CONTEMPORARY MATHEMATICS 484

Spectral Analysis in Geometry and

International Conference on the Occasion of Toshikazu Sunada's 60th Birthday August 6–10, 2007 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: ) 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.

x Program of Conference

August 6 (Mon) 10:00–10:50 M. Shubin (Northeastern University) KdV, mKdV and eigenfunctions of Schr¨odinger operators 11:00–11:50 T. Kobayashi () On a conjecture of Sunada and theory of discretely decomposable restrictions of unitary representations 13:30–14:20 M. Kanai (Nagoya University) Rigidity of the Weyl chamber flow, and the vanishing theorems of Weil and Matsushima 14:30–15:20 H. Moriyoshi (Keio University) Twisted Index Theorem and its geometric applications 15:50–16:20 J. Masamune (Worcester Polytechnic Institute) Conservative principle of differential forms 16:30–17:20 P. Buser (EPF Lausanne) Pictures of Riemann surfaces August 7 (Tue) 10:00–10:50 C. Gordon (Dartmouth College) Sunada’s Isospectrality Technique: Two Decades Later 11:00–11:50 J. Noguchi (University of Tokyo) Value distribution and distribution of rational points 13:30–14:20 N. Kurokawa (Tokyo Institute of Technology) Spectra of Alternating Hilbert operators 14:30–15:20 A. Katsuda (Okayama University) An overview of Sunada’s works 15:40–16:30 T. Sunada () Abel-Jacobi maps in graph theory August 8 (Wed) 10:00–10:50 L. Saloff-Coste (Cornell University) Property RD and random walks 11:00–11:50 K. Sugiyama (Chiba University) A geometric analog of the Iwasawa conjecture for a hyperbolic threefold 13:30–14:20 W. Woess (Graz university of Technology) Horocyclic products of trees and of hyperbolic spaces

xi xii PROGRAM OF CONFERENCE

14:30–15:20 M. Morishita (Kyushu University) Chern-Simons variation and Hida theory 15:50–16:20 I. Suan (University of the Philippines) Non-uniform Bounds in Discretized Normal Approximation 16:30–17:20 P. Kuchment (Texas A&M University) Liouville theorems on abelian coverings of graphs and manifolds. August 9 (Thu) 10:00–10:50 J. Br¨uning (Humboldt University) Boundary value problems for Dirac operators 11:00–11:50 M. Pollicott (Warwick University) The dynamical approach to geometric zeta functions 13:30–14:50 T. Morita (Hiroshima University) Renormalized RVZ inductions 14:30–14:50 M. Minamide (Nagoya University) The zero-free region of the derivative of Selberg zeta functions 14:50–15:10 L. Faina (University of the Philippines) Qualitative Analysis of A Three-Dimensional, Single-Strain HIV Model 16:00–16:30 M. Rey (University of the Philippines) Poisson Approximation for Unbounded Functions 16:30–17:00 S. Ishiwata (Tsukuba University) Random walks on nilpotent covering graphs August 10 (Fri) 10:00–10:50 Z. Rudnick (Tel Aviv University) Nodal lines of eigenfunctions of the Laplacian on the torus 11:00–11:30 Y. Miyanishi (The University of the Air) Some remarks on the behavior of semiclassical eigenfunctions with singular potential 11:40–12:30 S. Zelditch (Johns Hopkins University) Geodesic flow and nodal lines of eigenfunctions Titles in This Series

484 Motoko Kotani, Hisashi Naito, and Tatsuya Tate, Editors, Spectral analysis in geometry and number theory, 2009 483 Vyacheslav Futorny, Victor Kac, Iryna Kashuba, and Efim Zelmanov, Editors, Algebras, representations and applications, 2009 482 Kazem Mahdavi and Deborah Koslover, Editors, Advances in quantum computation, 2009 481 Aydın Aytuna, Reinhold Meise, Tosun Terzio˘glu, and Dietmar Vogt, Editors, Functional analysis and complex analysis, 2009 480 Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, and Pramod Kanwar, Editors, Rings, modules and representations, 2008 479 Timothy Y. Chow and Daniel C. Isaksen, Editors, Communicating mathematics, 2008 478 Zongzhu Lin and Jianpan Wang, Editors, Representation theory, 2008 477 Ignacio Luengo, Editor, Recent Trends in Cryptography, 2008 476 Carlos Villegas-Blas, Editor, Fourth summer school in analysis and mathematical physics: Topics in spectral theory and quantum mechanics, 2008 475 Jean-Paul Brasselet, Jos´e Luis Cisneros-Molina, David Massey, Jos´eSeade, and Bernard Teissier, Editors, Singularities II: Geometric and topological aspects, 2008 474 Jean-Paul Brasselet, Jos´e Luis Cisneros-Molina, David Massey, Jos´eSeade, and Bernard Teissier, Editors, Singularities I: Algebraic and analytic aspects, 2008 473 Alberto Farina and Jean-Claude Saut, Editors, Stationary and time dependent Gross-Pitaevskii equations, 2008 472 James Arthur, Wilfried Schmid, and Peter E. Trapa, Editors, Representation Theory of Real Reductive Lie Groups, 2008 471 Diego Dominici and Robert S. Maier, Editors, Special functions and orthogonal polynomials, 2008 470 Luise-Charlotte Kappe, Arturo Magidin, and Robert Fitzgerald Morse, Editors, Computational group theory and the theory of groups, 2008 469 Keith Burns, Dmitry Dolgopyat, and Yakov Pesin, Editors, Geometric and probabilistic structures in dynamics, 2008 468 Bruce Gilligan and Guy J. Roos, Editors, Symmetries in complex analysis, 2008 467 Alfred G. No¨el, Donald R. King, Gaston M. N’Gu´er´ekata, and Edray H. Goins, Editors, Council for African American researchers in the mathematical sciences: Volume V, 2008 466 Boo Cheong Khoo, Zhilin Li, and Ping Lin, Editors, Moving interface problems and applications in fluid dynamics, 2008 465 Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, and Elham Izadi, Editors, Curves and Abelian varieties, 2008 464 Gestur Olafsson,´ Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, Peter R. Massopust, Eric Todd Quinto, and Boris Rubin, Editors, Radon transforms, geometry, and wavelets, 2008 463 Kristin E. Lauter and Kenneth A. Ribet, Editors, Computational arithmetic geometry, 2008 462 Giuseppe Dito, Hugo Garc´ıa-Compe´an, Ernesto Lupercio, and Francisco J. Turrubiates, Editors, Non-commutative geometry in mathematics and physics, 2008 461 Gary L. Mullen, Daniel Panario, and Igor Shparlinski, Editors, Finite fields and applications, 2008 460 Megumi Harada, Yael Karshon, Mikiya Masuda, and Taras Panov, Editors, Toric topology, 2008 459 Marcelo J. Saia and Jos´e Seade, Editors, Real and complex singularities, 2008 TITLES IN THIS SERIES

