CONTEMPORARYMATHEMATICS

American Mathematical Society

484

Spectral Analysis in Geometry and Number TheoryInternational 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

Spectral Analysis in Geometry and Number Theory

http://dx.doi.org/10.1090/conm/484

Professor Toshikazu Sunada

American Mathematical SocietyProvidence, Rhode Island

CONTEMPORARYMATHEMATICS

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

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’s60th birthday, August 6–10, 2007, Nagoya University, Nagoya, Japan / Motoko Kotani, HisashiNaito, 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 2009516—dc22 2008046241

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10 9 8 7 6 5 4 3 2 1 14 13 12 11 10 09

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 SunadaAtsushi Katsuda and Polly Wee Sy 3

An Overview of Sunada’s Work up to Age 60Atsushi Katsuda and Polly Wee Sy 7

Articles

Sunada’s Isospectrality Technique: Two Decades LaterCarolyn Gordon 45

A Central Limit Theorem on Modified Graphs of Nilpotent Covering GraphsSatoshi Ishiwata 59

Hidden Symmetries and Spectrum of the Laplacian on an IndefiniteRiemannian Manifold

Toshiyuki Kobayashi 73

Spectra of Alternating Hilbert OperatorsNobushige Kurokawa and Hiroyuki Ochiai 89

A Liouville Property and its Application to the Laplacian of an Infinite GraphJun Masamune 103

A Note on Zero-Free Regions for the Derivative of Selberg Zeta FunctionsMakoto Minamide 117

Chern-Simons Variation and Deligne CohomologyMasanori Morishita and Yuji Terashima 127

Renormalized Rauzy-Veech-Zorich InductionsTakehiko Morita 135

v

vi CONTENTS

Visualization of Standard Realized Crystal LatticesHisashi Naito 153

Value Distribution and Distribution of Rational PointsJunjiro Noguchi 165

Limiting Distributions for Geodesics Excursions on the Modular SurfaceMark Pollicott 177

On the Statistics of the Minimal Solution of a Linear Diophantine Equationand Uniform Distribution of the Real Part of Orbits in Hyperbolic Spaces

Morten S. Risager and Zeév Rudnick 187

Computations of Spectral Radii on G-SpacesLaurent Saloff-Coste and Wolfgang Woess 195

Lengths, Quasi-Morphisms and Statistics for Free GroupsMatthew Horsham and Richard Sharp 219

Semiclassical Asymptotics on Manifolds with BoundaryNilufer Koldan, Igor Prokhorenkov, and Mikhail Shubin 239

On Geometric Analogues of the Birch and Swinnerton-Dyer Conjecture forLow Dimensional Hyperbolic Manifolds

Ken-ichi Sugiyama 267

Ray-Singer Zeta Functions for Compact Flat ManifoldsToshikazu Sunada and Hajime Urakawa 287

Bernstein Measures on Convex PolytopesTatsuya Tate 295

Real and Complex Zeros of Riemannian Random WavesSteve Zelditch 321

Preface

This volume is an outgrowth of the International Conference on “SpectralAnalysis in Geometry and Number Theory”, a conference in honor of ProfessorToshikazu Sunada on the occasion of his sixtieth birthday. It took place at theNagoya University, Japan from August 6 to August 10, 2007. During the confer-ence, twenty-seven invited talks were presented and over a hundred participantsgathered 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 madesignificant contributions to geometry. Nevertheless, it is also apparent that he hasan influence over a wide range of subjects which is evident from the variety oftalks presented in the conference and articles written for this volume. He has shedlight, in a very novel and unique way, the crucial interactions among differentialgeometry, number theory, probability theory, discrete analysis, and graph theory.Actually it was his dream, according to him, to provide a geometric model ofthe 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 Workup to Age 60” in this volume. With his enthusiastic effort to keep on courageouslychallenging a new field, it is our pleasure to predict that his publication lists wouldsurely continue to grow.

Last but not the least, we would like to express our sincere gratitude to all whomade this volume possible. Through our editorial work, we become fully aware ofthe high respect for Professor Sunada from all of the contributors as shown in theircarefully prepared presentations for the conference and manuscripts submitted forthis volume. Special thanks go to Dr. Atushi Katsuda and Dr. Polly W. Sy whospent many hours writing up the overview of Professor Sunada’s life and work.

Motoko KotaniOn behalf of the editors.

September 7th, 2008

vii

Acknowledgment

The International Conference on Spectral Analysis in Geometry and NumberTheory was held in Nagoya on August 6–10, 2007. This Conference was dedicatedto Professor Sunada to honor his accomplishments in mathematics and to celebratehis sixtieth birthday.

