Class Groups Of Number Fields And Related Topics
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Class Groups Of Number Fields And Related Topics
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Author : Kalyan Chakraborty
language : en
Publisher: Springer Nature
Release Date : 2024-12-02
Class Groups Of Number Fields And Related Topics written by Kalyan Chakraborty and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-02 with Mathematics categories.
This book collects original research papers and survey articles presented at two conferences on the same theme: the International Conference on Class Groups of Number Fields and Related Topics, held at Kerala School of Mathematics, Kozhikode, Kerala, India, from 21–24 October 2021 and then from 21–24 November 2022. It presents the fundamental research problems that arise in the study of class groups of number fields and related areas. The book also covers some new techniques and tools to study these problems. Topics in this book include class groups of number fields, units, Ankeny–Artin–Chowla conjecture, Iwasawa theory, elliptic curves, Diophantine equations, partition functions, Diophantine tuples, congruent numbers, Carmichael ideals in a number field and their connection with class groups. This book will be a valuable resource for graduate students and researchers in mathematics interested in class groups of number fields and their connections to other branches of mathematics. It also attracts new researchers to the field and young researchers will benefit immensely from the diverse problems discussed in this book. All the contributing authors are leading academicians, scientists and profound researchers. This book is dedicated to Prof. Michel Waldschmidt, a renowned French number theorist, on his 75th birthday.
Class Groups Of Number Fields And Related Topics
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Author : Kalyan Chakraborty
language : en
Publisher: Springer
Release Date : 2020-03-11
Class Groups Of Number Fields And Related Topics written by Kalyan Chakraborty and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-11 with Mathematics categories.
This book gathers original research papers and survey articles presented at the “International Conference on Class Groups of Number Fields and Related Topics,” held at Harish-Chandra Research Institute, Allahabad, India, on September 4–7, 2017. It discusses the fundamental research problems that arise in the study of class groups of number fields and introduces new techniques and tools to study these problems. Topics in this book include class groups and class numbers of number fields, units, the Kummer–Vandiver conjecture, class number one problem, Diophantine equations, Thue equations, continued fractions, Euclidean number fields, heights, rational torsion points on elliptic curves, cyclotomic numbers, Jacobi sums, and Dedekind zeta values. This book is a valuable resource for undergraduate and graduate students of mathematics as well as researchers interested in class groups of number fields and their connections to other branches of mathematics. New researchers to the field will also benefit immensely from the diverse problems discussed. All the contributing authors are leading academicians, scientists, researchers, and scholars.
Advanced Topics In Computational Number Theory
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Author : Henri Cohen
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-10-29
Advanced Topics In Computational Number Theory written by Henri Cohen 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 2012-10-29 with Mathematics categories.
The computation of invariants of algebraic number fields such as integral bases, discriminants, prime decompositions, ideal class groups, and unit groups is important both for its own sake and for its numerous applications, for example, to the solution of Diophantine equations. The practical com pletion of this task (sometimes known as the Dedekind program) has been one of the major achievements of computational number theory in the past ten years, thanks to the efforts of many people. Even though some practical problems still exist, one can consider the subject as solved in a satisfactory manner, and it is now routine to ask a specialized Computer Algebra Sys tem such as Kant/Kash, liDIA, Magma, or Pari/GP, to perform number field computations that would have been unfeasible only ten years ago. The (very numerous) algorithms used are essentially all described in A Course in Com putational Algebraic Number Theory, GTM 138, first published in 1993 (third corrected printing 1996), which is referred to here as [CohO]. That text also treats other subjects such as elliptic curves, factoring, and primality testing. Itis important and natural to generalize these algorithms. Several gener alizations can be considered, but the most important are certainly the gen eralizations to global function fields (finite extensions of the field of rational functions in one variable overa finite field) and to relative extensions ofnum ber fields. As in [CohO], in the present book we will consider number fields only and not deal at all with function fields.
Number Fields
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Author : Daniel A. Marcus
language : en
Publisher: Springer
Release Date : 2018-07-05
Number Fields written by Daniel A. Marcus and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-07-05 with Mathematics categories.
Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.
Problems On Mapping Class Groups And Related Topics
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Author : Benson Farb
language : en
Publisher: American Mathematical Soc.
Release Date : 2006-09-12
Problems On Mapping Class Groups And Related Topics written by Benson Farb and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-09-12 with Mathematics categories.
The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.
Number Theory
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Author : Henri Cohen
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-17
Number Theory written by Henri Cohen 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 2008-12-17 with Mathematics categories.
This book deals with several aspects of what is now called "explicit number theory." The central theme is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.
Classical Groups And Related Topics
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Author : Alexander Hahn
language : en
Publisher: American Mathematical Soc.
Release Date : 1989
Classical Groups And Related Topics written by Alexander Hahn and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989 with Mathematics categories.
During his lifetime, L. K. Hua played a leading role in and exerted a great influence upon the development in China of modern mathematics, both pure and applied. His mathematical career began in 1931 at Tsinghua University where he continued as a professor for many years. Hua made many significant contributions to number theory, algebra, geometry, complex analysis, numerical analysis, and operations research. In particular, he initiated the study of classical groups in China and developed new matrix methods which, as applied by him as well as his followers, were instrumental in the successful attack of many problems. To honor his memory, a joint China-U.S. conference on Classical Groups and Related Topics was held at Tsinghua University in Beijing in May 1987. This volume represents the proceedings of that conference and contains both survey articles and research papers focusing on classical groups and closely related topics.
Lucant Lmfdb Computation And Number Theory
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Author : John Cremona
language : en
Publisher: American Mathematical Soc.
Release Date : 2024-03-22
Lucant Lmfdb Computation And Number Theory written by John Cremona and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-03-22 with Mathematics categories.
This book will be published Open Access with a Creative Commons Attribution 4.0 International License (CC BY 4.0). The eBook can be downloaded electronically for free. This volume contains the proceedings of the LuCaNT (LMFDB, Computation, and Number Theory) conference held from July 10–14, 2023, at the Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, Rhode Island and affiliated with Brown University. This conference provided an opportunity for researchers, scholars, and practitioners to exchange ideas, share advances, and collaborate in the fields of computation, mathematical databases, number theory, and arithmetic geometry. The papers that appear in this volume record recent advances in these areas, with special focus on the LMFDB (the L-Functions and Modular Forms Database), an online resource for mathematical objects arising in the Langlands program and the connections between them.
Quadratic Number Fields
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Author : Franz Lemmermeyer
language : en
Publisher: Springer Nature
Release Date : 2021-09-18
Quadratic Number Fields written by Franz Lemmermeyer and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09-18 with Mathematics categories.
This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.
Cohomology Of Number Fields
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Author : Jürgen Neukirch
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-09-26
Cohomology Of Number Fields written by Jürgen Neukirch 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 2013-09-26 with Mathematics categories.
This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.