458 Jinho Baik, Thomas Kriecherbauer, Luen-Chau Li, Kenneth D. T-R McLaughlin, and Carlos Tomei, Editors, Integrable systems and random matrices, 2008 457 Tewodros Amdeberhan and Victor H. Moll, Editors, Tapas in experimental mathematics, 2008 456 S. K. Jain and S. Parvathi, Editors, Noncommutative rings, group rings, diagram algebras and their applications, 2008 455 Mark Agranovsky, Daoud Bshouty, Lavi Karp, Simeon Reich, David Shoikhet, and Lawrence Zalcman, Editors, Complex analysis and dynamical systems III, 2008 454 Rita A. Hibschweiler and Thomas H. MacGregor, Editors, Banach spaces of analytic functions, 2008 453 Jacob E. Goodman, J´anos Pach, and Richard Pollack, Editors, Surveys on Discrete and Computational Geometry–Twenty Years Later, 2008 452 Matthias Beck, Christian Haase, Bruce Reznick, Mich`ele Vergne, Volkmar Welker, and Ruriko Yoshida, Editors, Integer points in polyhedra, 2008 451 David R. Larson, Peter Massopust, Zuhair Nashed, Minh Chuong Nguyen, Manos Papadakis, and Ahmed Zayed, Editors, Frames and operator theory in analysis and signal processing, 2008 450 Giuseppe Dito, Jiang-Hua Lu, Yoshiaki Maeda, and Alan Weinstein, Editors, Poisson geometry in mathematics and physics, 2008 449 Robert S. Doran, Calvin C. Moore, and Robert J. Zimmer, Editors, Group representations, ergodic theory, and mathematical physics: A tribute to George W. Mackey, 2007 448 Alberto Corso, Juan Migliore, and Claudia Polini, Editors, Algebra, geometry and their interactions, 2007 447 Fran¸cois Germinet and Peter Hislop, Editors, Adventures in mathematical physics, 2007 446 Henri Berestycki, Michiel Bertsch, Felix E. Browder, Louis Nirenberg, Lambertus A. Peletier, and Laurent V´eron, Editors, Perspectives in Nonlinear Partial Differential Equations, 2007 445 Laura De Carli and Mario Milman, Editors, Interpolation Theory and Applications, 2007 444 Joseph Rosenblatt, Alexander Stokolos, and Ahmed I. Zayed, Editors, Topics in harmonic analysis and ergodic theory, 2007 443 Joseph Stephen Verducci and Xiaotong Shen, Editors, Prediction and discovery, 2007 442 Yi-Zhi Huang and Kailash C Misra, Editors, Lie algebras, vertex operator algbras and their applications, 2007 441 Louis H. Kauffman, David E. Radford, and Fernando J. O. Souza, Editors, Hopf algebras and generalizations, 2007 440 Fernanda Botelho, Thomas Hagen, and James Jamison, Editors, Fluids and Waves, 2007 439 Donatella Danielli, Editor, Recent developments in nonlinear partial differential equations, 2007 438 Marc Burger, Michael Farber, Robert Ghrist, and Daniel Koditschek, Editors, Topology and robotics, 2007

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/. This volume is an outgrowth of an international conference in honor of Toshikazu Sunada, on the occasion of his sixtieth birthday, which took place at Nagoya University, Japan, in 2007. Sunada’s research covers a wide spectrum of spectral analysis, including interactions among geometry, number theory, dynamical systems, probability theory and mathematical physics. Readers will find papers on trace formulae, isospectral problems, zeta functions, quantum ergodicity, random waves, discrete geometric analysis, value distribution, and semiclassical analysis. This volume also contains an article that presents an overview of Sunada’s work in mathematics up to the age of sixty.

CONM/484 AMS on the Web www.ams.org