Spectral Theory of Automorphic Functions and Its Applications

188 Pages·1990·6.168 MB·English

by  Alexei B. Venkov

Spectral Theory of Automorphic Functions
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.S.R.
Yu. I. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.
M. C. POLYVANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklov Institute of Mathematics, Moscow, U.S.S.R.
Volume 51
Spectral Theory of
Automorphic Functions
and Its Applications
by
ALEXEI B. VENKOV
Steklov Mathematical Institute. Leningrad. US.S.R.
KLUWER ACADEMIC PUBLISHERS
DORDRECHT / BOSTON / LONDON
Library of Congress Cataloging-in-Publication Data
Venkov. A. B.
[Spektral 'nara teorira avtomorfnykh funkiSil. English]
Spectral theory of automorphic functions and its applications /
Alexel B. Venkov.
p. cm. (Mathematics and its applications (Soviet series)
51)
Trans 1a t i on of: Spek tra 1 'na ra tear ira avtomorfnykh funk is i 1.
[SBN-[3:978-94-010-7344-8
1. Spectral theory (Mathematics) 2. Functions. Automorphic.
I. Tltle. II. Serles' Mathematics and its applicatlons (Kluwer
Academlc Publishersl. SOvlet serles ; 51.
OA322.V4613 1990
515' .9--dc20 89-20006
ISBN-13:978-94-010-7344-8 e-ISBN-13:978-94-009-1892-4
DOl: 10.1007/978-94-009-1892-4
Translated by N. B. Lebedinskaya
Published by Kluwer Academic Publishers,
P.O. Box 17,3300 AA Dordrecht, The Netherlands.
Kluwer Academic Publishers incorporates
the publishing programmes of
D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.
Sold and distributed in the U.S.A. and Canada
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In all other countries, sold and distributed
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Printed on acid-free paper
All Rights Reserved
© 1990 by Kluwer Academic Publishers, Dordrecht, The Netherlands.
Softcover reprint of the hadrcover 1s t edition 1990
No part of the material protected by this copyright notice may be reproduced or utilized in any form
or by any means, electronic or mechanical, including photocopying, recording, or by any informa
tion storage and retrieval system, without written permission from the copyright owner.
The crux of events is in events themselves, and the words
about them have meaning. The crux of Tao is not in Tao
and the words about it are meaningless. If one knows that
words are meaningless then each word will tum one to Tao.
If one does not know that words are meaningless then even
the most refined words will be all the same as an eyesore.
From 'Guan In-tzy'
SERIES EDITOR'S PREFACE
'Et moi, ... , si j'avait su comment en revcnrr, One service mathematics has rendered the
je n'y serais point aile.' human race. It has put common sense back.
Jules Verne where it belongs, on the topmost shelf next
to the dusty canister labelled 'discarded non
The series is divergent; therefore we may be sense'.
able to do something with it. Eric T. Bell
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non
linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for
other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such statements as:
'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com
puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And
all statements obtainable this way form part of the raison d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one hundred
volumes have appeared it seems opportune to reexamine its scope. At the time I wrote
"Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the 'tree' of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related. Further, the kind and level of sophistication
of mathematics applied in various sciences has changed drastically in recent years:
measure theory is used (non-trivially) in regional and theoretical economics; algebraic
geometry interacts with physics; the Minkowsky lemma, coding theory and the structure
of water meet one another in packing and covering theory; quantum fields, crystal
defects and mathematical programming profit from homotopy theory; Ue algebras are
relevant to filtering; and prediction and electrical engineering can use Stein spaces. And
in addition to this there are such new emerging subdisciplines as 'experimental
mathematics', 'CFD', 'completely integrable systems', 'chaos, synergetics and large-scale
order', which are almost impossible to fit into the existing classification schemes. They
draw upon widely different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems rather
fragmented and that to find, see, and exploit the deeper underlying interrelations more effort is
needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will
continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples of
interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modu
lar functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more)
all come together. And to the examples of things which can be usefully applied let me add the topic
'finite geometry'; a combination of words which sounds like it might not even exist, let alone be
applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via
finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to
be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And,