As can be seen from the Brief Profile given by Katsuda and Sy, ProfessorSunada was not 60 years old on the occasion of the Conference. However, by useof 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 passingof New Year’s Day, one year is added to his age. In this way, 2007 was ProfessorSunada’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 asone life cycle, because it signifies the completion of the cycle of the Zodiac; thatis, the 60 combinations of the 12 animal signs and the 10 element signs. The 60thbirthday therefore celebrates the point in a human’s life when his personal calendarreturns 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 forProfessor Sunada, I wish for him that, whatever he wants to achieve most in hisnew 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 liketo thank to all the invited speakers and participants who had come to pay tribute toour distinguished scholar and mentor. Moreover, I gratefully make acknowledgmentto Ms. Kazuko Kozaki, Prof. Masashi Kubo and other staffs of Graduated Schoolof Mathematics, Nagoya University for their help and support in the coordinationand registration for the Conference. I would also like to acknowledge the financialsupport 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 ADACHISeptember 7, 2008 Vice Chairman

Organizing Committee

ix

x

International Conference

“Spectral Analysis in Geometry and Number Theory”

Aug. 6th - Aug. 10th, 2007Nagoya 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 forScientific Researches.

Program of Conference

August 6 (Mon)

10:00–10:50 M. Shubin (Northeastern University)KdV, mKdV and eigenfunctions of Schrödinger operators

11:00–11:50 T. Kobayashi (University of Tokyo)On a conjecture of Sunada and theory of discretely decomposable restrictions ofunitary 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 (Meiji University)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. Brüning (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 ingeometry 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 quantumcomputation, 2009

481 Aydın Aytuna, Reinhold Meise, Tosun Terzioğlu, and Dietmar Vogt, Editors,

Functional analysis and complex analysis, 2009

480 Nguyen Viet Dung, Franco Guerriero, Lakhdar Hammoudi, and PramodKanwar, 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 mathematicalphysics: Topics in spectral theory and quantum mechanics, 2008

475 Jean-Paul Brasselet, José Luis Cisneros-Molina, David Massey, José Seade,and Bernard Teissier, Editors, Singularities II: Geometric and topological aspects, 2008

474 Jean-Paul Brasselet, José Luis Cisneros-Molina, David Massey, José Seade,and Bernard Teissier, Editors, Singularities I: Algebraic and analytic aspects, 2008

473 Alberto Farina and Jean-Claude Saut, Editors, Stationary and time dependentGross-Pitaevskii equations, 2008

472 James Arthur, Wilfried Schmid, and Peter E. Trapa, Editors, RepresentationTheory of Real Reductive Lie Groups, 2008

471 Diego Dominici and Robert S. Maier, Editors, Special functions and orthogonalpolynomials, 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. Noël, Donald R. King, Gaston M. N’Guérékata, and Edray H. Goins,Editors, Council for African American researchers in the mathematical sciences: VolumeV, 2008

466 Boo Cheong Khoo, Zhilin Li, and Ping Lin, Editors, Moving interface problems andapplications in fluid dynamics, 2008

465 Valery Alexeev, Arnaud Beauville, C. Herbert Clemens, and Elham Izadi,Editors, Curves and Abelian varieties, 2008

464 Gestur Ólafsson, Eric L. Grinberg, David Larson, Palle E. T. Jorgensen, PeterR. Massopust, Eric Todd Quinto, and Boris Rubin, Editors, Radon transforms,geometry, and wavelets, 2008

463 Kristin E. Lauter and Kenneth A. Ribet, Editors, Computational arithmeticgeometry, 2008

462 Giuseppe Dito, Hugo Garćıa-Compeán, 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 andapplications, 2008

460 Megumi Harada, Yael Karshon, Mikiya Masuda, and Taras Panov, Editors,Toric topology, 2008

459 Marcelo J. Saia and José Seade, Editors, Real and complex singularities, 2008

TITLES IN THIS SERIES

458 Jinho Baik, Thomas Kriecherbauer, Luen-Chau Li, Kenneth D. T-RMcLaughlin, and Carlos Tomei, Editors, Integrable systems and random matrices,2008

457 Tewodros Amdeberhan and Victor H. Moll, Editors, Tapas in experimentalmathematics, 2008

456 S. K. Jain and S. Parvathi, Editors, Noncommutative rings, group rings, diagramalgebras 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 ofanalytic functions, 2008

453 Jacob E. Goodman, János Pach, and Richard Pollack, Editors, Surveys onDiscrete and Computational Geometry–Twenty Years Later, 2008

452 Matthias Beck, Christian Haase, Bruce Reznick, Michèle Vergne, VolkmarWelker, 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 inanalysis 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 andtheir interactions, 2007

447 François 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 Véron, Editors, Perspectives in NonlinearPartial 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 inharmonic 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 algbrasand their applications, 2007

441 Louis H. Kauffman, David E. Radford, and Fernando J. O. Souza, Editors, Hopfalgebras and generalizations, 2007

440 Fernanda Botelho, Thomas Hagen, and James Jamison, Editors, Fluids andWaves, 2007

439 Donatella Danielli, Editor, Recent developments in nonlinear partial differentialequations, 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 theAMS 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.

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