The Arithmetic and Spectral Analysis of Poincare Series

181 Pages·1990·6.26 MB·English

by  James W. Cogdell

The Arithmetic and
Spectral Analysis of
Poincaré Series
James W. Cogdell
Oklahoma State University
Stillwater, Oklahoma
llya Piatetski-Shapiro
Tel Aviv University
Tel Aviv, Israel
and
Yale University
New Haven, Connecticut
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Library of Congress Cataloging-in-Publication Data
Cogdell, James W.
The arithmetic and spectral analysis of Poincaré series / James W.
Cogdell, llya Piatetski-Shapiro.
p. cm. - (Perspectives in mathematics: vol. 13)
Includes bibliographical references.
ISBN 0-12-178590-4 (alk. paper)
1. Poincaré series. 2. Spectral theory (Mathematics)
I. Piatetski-Shapiro, llya, 1929- II. Title. III. Series.
QA295.C56 1990
519.5-dc20 90-482
CIP
Printed in the United States of America
90 91 92 93 9 8 7 6 5 4 3 2 1
ACKNOWLEDGMENTS
The two articles in these notes deal with the spectral properties of
Poincaré series and their relation with Kloosterman sums.
The first article deals with Poincaré series for an arbitrary Fuchsian
group of the first kind Γ in PGL^R). The object of this paper is to
prove Kuznetsov's formula and its generalization in terms of the spec-
tral analysis of Poincaré series in the framework of representation theory.
This method leads to a conceptual proof of this formula and avoids the
detailed analysis of the special functions involved which is inherent in
a more calssical approach. An analysis of the spectral expansion of the
Kloosterman-Selberg zeta function is also included. We would like to
thank Joseph Bernstein, Henryk Iwaniec, and Peter Sarnak for many
helpful conversations during the preparation of this manuscript. We
would also like to thank The Institute for Advanced Studies of The He-
brew University for their hospitality during their special year devoted to
Automorphic Forms and L-functions, 1987-88, and Stanford University
for their hospitality over the summers of 1988 and 1989.
The second article develops the adellic theory of Poincaré series and
Kloosterman sums over a global function field. The main result here is
to show that in this context the analogue of the Linnik conjecture can
be derived from the Ramanujan conjecture over function fields, which
has been proven by Drinfeld. This article is made up of the lectures
given by the second author at Yale University in the Fall of 1987. They
were written up by Ze'ev Rudnik. We are very grateful to him for his
assistance in this project.
During the preparation of this work both authors were partially funded
by the National Science Foundation under grants DMS 88-03085 and
DMS 88-07336, respectively. We would like to take this opportunity to
thank the NSF for its support.
2
INTRODUCTION
In 1965 A. Selberg introduced a very interesting class of series, which
he called Poincaré series [25]. The simplest examples of these for Γ =
SL (1) is
2
P(z,s)= £ (Im( z))V^.
7
7€Γοο\Γ
Here z G S), the Poincaré upper half plane, and
r - { (J l)-.nez}.
This construction led him to introduce a very interesting zeta function,
c>0
where
s(c) = Σ e y < )
z,y (mod c)
xy~l (mod c)
are the simplest of the Kloosterman sums. Zeta functions of this type and
their generalizations are called Kloosterman-Selberg zeta functions. As
Selberg realized, the analytic properties of these functions are intimately
tied to the spectral theory of the Laplacian on T\S). For more details,
see [12,13,25] or Sections 7 and 10 below.
N. Kuznetsov found another interesting application of Poincaré series
[17]. He proved the first version, for Γ = SL (1), of the formula which
2
has subsequently been referred to as Kuznetsov's formula. For a precise
formulation see [17] and Section 6 below. By use of this formula he
proved a very interesting result in the direction of the Linnik-Selberg
5
conjecture. Recall that the Linnik-Selberg conjecture in its simplest
version says that
Σ «si«.·
x>c>0
for any e > 0 and x sufficiently large. Kuznetsov gave the estimate
S(c)^A+<
Σ -5r<
χβ
C
x>c>0
which is the best result to date.
Kuznetsov's original proof of his formula was very difficult. He used
a very detailed and delicate study of the special functions involved. We
will explain here how to avoid this. In fact, the formula reflects, in
a straightforward way, the structure of the unitary representations of
PGL2(R). The objective of this manuscript is to prove Kuznetsov's
formula and its generalization in the framework of representation theory.
This method lets us prove a Kuznetsov formula for an arbitrary Fuchsian
group of the first kind Γ C PGL {R).
2
Our method is a straightforward generalization of Selberg's original
analysis [25]. We begin with a general class of Poincaré series Pf(g)