accordingly, the applied mathematician needs to be aware of much more. Besides analysis and
numerics, the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world and the
vii
viii SERIES EDITOR'S PREFACE
extra mathematical sophistication that this requires. For that is where the rewards are. Linear
models are honest and a bit sad and depressing: proportional efforts and results. It is in the non
linear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreci
ate what I am hinting at: if electronics were linear we would have no fun with transistors and com
puters; we would have no TV; in fact you would not be reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace
and anticommuting integration, p-adic and ultrametric space. All three have applications in both
electrical engineering and physics. Once, complex numbers were equally outlandish, but they fre
quently proved the shortest path between 'real' results. Similarly, the first two topics named have
already provided a number of 'wormhole' paths. There is no telling where all this is leading -
fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises five sub
series: white (Japan), yellow (China), red (ljSSR), blue (Eastern Europe), and green (everything
else), still applies. It has been enlarged a bit to include books treating of the tools from one subdis
cipline which are used in others. Thus the series stilI aims at books dealing with:
- a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
- new applications of the results and ideas from one area of scientific endeavour into another;
- influences which the results, problems and concepts of one field of enquiry have, and have had,
on the development of another.
Automorphic functions, i.e. 'nice' functions (in the analytic sense) which are invariant with respect
to some discretely acting group, have been around for a quite long time. The subject has always
been a meeting ground for various different areas in mathematics such as number theory, dif
ferential equations, harmonic analysis, algebraic groups,... Since about 1956, a further branch of
mathematics has become strongly involved: spectral theory; more precisely the spectral theory of
automorphic Laplacians such as the Laplace Beltrami operator on the Lobachevsky plane. As the
author writes there is now a recognizable area of mathematics which can be called Selberg theory
though it is still very much in development and new applications to or relations with other parts of
mathematics come up all the time (for instance, rather recently with string theory, quantum fields
and quantum statistical mechanics).
I am therefore happy that this timely authoritative monograph on the present state of the spec
tral theory of automorphic functions is now appearing and I am most pleased that this takes place
within the framework of this series.
The shortest path between two truths in the N ever lend books, for no one ever returns
real domain passes through the complex them; the only books I have in my library
domain. are books that other folk have lent me.
J. Hadamard Anatole France
La physique ne nous donne pas seulement The function of an expert is not to be more
l' occasion de resoudre des problemes ... elle right than other people, but to be wrong for
nous fait pressentir la solution. more sophisticated reasons.
H. Poincare David Butler
Bussum, August 1990 Michiel Hazewinkel
Table of Contents
Preface Xl
List of Notations xiii
Chapter 1. Introduction 1
Chapter 2. What Does One Need Automorphic Functions for?
Some Remarks for a Pragmatic Reader 5
Chapter 3. Harmonic Analysis of Periodic Functions.
The Hardy-Voronol Formula 9
Chapter 4. Expansion in Eigenfunctions of the Automorphic
Laplacian on the Lobachevsky Plane 13
Chapter 5. Harmonic Analysis of Automorphic Functions.
Estimates for Fourier Coefficients of Parabolic Forms
of Weight Zero 28
Chapter 6. The Selberg Trace Formula for Fuchsian Groups
of the First Kind 38
Chapter 7. The Theory of the Selberg Zeta-Function 47
Chapter 8. Problems in the Theory of the Discrete Spectrum
of Automorphic Laplacians 55
Chapter 9. The Spectral Moduli Problem 71
Chapter 10. Automorphic Functions and the Kummer Problem 76
Chapter 11. The Selberg Trace Formula on the Reductive
Lie Groups 89
Chapter 12. Automorphic Functions, Representations and
L-functions 99
Chapter 13. Remarks and Comments. Annotations to the Cited
Literature 108
IX
x CONTENTS
References 119
Appendix 1. Monodromy Groups and Automorphic Functions 126
Appendix 2. Automorphic Functions for Effective Solutions of
Certain Issues of the Riemann-Hilbert Problem 140
Author Index 172
Subject Index 175
Preface
This book consists of three independent parts supplementing one another. The
first main part is a large survey written especially for this edition. The second part
(Appendix 1), 'Monodromy Groups and Automorphic Functions', is also an original
survey. Finally, the third part (Appendix 2), 'Automorphic Functions for Effective
Solutions of Certain Issues of the Riemann-Hilbert Problem', is a slightly modified
paper published earlier in 'Zapisky nauchnykh seminarov LOM!', 162 (1987), pp.
5-42.
Xl