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Author: John Coates Publisher: Springer Science & Business Media ISBN: 3540330690 Category : Mathematics Languages : en Pages : 120
Book Description
Written by two leading workers in the field, this brief but elegant book presents in full detail the simplest proof of the "main conjecture" for cyclotomic fields. Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. From the reviews: "The text is written in a clear and attractive style, with enough explanation helping the reader orientate in the midst of technical details." --ZENTRALBLATT MATH
Author: John Coates Publisher: Springer Science & Business Media ISBN: 3540330690 Category : Mathematics Languages : en Pages : 120
Book Description
Written by two leading workers in the field, this brief but elegant book presents in full detail the simplest proof of the "main conjecture" for cyclotomic fields. Its motivation stems not only from the inherent beauty of the subject, but also from the wider arithmetic interest of these questions. From the reviews: "The text is written in a clear and attractive style, with enough explanation helping the reader orientate in the midst of technical details." --ZENTRALBLATT MATH
Author: S. Lang Publisher: Springer ISBN: Category : Mathematics Languages : en Pages : 282
Book Description
Kummer's work on cyclotomic fields paved the way for the development of algebraic number theory in general by Dedekind, Weber, Hensel, Hilbert, Takagi, Artin and others. However, the success of this general theory has tended to obscure special facts proved by Kummer about cyclotomic fields which lie deeper than the general theory. For a long period in the 20th century this aspect of Kummer's work seems to have been largely forgotten, except for a few papers, among which are those by Pollaczek [Po], Artin-Hasse [A-H] and Vandiver [Va]. In the mid 1950's, the theory of cyclotomic fields was taken up again by Iwasawa and Leopoldt. Iwasawa viewed cyclotomic fields as being analogues for number fields of the constant field extensions of algebraic geometry, and wrote a great sequence of papers investigating towers of cyclotomic fields, and more generally, Galois extensions of number fields whose Galois group is isomorphic to the additive group of p-adic integers. Leopoldt concentrated on a fixed cyclotomic field, and established various p-adic analogues of the classical complex analytic class number formulas. In particular, this led him to introduce, with Kubota, p-adic analogues of the complex L-functions attached to cyclotomic extensions of the rationals. Finally, in the late 1960's, Iwasawa [Iw 1 I] . made the fundamental discovery that there was a close connection between his work on towers of cyclotomic fields and these p-adic L-functions of Leopoldt-Kubota.
Author: Takashi Aoki Publisher: Springer Science & Business Media ISBN: 0387249818 Category : Mathematics Languages : en Pages : 228
Book Description
This volume contains papers by invited speakers of the symposium "Zeta Functions, Topology and Quantum Physics" held at Kinki U- versity in Osaka, Japan, during the period of March 3-6, 2003. The aims of this symposium were to establish mutual understanding and to exchange ideas among researchers working in various fields which have relation to zeta functions and zeta values. We are very happy to add this volume to the series Developments in Mathematics from Springer. In this respect, Professor Krishnaswami Alladi helped us a lot by showing his keen and enthusiastic interest in publishing this volume and by contributing his paper with Alexander Berkovich. We gratefully acknowledge financial support from Kinki University. We would like to thank Professor Megumu Munakata, Vice-Rector of Kinki University, and Professor Nobuki Kawashima, Director of School of Interdisciplinary Studies of Science and Engineering, Kinki Univ- sity, for their interest and support. We also thank John Martindale of Springer for his excellent editorial work.
Author: Lawrence C. Washington Publisher: Springer Science & Business Media ISBN: 1468401335 Category : Mathematics Languages : en Pages : 401
Book Description
This book grew. out of lectures given at the University of Maryland in 1979/1980. The purpose was to give a treatment of p-adic L-functions and cyclotomic fields, including Iwasawa's theory of Zp-extensions, which was accessible to mathematicians of varying backgrounds. The reader is assumed to have had at least one semester of algebraic number theory (though one of my students took such a course concurrently). In particular, the following terms should be familiar: Dedekind domain, class number, discriminant, units, ramification, local field. Occasionally one needs the fact that ramification can be computed locally. However, one who has a good background in algebra should be able to survive by talking to the local algebraic number theorist. I have not assumed class field theory; the basic facts are summarized in an appendix. For most of the book, one only needs the fact that the Galois group of the maximal unramified abelian extension is isomorphic to the ideal class group, and variants of this statement. The chapters are intended to be read consecutively, but it should be possible to vary the order considerably. The first four chapters are basic. After that, the reader willing to believe occasional facts could probably read the remaining chapters randomly. For example, the reader might skip directly to Chapter 13 to learn about Zp-extensions. The last chapter, on the Kronecker-Weber theorem, can be read after Chapter 2.
Author: Lawrence C. Washington Publisher: Springer Science & Business Media ISBN: 1461219345 Category : Mathematics Languages : en Pages : 504
Book Description
This text on a central area of number theory covers p-adic L-functions, class numbers, cyclotomic units, Fermat’s Last Theorem, and Iwasawa’s theory of Z_p-extensions. This edition contains a new chapter on the work of Thaine, Kolyvagin, and Rubin, including a proof of the Main Conjecture, as well as a chapter on other recent developments, such as primality testing via Jacobi sums and Sinnott’s proof of the vanishing of Iwasawa’s f-invariant.
Author: John Coates Publisher: Cambridge University Press ISBN: 1316241300 Category : Mathematics Languages : en Pages : 317
Book Description
There are still many arithmetic mysteries surrounding the values of the Riemann zeta function at the odd positive integers greater than one. For example, the matter of their irrationality, let alone transcendence, remains largely unknown. However, by extending ideas of Garland, Borel proved that these values are related to the higher K-theory of the ring of integers. Shortly afterwards, Bloch and Kato proposed a Tamagawa number-type conjecture for these values, and showed that it would follow from a result in motivic cohomology which was unknown at the time. This vital result from motivic cohomology was subsequently proven by Huber, Kings, and Wildeshaus. Bringing together key results from K-theory, motivic cohomology, and Iwasawa theory, this book is the first to give a complete proof, accessible to graduate students, of the Bloch–Kato conjecture for odd positive integers. It includes a new account of the results from motivic cohomology by Huber and Kings.
Author: Minking Eie Publisher: World Scientific ISBN: 9814472646 Category : Mathematics Languages : en Pages : 313
Book Description
This is the first book on the theory of multiple zeta values since its birth around 1994. Readers will find that the shuffle products of multiple zeta values are applied to complicated counting problems in combinatorics, producing numerous interesting identities that are ready to be used. This will provide a powerful tool to deal with problems in multiple zeta values, both in evaluations and shuffle relations. The volume will benefit graduate students doing research in number theory.
Author: John Coates
Author: John Coates
Author: S. Lang
Author: Robert Spira
Author: Takashi Aoki
Author: Dylan Attwell-Duval
Author: Lawrence C. Washington
Author: Lawrence C. Washington
Author: 
Author: John Coates
Author: Minking Eie