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An Introduction to the Theory of Local Zeta Functions AMS/IP Studies in Advanced Mathematics S.-T. Yau, Series Editor An Introduction to the Theory of Local Zeta Functions Jun-ichi Igusa American Mathematical Society • International Press Selected Titles in This Series 14 Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, 2000 13 Vasilios Alexiades and George Siopsis, Editors, Trends in Mathematical Physics, 1999 12 Sheng Gong, The Bieberbach Conjecture, 1999 11 Shinichi Mochizuki, Foundations of p-adic Teichm¨uller Theory, 1999 10 Duong H. Phong, Luc Vinet, and Shing-Tung Yau, Editors, Mirror Symmetry III, 1999 9 Shing-Tung Yau, Editor, Mirror Symmetry I, 1998 8 J¨urgen Jost, Wilfrid Kendall, Umberto Mosco, Michael R¨ockner, and Karl-Theodor Sturm, New Directions in Dirichlet Forms, 1998 7 D. A. Buell and J. T. Teitelbaum, Editors, Computational Perspectives on Number Theory, 1998 6 Harold Levine, Partial Differential Equations, 1997 5 Qi-keng Lu, Stephen S.-T. Yau, and Anatoly Libgober, Editors, Singularities and Complex Geometry, 1997 4 Vyjayanthi Chari and Ivan B. Penkov, Editors, Modular Interfaces: Modular Lie Algebras, Quantum Groups, and Lie Superalgebras, 1997 3 Xia-Xi Ding and Tai-Ping Liu, Editors, Nonlinear Evolutionary Partial Differential Equations, 1997 2.2 William H. Kazez, Editor, Geometric Topology, 1997 2.1 William H. Kazez, Editor, Geometric Topology, 1997 1 B. Greene and S.-T. Yau, Editors, Mirror Symmetry II, 1997 An Introduction to the Theory of Local Zeta Functions https://doi.org/10.1090/amsip/014 AMS/IP Studies in Advanced Mathematics Volume 14 An Introduction to the Theory of Local Zeta Functions Jun-ichi Igusa American Mathematical Society • International Press Shing-Tung Yau, Managing Editor 2000 Mathematics Subject Classification. Primary 11Sxx, 11S40, 11Mxx, 11Gxx, 14Gxx. Library of Congress Cataloging-in-Publication Data Igusa, Jun-ichi, 1924– An introduction to the theory of local zeta functions / Jun-ichi Igusa. p. cm. — (AMS/IP studies in advanced mathematics, ISSN 1089-3288 ; v. 14) Includes bibliographical references and index. ISBN 0-8218-2015-X (hard cover; alk. paper) ISBN 978-0-8218-2907-3 (soft cover; alk. paper) 1. Functions, Zeta. I. Title. II. Series. QA351 .I38 2000 515.56–dc21 99-087031 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. c 2000 by the American Mathematical Society and International Press. All rights reserved. Reprinted by the American Mathematical Society, 2007. The American Mathematical Society and International Press retain all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ Visit the International Press home page at URL: http://www.intlpress.com/ 10987654321 121110090807 Contents 1 Preliminaries 1 1.1Reviewofsomebasictheorems................... 1 1.2Noetherianrings........................... 5 1.3Hilbert’stheorems.......................... 8 2 Implicit function theorems and K-analytic manifolds 15 2.1Implicitfunctiontheorem...................... 15 2.2Implicitfunctiontheorem(non-archimedeancase)........ 21 2.3Weierstrasspreparationtheorem.................. 24 2.4K-analyticmanifoldsanddifferentialforms............ 28 2.5Criticalsetsandcriticalvalues................... 32 3 Hironaka’s desingularization theorem 35 3.1Monoidaltransformations...................... 35 3.2 Hironaka’s desingularization theorem (analyticform)............................ 38 3.3Desingularizationofplanecurves.................. 40 4 Bernstein’s theory 45 4.1 Bernstein’s polynomial bf (s)..................... 45 4.2 Some properties of bf (s)....................... 47 4.3Reductionoftheproof........................ 49 4.4 A general theorem on D-modules.................. 52 4.5Completionoftheproof....................... 55 5 Archimedean local zeta functions 59 5.1 The group Ω(K×) .......................... 59 5.2 Schwartz space S(Kn)........................ 61 5.3 Local zeta function ZΦ(ω)...................... 67 5.4 Complex power ω(f)viadesingularization............. 73 5.5Anapplication............................ 77 6 Prehomogeneous vector spaces 83 6.1 Sato’s b-function b(s)......................... 83 6.2TheΓ-function(adigression).................... 87 6.3 b(s)=bf (s)andtherationalityofthezeros............ 91 vii viii CONTENTS 7 Totally disconnected spaces and p-adic manifolds 97 7.1Distributionsintotallydisconnectedspaces............ 