehensive and elementary account of the new Iwasawa theory innovated via the deformation theory of modular forms and Galois representations. The deformation theory of modular forms is developed by generalizing the cohomological approach discovered in the author's 2019 AMS Leroy P Steele Prize-winning article without using much algebraic geometry.Starting with a description of Iwasawa's classical results on his proof of the main conjecture under the Kummer-Vandiver conjecture (which proves cyclicity of his Iwasawa module more than just proving his main conjecture), we describe a generalization of the method proving cyclicity to the adjoint Selmer group of every ordinary deformation of a two-dimensional Artin Galois representation.The fundamentals in the first five chapters are as follows:Many open problems are presented to stimulate young researchers pursuing their field of study.