on T\G. We then compute the Fourier coefficient, or associated Whit-
taker function, in two different ways, first unfolding Pf(g) to obtain a
weighted sum of Kloosterman sums, and secondly writing Pf(g) in terms
of its spectral expansion in L2(T\G) and then computing the Fourier co-
efficient. This yields what we call a "soft" Kloosterman-spectral formula
(Theorem 2.1). By specifying the Poincaré series more narrowly we can
explicitly compute the Fourier coefficients of the spectral components in
terms of basic representation theory of PGL2(R)- We compute these
6
using the Whittaker and Kirillov models of unitary representations of
PC?Z (R)· In these computations the Bessel function J-x(y) of a rep-
2
resentation naturally arises along with certain coefficients obtained as
ratios of the Petersson inner product on T\G and the canonical inner
product on the Kirillov model of a unitary representation. In this way
we arrive at a detailed Kloosterman-spectral formula (Theorem 5.1).
This is the general formulation of Kuznetsov's result. If we evaluate
the coefficients and the Bessel functions in terms of classical parameters
(and restrict to Γ = PSL2(Z)) we arrive at Kuznetsov's original formula
(Theorem 6.1).
The Kloosterman-Selberg zeta function can be obtained naturally in
this context as the Mellin transform of the Whittaker function of an
appropriate Poincaré series. If we apply this Mellin transform to the
Kloosterman-spectral formula we obtain a spectral expansion of the
Kloosterman-Selberg zeta function in terms of the 7-functions η of rep-
π
resentations (Theorem 10.1). In this way we recover Selberg's relation
between the analytic properties of this zeta function and spectrum of
X2(r\G), as well as generalize the spectral expansion of the zeta func-
tion for PSL (Z) given by Kuznetsov.
2
There have been many approaches to the relation between Klooster-
man sums and the spectrum of L2(T\G). Along classical lines, besides
the work of Selberg and Kuznetsov, much has been done by Bruggeman
[2,3], Proskurin [24], and Deshouillers and Iwaniec [5]. In a represen-
tation theoretic approach, Miatello and Wallach [20] have developed a
Kuznetsov formula for arbitrary groups of real rank 1. This type of for-
7
mula can also be obtained from Jacquet's relative trace formula [14] and
has been used by Ye to give another proof of quadratic base change [34].
As far as new applications of these formulas, the most striking are due
to Deshouillers and Iwaniec [5].
8
o. NOTATION AND CONVENTIONS
0.1. Throughout, we let G = PGL (R). We will denote elements of G
2
by a representative matrix in 6?£ (R)> with equivalence under multipli-
2
cation by the center as understood. We let
Let B = NA be a Borel subgroup of G. Let w = I 1 be the Weyl
element of G. Then we have the Bruhat decomposition G — B \jNwB.
Fix a maximal compact subgroup K C G.
Let Γ C G be a discrete subgroup such that T\G is not compact,
but T\G still has finite invariant volume. By conjugating Γ inside G if
necessary we may assume that Γ Π N = Γ^ = ^ 1 J | n E Z >, i.e.,
n 1
that oo is a cusp of width 1 for Γ. Let C denote the set of cusps of Γ.
For each cusp o E C, we let g E G such that g · a = oo.
a a
If P is any parabolic subgroup of (?, P = MU its Langlands decompo-
sition with M the Levi factor and U the unipotent radical, we say that
P is Γ-rational if (Γ Π U)\U is compact.
We will let Q denote the Lie algebra of G and U(g) its universal en-
veloping algebra. We will use a similar convention for any Lie group.
0.2. Let us recall the unitary representations of G. If μι,μ2 are any two
characters of Rx, we may define the normalized induced representation
Ind(/xi,/x ) of GL (R) as in Jacquet-Langlands [14]. This representation
2 2
will factor to G = PGL (R) iff μι = μ 1 · Let us denote Ind(/x,/x_1) =
2 2
π(μ) when this is irreducible. If Ind(/i,/x -1) is reducible, it will have a
unique infinite dimensional subquotient. We will denote this by σ(μ).
Then the irreducible infinite dimensional unitary representations of G
are:
(i) The unitary principal series ττ(μ) with μ(χ) — \x\ tr or μ(χ) =
sgn(x)|a;|ir.
(ii) The complementary series ττ(μ) with μ(χ) = \x\r or μ(χ) =
sgn(ic)|a?|r with — | < r < |.
(iii) The discrete series σ(μ) with μ(#) = x d~* with d £ Z,ci > 0.
We have the equivalences π(μ) ~ π(μ -1),σ(μ) ~ σ(μ-1).
Note that G has a non-trivial character χ : G —* {±1} defined by
x(g) = sgn(det^r). Then we have π(μ) ® X — π(μχ) and σ(μ) ® χ ~
σ(μχ) ~σ(μ).
For convenience, let us then write
<r) = π(| Γ)
τφΥ) = π(| |ir)
<r(«Q = <r(|f-*)
so that the irreducible infinite dimensional unitary representations of
PGL (R) are
2
(i) π(ζ>),χ (g) π(ζ>) for r > 0.
(ii) 7r(r),x ®π(τ·) for 0 < r < |.
(iii) σ(Λ) for d= 1,2,3,....
10