E-Book Overview
This book, together with the companion volume, Fermat's Last Theorem: The proof, presents in full detail the proof of Fermat's Last Theorem given by Wiles and Taylor. With these two books, the reader will be able to see the whole picture of the proof to appreciate one of the deepest achievements in the history of mathematics. Crucial arguments, including the so-called - trick, theorem, etc., are explained in depth. The proof relies on basic background materials in number theory and arithmetic geometry, such as elliptic curves, modular forms, Galois representations, deformation rings, modular curves over the integer rings, Galois cohomology, etc. The first four topics are crucial for the proof of Fermat's Last Theorem; they are also very important as tools in studying various other problems in modern algebraic number theory. The remaining topics will be treated in the second book to be published in the same series in 2014. In order to facilitate understanding the intricate proof, an outline of the whole argument is described in the first preliminary chapter, and more details are summarized in later chapters.
E-Book Content
Fermat's Last Theorem Basic Tools
Translations of
MATHEMATICAL MONOGRAPHS
Volume 243
Fermat's Last Theorem Basic Tools Takeshi Saito Translated from the Japanese by Masato Kuwata
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American Mathematical Society Providence, Rhode Island
FERUMA YOSO (Fermat Conjecture)
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by Takeshi Saito
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© 2009 by Takeshi Saito First published 2009 by lwanami Shoten, Publishers, Tokyo. T his English language edition published in 2013 by the American Mathematical Society, Providence by arrangement with the author c/o lwanami Shoten, Publishers, Tokyo Translated from the Japanese by Masato Kuwata 2010
Mathematics Subject Classification.
Primary 11D41;
Secondary 11G05, llFll, 11F80, 11Gl8.
Library of Congress Cataloging-in-Publication Data Saito, Takeshi , 196 1Fermat's last theorem: basic tools / Takeshi Saito ; translated by Masato Kuwata.-English language edition. pages cm.- ( Translations of mathematical monographs ; volume 243 ) First published by Iwanami Shoten, Publishers , Tokyo, 2009 . Includes bibliographical references and index. ISBN 978-0-82 18-9848-2 ( alk. paper ) 1. Fermat's last theorem. 2. Number theory. 3. Algebraic number theory. I. Title . II . Title: Fermat's last theorem: basic tools .
Q A244.S2513 2013 5 12.7'4-dc23
2013023932
©
2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America.
i§ The paper used in this book
is acid-free and falls within the guidelines
established to ensure permanence and durability.
Information on copying and reprinting can be found in the back of this volume. Visit the AMS home page at http: I /wW'il. ams. org/ 10 9 8 7 6 5 4 3 2 1
18 17 16 15 14 13
Contents
Preface
ix
Preface to the English Edition
xv
Chapter 0. Synopsis 0. 1 . Simple paraphrase 0.2. Elliptic curves 0.3. Elliptic curves and modular forms 0.4. Conductor of an elliptic curve and level of a modular form 0.5. £-torsion points of elliptic curves and modular forms
1 1 3 5 7 9
Chapter l . Elliptic curves 1 . 1 . Elliptic curves over a field 1 . 2 . Reduction mod p 1.3. Morphisms and the Tate modules 1 .4. Elliptic curves over an arbitrary scheme 1.5. Generalized elliptic curves
13 13 15 22 26 29
Chapter 2. Modular forms 2 . 1 . The j-invariant 2.2. Moduli spac