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Contemporary Mathematics 484 CONTEMPORARY MATHEMATICS 484 Spectral Analysis in Geometry and Number Theory International Conference on the Occasion of Toshikazu Sunada's 60th Birthday August 6–10, 2007 Nagoya University Nagoya, Japan Motoko Kotani Hisashi Naito Tatsuya Tate Editors American Mathematical Society http://dx.doi.org/10.1090/conm/484 Spectral Analysis in Geometry and Number Theory Professor Toshikazu Sunada CONTEMPORARY MATHEMATICS 484 Spectral Analysis in Geometry and Number Theory International Conference on the Occasion of Toshikazu Sunada's 60th Birthday August 6–10, 2007 Nagoya University Nagoya, Japan Motoko Kotani Hisashi Naito Tatsuya Tate Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 58J50, 11M36, 37C30; Secondary 35P05, 60J60. Photographs courtesy of Toshikazu Sunada Library of Congress Cataloging-in-Publication Data Spectral analysis in geometry and number theory : a conference in honor of Toshikazu Sunada’s 60th birthday, August 6–10, 2007, Nagoya University, Nagoya, Japan / Motoko Kotani, Hisashi Naito, Tatsuya Tate, editors. p. cm. — (Contemporary mathematics ; v. 484) Includes bibliographical references. ISBN 978-0-8218-4269-0 (alk. paper) 1. Spectral geometry—Congresses. 2. Number theory—Congresses. I. Sunada, T. (Toshikazu), 1948– II. Kotani, Motoko, 1960– III. Naito, Hisashi, 1961– IV. Tate, Tatsuya, 1971– QA614.95.S64 2009 516—dc22 2008046241 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Math- ematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2009 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. 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/ 10987654321 141312111009 Contents Preface vii Acknowledgment ix International Conference x Program of Conference xi A Short Biography and the Work of Professor Sunada Brief Profile of Professor Toshikazu Sunada Atsushi Katsuda and Polly Wee Sy 3 An Overview of Sunada’s Work up to Age 60 Atsushi Katsuda and Polly Wee Sy 7 Articles Sunada’s Isospectrality Technique: Two Decades Later Carolyn Gordon 45 A Central Limit Theorem on Modified Graphs of Nilpotent Covering Graphs Satoshi Ishiwata 59 Hidden Symmetries and Spectrum of the Laplacian on an Indefinite Riemannian Manifold Toshiyuki Kobayashi 73 Spectra of Alternating Hilbert Operators Nobushige Kurokawa and Hiroyuki Ochiai 89 A Liouville Property and its Application to the Laplacian of an Infinite Graph Jun Masamune 103 A Note on Zero-Free Regions for the Derivative of Selberg Zeta Functions Makoto Minamide 117 Chern-Simons Variation and Deligne Cohomology Masanori Morishita and Yuji Terashima 127 Renormalized Rauzy-Veech-Zorich Inductions Takehiko Morita 135 v vi CONTENTS Visualization of Standard Realized Crystal Lattices Hisashi Naito 153 Value Distribution and Distribution of Rational Points Junjiro Noguchi 165 Limiting Distributions for Geodesics Excursions on the Modular Surface Mark Pollicott 177 On the Statistics of the Minimal Solution of a Linear Diophantine Equation and Uniform Distribution of the Real Part of Orbits in Hyperbolic Spaces Morten S. Risager and Zeev´ Rudnick 187 Computations of Spectral Radii on G-Spaces Laurent Saloff-Coste and Wolfgang Woess 195 Lengths, Quasi-Morphisms and Statistics for Free Groups Matthew Horsham and Richard Sharp 219 Semiclassical Asymptotics on Manifolds with Boundary Nilufer Koldan, Igor Prokhorenkov, and Mikhail Shubin 239 On Geometric Analogues of the Birch and Swinnerton-Dyer Conjecture for Low Dimensional Hyperbolic Manifolds Ken-ichi Sugiyama 267 Ray-Singer Zeta Functions for Compact Flat Manifolds Toshikazu Sunada and Hajime Urakawa 287 Bernstein Measures on Convex Polytopes Tatsuya Tate 295 Real and Complex Zeros of Riemannian Random Waves Steve Zelditch 321 Preface This volume is an outgrowth of the International Conference on “Spectral Analysis in Geometry and Number Theory”, a conference in honor of Professor Toshikazu Sunada on the occasion of his sixtieth birthday. It took place at the Nagoya University, Japan from August 6 to August 10, 2007. During the confer- ence, twenty-seven invited talks were presented and over a hundred participants gathered from all over the world to pay tribute to our respected scholar and col- league. Professor Toshikazu Sunada who considers himself as a geometer, has made significant contributions to geometry. Nevertheless, it is also apparent that he has an influence over a wide range of subjects