97 7.2Thecaseofhomogeneousspaces..................101 7.3Structureofeigendistributions...................106 7.4 Integration on p-adicmanifolds...................108 7.5 Serre’s theorem on compact p-adicmanifolds...........113 7.6Integrationoverthefibers......................114 8 Local zeta functions (p-adic case) 117 8.1 Selfduality of K andsomelemmas.................117 8.2 p-adic zeta function ZΦ(ω)......................120 ∗ ∗ 8.3 Weil’s functions FΦ(i)andFΦ(i )..................125 8.4 Relation of FΦ(i)andZΦ(ω) ....................129 8.5 Poles of ω(f) for a group invariant f ................134 9 Some homogeneous polynomials 137 9.1QuadraticformsandWitt’stheorem................137 9.2Quadraticformsoverfinitefields..................141 9.3Classicalgroupsoverfinitefields..................145 9.4CompositionandJordanalgebras..................149 9.5NormformsandFreudenthalquartics...............154 9.6Gauss’identityanditscorollaries..................160 10 Computation of Z(s) 163 10.1 Z(ω)insomesimplecases......................163 10.2 A p-adicstationaryphaseformula.................167 10.3Akeylemma.............................173 10.4 Z(s)foraFreudenthalquartic...................178 10.5 Z(s) for the Gramian det(txhx)...................184 10.6Anintegrationformula........................188 10.7 Z(s)fordet(txhx)inproductforms................193 11 Theorems of Denef and Meuser 199 11.1Regularlocalrings..........................199 11.2Geometriclanguage.........................202 11.3 Hironaka’s desingularization theorem (algebraicform)............................205 11.4 Weil’s zeta functions over finite fields ................210 11.5 Degree of Z(s) ............................214 11.6 The field Ke (adigression)......................217 11.7 Functional equation of Z(s).....................221 Bibliography 227 Index 231 INTRODUCTION Local zeta functions are relatively new mathematical objects. The first general theorems were proved from 1968 to 1973. Since then, especially during the last fifteen years, remarkable results have been obtained, allowing one to call the accu- mulation a “theory.” Nevertheless, there remain several challenging problems whose solution will make the theory much richer. The purpose of this book is to introduce the readers to this theory. The book is written in such a way that it should be appropriate for those who have mastered the “basic courses” taught in America for first year graduate students. Assuming the reader has this background, nearly all material will be explained with detailed definitions and proofs. There are, however, two exceptions. We shall use Hironaka’s desingularization theorem and the func- tional equations of Weil’s zeta functions over finite fields. We shall explain these theorems by examples so that the readers can accept them with some understand- ing. The references are given primarily to indicate our indebtedness to the authors and not for the readers to consult. Since local zeta functions are new, we shall define them briefly with details given in the text. If k is a number field, any completion of k is called a local field. Every local field K carries a Haar measure, and the rate of measure change under the × multiplication by a in K = K\{0} defines its absolute value |a|K ; it is completed n by |0|K =0.IfnowX = K for some n ≥ 1, f is a K-valued non-constant polynomial function on X, and Φ is in the Schwartz-Bruhat space S(x)ofX,then | |s ZΦ(s)= f(x) K Φ(x) dx, Re(s) > 0 X is called a local zeta function. If Φ is the standard function on X, i.e., exp(−πtxx) R − t C n for K = ,exp( 2π xx¯)forK = , and the characteristic function of OK for a p-adic field K with OK as its maximal compact subring, then we drop the subscript Φ. Furthermore, we normalize the Haar measure dx on X so that Z(s) tends to → · |·|s C 1ass 0. The set of ωs( )= K for all s in forms the identity component ix x INTRODUCTION of the group Ω(K×) of all continuous homomorphisms ω from K× to C×.By replacing ωs by ω we get a more general local zeta function ZΦ(ω). In view of the fact that |ω(·)| = ωσ(ω)(·) for a unique σ(ω)inR satisfying σ(ωs)=Re(s), we define the right-half plane in Ω(K×)byσ(ω) > 0. Then the first general theorems are as follows: ZΦ(ω), which is clearly holomorphic on the right-half plane, has a meromorphic continuation to the whole Ω(K×). Furthermore, in
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