Cyclotomic Fields and Zeta Values
Cyclotomic fields have always occupied a central place in number theory, and the so called "main conjecture" on cyclotomic fields is arguably the deepest and most beautiful theorem known about them. It is also the simplest example of a vast array of subse
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Springer Monographs in Mathematics
Cyclotomic Fields and Zeta Values
Springer Monographs in Mathematics
J. Coates . R. Sujatha
Cy clotom ic Field s and Zeta V alues
123
J. C oates Centre for Mathematical Sciences DPMMS Wilberforce Road Cambridge, CB3 0WB, England e-mail: J.H .C [email protected] R . S ujatha School of Mathematics Tata Institute of F undamental Research Homi Bhabha Road, Colaba Mumbai 400 005, India e-mail: [email protected]
Library of Congress Control Number: 2006927549
Mathematics Subject Classification (2000): 11R18, 11R23
ISSN 1439-7382 ISBN-10 3-540-33068-2 Springer Berlin Heidelberg New York ISBN-13 978-3-540-33068-4 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. A X macro package Typesetting by the authors and SPi using a Springer LT E Cover design: Erich Kirchner, Heidelberg, Germany
Printed on acid-free paper
SPIN: 11685852
41/3100/SPi - 5 4 3 2 1 0
Preface
Chihayaburu Kami no igaki ni Hau kuzu mo Aki ni wa aezu Utsuroinikeri
Mighty they are The gods within this sacred shrineYet even the vines Creeping in the precincts could not hold Against the autumn’s tingeing of their leaves. – Ki no Tsurayaki (Kokinshu, V : 262).
This little book is intended for graduate students and the non-expert in Iwasawa theory. Its aim is to present in full detail the simplest proof of the important theorem on cyclotomic fields, which is often called “the main conjecture”. We have thought it worthwhile to write such a book, not only because this theorem is arguably the deepest and most beautiful known result about the arithmetic of cyclotomic fields, but also because it is the simplest example of a vast array of subsequent, unproven “main conjectures” in modern arithmetic geometry involving the arithmetic behaviour of motives over p-adic Lie extensions of number fields (see [CFKSV]). These main conjectures are concerned with what one might loosely call the exact formulae of number theory which conjecturally link the special values of zeta and L-functions to purely arithmetic expressions (the most celebrated example being the conjecture of Birch and
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