which is evident from the variety of talks presented in the conference and articles written for this volume. He has shed light, in a very novel and unique way, the crucial interactions among differential geometry, number theory, probability theory, discrete analysis, and graph theory. Actually it was his dream, according to him, to provide a geometric model of the class field theory and yield a general method, which is now known as the ”Sunada triples method”, to construct isospectral manifolds in his famous paper “Riemannian coverings and isospectral manifolds”, Annals of Mathematics in 1985. This is just one example of how geometrical ideas are always important in his works, and other examples are explained in the article ”An Overview of Sunada’s Work up to Age 60” in this volume. With his enthusiastic effort to keep on courageously challenging a new field, it is our pleasure to predict that his publication lists would surely continue to grow. Last but not the least, we would like to express our sincere gratitude to all who made this volume possible. Through our editorial work, we become fully aware of the high respect for Professor Sunada from all of the contributors as shown in their carefully prepared presentations for the conference and manuscripts submitted for this volume. Special thanks go to Dr. Atushi Katsuda and Dr. Polly W. Sy who spent many hours writing up the overview of Professor Sunada’s life and work. Motoko Kotani On behalf of the editors. September 7th, 2008 vii Acknowledgment The International Conference on Spectral Analysis in Geometry and Number Theory was held in Nagoya on August 6–10, 2007. This Conference was dedicated to Professor Sunada to honor his accomplishments in mathematics and to celebrate his sixtieth birthday. As can be seen from the Brief Profile given by Katsuda and Sy, Professor Sunada was not 60 years old on the occasion of the Conference. However, by use of the traditional Japanese way of counting one’s age, he was one year old at birth. On his first New Year’s Day, he became two years old. Thereafter, on each passing of New Year’s Day, one year is added to his age. In this way, 2007 was Professor Sunada’s 60th year. The traditional Japanese calendar, which was based on the Chinese calendar, was organized on 60-year cycles. In Japan, a period of 60 years is considered as one life cycle, because it signifies the completion of the cycle of the Zodiac; that is, the 60 combinations of the 12 animal signs and the 10 element signs. The 60th birthday therefore celebrates the point in a human’s life when his personal calendar returns to its starting point. It is also a recognition of his “second infancy”. As today, 7th of September, 2008, marks the first day of a new life cycle for Professor Sunada, I wish for him that, whatever he wants to achieve most in his new life cycle, it comes to him, just in the way he imagined it, or even better. Lastly, on behalf of the Organizing Committee of this Conference, I would like to thank to all the invited speakers and participants who had come to pay tribute to our distinguished scholar and mentor. Moreover, I gratefully make acknowledgment to Ms. Kazuko Kozaki, Prof. Masashi Kubo and other staffs of Graduated School of Mathematics, Nagoya University for their help and support in the coordination and registration for the Conference. I would also like to acknowledge the financial support from Nagoya University, the Japan Association for Mathematical Sciences (JAMS) and the Japan Society for the Promotion of Sciences (JSPS) through Grant- in-Aids for Scientific Researches. Nagoya Toshiaki ADACHI September 7, 2008 Vice Chairman Organizing Committee ix International Conference “Spectral Analysis in Geometry and Number Theory” Aug. 6th - Aug. 10th, 2007 Nagoya University, Japan on the occasion of Toshikazu Sunada’s 60th birthday Organizing Committee Motoko Kotani (Chair: Tohoku University) Toshiaki Adachi (Nagoya Institute of Technology) Yusuke Higuchi (Showa University) Satoshi Ishiwata (Tsukuba University) Atsushi Katsuda (Okayama University) Hisashi Naito (Nagoya Univeristy) Polly Wee Sy (University of the Philippines) Tatsuya Tate (Nagoya Univeristy) Steven Zelditch (Johns Hopkins University) Cosponsor Nagoya University, Japan Association for Mathematical Sciences and the Japan Society for the Promotion of Sciences through Grant-in-Aids for Scientific Researches.
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