Table Of ContentLecture Notes ni
Mathematics
detidE yb .A Dold dna .B nnamkcE
548
Dennis .A Hejhal
The Selberg Trace Formula
for PSL )RI,2(
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galreV-regnirpS
Berlin. Heidelberg- New York 1976
Author
Dennis A. Hejhal
Department of Mathematics
Columbia University
New York, N.Y. 1002?/USA
AMS Subject Classifications (1970): 10D15, 10H05, 30A46, 30A58,
35 P 20
ISBN 3-540-07988-2 Springer-Verlag Berlin (cid:12)9 Heidelberg (cid:12)9 New York
ISBN 0-38?-0?988-2 Springer-Verlag New York (cid:12)9 Heidelberg (cid:12)9 Berlin
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ECAFERP
During the last 10 years or so, mathematicians have become increasingly fascinated
~th the Selberg trace formula. However, for various reasons, the number of detailed
sccounts of the trace formula has remained very small.
These notes were written to help remedy this situation. Their ~n purpose is to
provide a cotpreheaslve development of the trace formula for PSL(2,R). (~) VolL~e one
leals exclusively with the case of compact quotient space; the noncompact case rill ha
~onsldered in volume two which is now in preparation.
Although the trace formula can be developed much more generally, there are severe
Limitations on what is known for the hlgher-dlmenslonal groups. C~. Selberg1,pp.79-81.
Jnder these clrcu~stances, it makes sense to try to understand the simplest cases first.
yM basic philosophy in writing these notes has been very simple:
(a) try to keep things as classlcal and explicit as possible;
(b) keep a11 the basic ideas out in the open;
(c) supply complete proofs for everything (very much in the style of Landaul).
in approach based on (a)-(c) seems useful, especially if one wants to learn the basic
:echnlques.
The main chapter in volmne one is chapter 2. In this chapter, we study the trace
Formula using the techniques of analytic number theory. By focusing on the Selber8 zeta
Function Z(s), we can prove some deep results about the distribution of elgenvalues
md pseudoprlmes (lengths of closed geodesics) for a compact l~en~nn surface. This is
:he first time most of this material has appeared in print. Roughly speaklng: i/3 of 4t
sL fairly obvious, I/3 is due to Selberg (unpublished from 1950-53), and I/3 is new.
|ectlons 11-15 are included in the last category. It should also be mentioned that
uberl,2,5 independently obtained one of Selberg's estimates for the length spectrum,
~mely Q(x) = Ox3/4(lu x) -I/2 .
Although I have made a reasonable attempt to push the estimates in chapter 2 as f~
ts they can go, further improvements might still be possible. There are definitely
teveral loose-ends which deserve further investlsatton. However, as a general rule,
It) PSL(2,R) = SL(2,B)I((cid:127) ; there is ar index of notation on pp. 500~SD2
VI
I am inclined to say that the techniques are Just.as important as the theorems.
Another.lmportant feature of vol~me one is the derlvatlon'of a trace formula for.
(cid:12)9 odular correspondences, in which the analytic and non-8--lytic automorphic forms are
treated simultaneously. The -n-lyric component of this formula corresponds to the
well-known Eichler-Selberg trace formula.
These notes 8few out of an attempt to understand the relationship between trace
formulas and the Riemann zeta function ~(s). To make a long story very short: the
relationships considered in volume one proceed mainly from ~(s) to the trace formula,
while those in the other direction first appear in volume two. In other words:
volmne one
~(s) )sCZ~
,
volume two
A survey of these results has already appeared; see HeJhall,2.
While preparing volume one, I have benefitted from numerous dlscusslons wlth
Professor &tle Selberg. Our po/nts-of-vlew have turned out to be qulte similar. As a
kind of mathematical challenge, I thought it might be a good idea to work as Independently
as possible (especlally when proving the theorems). For this reason, the exposition of
Selberg's unpublished work is my own, and I take full responslb111ty for whatever fault, s
it may contain.
I would also like to express my deepest appreciatiDn to Professors E. Bombieri aud
G. Po/lya for some very helpful (and inspiring) conversations about trace formulas and
zeta functions. This research was supported in part by a fellowship from the Sloan
Foundation.
February 1976 D.A.HeJhal
TABLE OF CONTENTS
CHAPTER ONE. The Trace Formula for Compact Riemann Surfaces.
.i Preliminaries I
.2 Some elementary non-Euclldean geometry 5
.3 Polnt-palr invarlants 8
.4 Technicalities 15
.5 Setting up the trace formula 22
6. Carrying out the computation 24
7. Final form of the trace formula 30
Notes for chapter one 35
CHAPTER TWO. The Selberg Zeta Function.
.i Introduction 39
.2 Primitive applications of the trace formula 42
.3 Essentially primitive applications of the trace formula 53
.4 Basic properties of Z(s) 66
.5 The explicit formula for ~l(X) 82
.6 A more refined approach to the PNT 95
.7 Structure theorem for ~(T) 115
.8 An improved upper bound for S(T) 119
.9 Bounds for the integrated functions Sm(T) 140
i0. Integra I representations for Z'(s)/Z(s) and Log Z(s) 148
.II An explicit formula for ~(x) part one 156
.21 An explicit formula for ~(x) part two 175
13. Behavior of the expllcit formula near x = 1 189
.41 Some applications of the explicit formula to the PNT 214
15. Further applications (using a different method) 239
16. Irregularities in the distribution of pseudo-prlmes 255
17. Proof of an ./L result for S(T) 280
.81 Some comments about mean-square results 297
Notes for chapter two 316
CHAPTER THREE. The Trace Formula for Vector-Valued Functions.
326
.i Preliminary remarks
326
.2 The L2~C\ H,%) spectral theory part one
335
.3 The L2(P\ H,~,) spectral theory part two
345
.4 Development of the trace formula for L2(P\ H, ~)
350
.5 The trace formula for groups with elliptic elements
352
.6 Some miscellaneous comments
354
Notes for chapter three
CHAPTER FOUR. The Trace Formula for Automorphic Forms of Weight m.
I. Introduction 355
2. Basic definitions, results, and notation 356
3. Reeollectlon of some fundamental results of Roelcke 372
4. Setting up the trace formula for L2(~ ,m) 373
5. Recollection of some further results of Roelcke 407
6. Explicit computation of some convenient trace formulas 409
7. Main applications to analytic automorphic forms 430
8. Some remarks abo6t the general L2( ~ ,m) trace formula 434
9. Final form of the trace formula for m = 0 and m = 1 438
10. An over-vlew of the situation (some mlscellaneous remarks) 451
Notes for chapter four 453
CHAPTER FIVE. The Selberg Trace Formula for Modular Correspondences.
i. Introduction 462
2. Preliminaries and notation 462
.3 Setting up the trace formula 479
.4 The bootstrap lena 488
.5 Computation of some convenient trace formulas 489
6+ Applications to analytic automorphlc forms 491
.7 Remarks about the general formula for Tr(L~4~) 494
494
.8 Final form of Tr(Lg~) for m = 0 and m = 1
.9 The unified trace formula and some miscellaneous eo---ents 496
Notes for chapter five 499
I~EX OF NOTATION. 500
REFERENCES. 503
CHAPTER ONE
The Trace Formula for Compact Riemann Surfaces
1. Preliminaries.
We want to give a rigorous development of the Selberg trace formula for a com-
pact Riemann surface of genus g = 2 . To this end, we will assume that the reader
has some familiarity with Selberg i and either GGPS1,chapter I or Kubota~.
The small amount of EDP theory necessary in our approach can all be found in Courant-
Hilbertl,2, Garabedlanl, and Hilbertl.
We begin with some very basic facts. Let F denote a compact Riemann surface
of genus g iff 2 . We can therefore represent F as a quotient space ~ \ H , where
is a strictly hyperbolic Fuchsian group and H is the upper half-plane. It is
understood here that ~ ~ PSL(2,R) .
++--
Let ~ denote the "usual" fundamental polygon for F . Thus ~ = Wl#l~l~l
++-- + +
(cid:12)9 -- ~g~gWg~g , where the sides ~k ' ~k and ~k " ~k are identified in pairs.
It is unnecessary to assume that ~ is a non-Euclidean polygon. To be on the safe
side, however, we will assume that the sides of ~ are piecewlse smooth.
The upper half-plane H carries the Poincare metric ds = Idz~ /y , whose
corresponding area element is d~(z) = y-2dxdy . We recall that the Poincar~ metric
has Gaussian curvature K = -1 ; see Stokerl,p.185. Let ~: H ~F denote the
obvious projection (or unlversal covering) map. By projecting the Poincar~ metric
onto F via ~ , we make F into a compact Riemannian ~anifold. We assume thru-
ou~ the entire chapter that F carries the Polncare (cid:12)9 t metric. It follows that the
Gaussian curvature of F is still K = -i . By a classical application of the Gauss-
Bonnet theorem, we know that
Area(F) = /r = 4rr(g-l)
See, for example, Siegell,pp.28,51 and Stokerl,p.196.
Before going on, it should be noted that the automorphic group ~ is deter-
mined only up to a conjugation in PSL(2,R) . The Poincare metric on F is unaffec-
ted by such conjugations because of the well-known invariance properties of the
Polncar~ metric on N .
Our first real task is to develop an analogue of Fourier analysis on F . This
is done by studying the spectral decomposition of the Laplace-Beltrami operator for
F . One proceeds as follows.
Since F is a compact Riemannian manifold, the Hilbert space L2(F) makes
perfectly good sense. We can also define
L2(FtH) = {f : f ~ L2(~) , f(Tz) = f(z) for all Te ~ }
The functions f are understood to be defined on all of H , and can be called
automorphlc functions. The linear space L2(F\H) is obviously a Hilbert space when
given the inner product
(fl.f=) =
The choice of ~ in this integral is easily seen to be irrelevant. Since ~ is a
reasonable (compact) fundamental polygon for F ~ H , there will exist an obvious
three-way isometry L2(F ) ~ ~ L2( P ~ H) ~ :L2( ~ ) . The first part is induced by
the map T , while the second is induced by restricting f to ~ . Needless to say,
the llnear spaces ck(F) and ck(r \ H) have obvious meanings in the present context.
Let ~ be the Laplace-Beltrami operator for F : see, for example, Courant-
Hilbert1,pp.225,267 and Helgason1,p.387. If and ~ ~re local coordinates on
F such that ds 2 = Ed~ 2 + 2Fd~d~ + Gd~ 2 , then
E~- ~.F ~_
Let D be the corresponding operator on H under T . A trivial calculation shows
that
Du = y2(Uxx + uyy) - y2Au for z iff x + ly ~ H .
Since D corresponds to ~ , the operator D must automatically commute with the
action of ~ . The non-Euclidean Laplaclan D actually commutes with the action
of all PSL(2,~) , as is easily seen by a direct computation.
We can now apply chapter 18 of Hilbsrtl to study the elgenfunctlons of
quite classically. We simply write ~(W) = W + W for the local coordinate z -
xx yy
A
x + ly ~ H and, then, extend chapter 18 from the sphere C to F . This extension
involves only a minor modification in the construction of the parametrlx. It should
also be noted that (by the maxlmum principle) ~u = 0 on F iff u ~ constant.
Using Nilbertl,pp.225-238, we see that the spectral theory of ~ is very much
llke that encountered in the well-known case of vibrating membranes (plane domains):
(A) the spectrum of ~ will be discrete:
0 iff ~o <~I iff /~2 .... aril ~z n = c~ ;
n ~
(B) corresponding to the eigenvalues ~n ' we can construct an orthonormal basis
{qn~ for L2(F) so that ~ qn +/JAn ~n = 0 ;
(C) the normalized e~igenfunctions ~n are real-valued;
(D)
D@
(Bessel's Inequallty)
Note that the positivlty of the ~n follows from the maximum principle.
The whole point herels that the spectral theory of ~ can be reformulated
in terms of integral equations. More precisely, one obtains a Fredholm integral
equation whose kernel is sy~netrlc and has at most a logarithmic singularity.
According to Garabedian1,p.383 and Hilbert~ ,pp.234-235, we are now assured that
the Fourier series ~0Cn ~n of any C2(F) function must converge uniformly and
absolutely. This is essentially the classical Hilbert-Schmldt theorem. It should
also be noted that the elgenfunctions ~n are real-analytic: see, for example,
Garabedian1,pp.145,164.
To correspond to the notation used in Selberg1 , we shall now write
O-~o (cid:12)9 ~ 2~ , mll ~. "- ~
-~ "'"
n~
We shall also use the identification L2(P \ H)~---->L2(F) to transfer the functions
~n onto ~ ~ H . It is obvious that
The automorphic eigenfunctions ~n form an orthonormal basis for (2L r ~ H):
L2(P ~ H) = (cid:12)9 Z Wn
~0
PROPOSITION I.i (trivial bound).
N IAk( (cid:12)9 = T = O(T )2 as
Proof. Let s= E~ 2-)k~( - ~ (A) 2- Clearly, then, N(T)-IT -2 S .
ksl k::l
Therefore N CT) ~ I+ST z . S
We now have enough information about Fourier analysis on F with which to
proceed. A more detailed treatment of the spectral theory for L2(F \ H) can be
found in chapter .3
