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00
2
Part 1 ) is a consequence of (2.3. l O), and 4) is a consequence of part 4) of Theorem 2.2. 1 . To prove 2) and 3), we first verify that PROOF.
91 ( K) tJ;( Z ; s ; a ; I ) = ( 1/w( s ))tJ;(z ; s ; a; I ) .
(2 .3 . 1 1 )
In fact, we have the equality of operators
91 (K ) + 91 ( K) � (S ) = ;t (K ) + ;t ( K ) � (S ) + � (K ) + � ( K ) � (S) = ;t (K ) + ;t ( K ) � (S) + ?S ( s ) . (2 .3 . 1 2) We have used Theorem 2. 1 . 1
and (2.2.5).
Next,
w;t (K ) � ( s ) = w 2 [ ;t ( K ) + w;t ( K ) (} ( s )] ?S (s ) = w2 0(s ) ?S ( s ) = � ( s ) - w ?S ( s ) .
From this and from (2.3. 1 2) we obtain
W 91 ( K ) + w 91 ( K) � (S) = W;t(K)
+
Finally, if we use the equality
� ( s ).
W ;t ( K) cp ( Z ; s ; a ; I) = cp(z; s ; a; I ) ,
then from (2.3. 1 3) we obtain the desired assertion (2.3 . 1 1).
(2 .3 . 1 3 )
II . EXPANSION IN EIGENFUNCTIONS OF � ( r ; X )
38
The equality (2.3. 1 1) shows that l/; ( z; s; a; I ) satisfies the equation in 3) in a weak sense. Since the differential operator L is elliptic, it follows from the general regularity theorem (see [32], Appendix 4, §5, Theorem 4) that we first have part 2), and then also part 3) of the theorem. The proof is complete. We now describe a functional equation for the kernel r(z, z'; s) of the resolvent, considered as a function of s. This functional equation relates its values at nonsingu lar points s and I s which both belong to the strip 0 < Re s < 2. With this in mind we prove the following lemma. -
LEMMA 2.3.1 . Let q ( z , z' ; s) be the kernel of the operator O (s). Then O ( z , z'; s ) - O ( z, z' , 1
I
n
ka
�
3 being fixed. On the right side of (2.3. 15) we substitute the expression for fft (s) and fft (s ') in (2.2.5) in terms of the operators Q3 and O . We obtain the equ�lity -
-
91 (s ) - fft ( s ') = [ 1 + w( s ) ( 1 + w( s ) O ( s )) Q3 ( s ) ] X [ O ( s ) - O (s') ] [Q3 ( s ')( O ( s ')w ( s ') + 1 )w( s ') + �f] + ( w ( s ) - w( s ')) [( 1 + w( s ) O (s ) ) Q3 ( s ) O ( s ') + O ( s ) 'B ( s')( O ( s ')w( s ') + 1 ) + ( 1 + w( s ) 0 (s )) Q3 ( s )(1 + w ( s ) O ( s ) + w( s ') O ( s ')) X Q3 (s')( O ( s ')w( s ') + 1 ) ] . ( 2.3 . 16 )
§2.3. EIGENFUNCTIONS OF THE CONTINUOUS SPECTRUM
39
In deriving (2.3.16) we also used the Hilbert identity for £l (s) at the points s and s ' :
£l ( s ) - £l ( s ') = [w ( s ) - w( s ')] £l ( s ) £l ( s ') , an identity which, in turn, follows from (2.2. 14). The right side of (2.3.16), regarded
as a composition of integral operators, permits a meromorphic continuation in s and s ' to the strip 0 < Re s, Re s ' < 2. Now setting s ' 1 - s and using Lemma 2.3.1, we obtain the desired formula (2.3.14). The proof is complete. We proceed to study the properties of the second coefficients in the asymptotic expansion of the vector-valued functions lj;(z; s; a ; I ), 1 � a � n, 1 � 1 � k a ' in a neighborhood of the cusps of F (see part 4) of Theorem 2.3.2). As we shall later see, the set of these coefficients, arranged in a matrix, plays an important role throughout the spectral theory of automorphic functions. Just as in the scalar theory, this matrix corresponds to the scattering operator in the perturbation theory of the continuous spectrum of selfadjoint operators (see [9], §4). In the notation of part 4) of Theorem 2.3.2 we set
=
@5 al,,Bk (S ) = oa,B ( ek ( /3 ) , e/ (a) ) a 2 S- 1 �: � 1 s 2 ) + ;: � 1 l I a II aqJ ( y, s ) ( e k ( /3 ) , b,BA z , z ' ; ( s ) e/ ( a )) ) qJ ( y ', s ) djl ( z ) djl ( z ') , 1 � a � n , 1 � /3 � n , 1 � l � k a ' 1 � k � k,B ' (2.3 . 17 )
ff
Theorem 2.3.1 implies the following theorem.
2.3.4. Each function ® al,{3k ( s) has the following properties: 1) It is analytic in the strip 0 < Re s < 2, except at the singular points of the operator �(s), where it has poles offinite multiplicity. 2) It does not have poles on the line Re s = 1 /2. 3) It has at most simple poles at the real singular points of � (s) for which 1 /2 < s � 1 . THEOREM
We now prove that the set of functions (2.3. 17) has two important properties, which characterize it as a matrix of order kef; X) = 2.� k a (see §1 .2).
2.3.5. The set offunctions ® al, {3k( s) has the following properties: 1 ) ® al, ,B k( S ) = ® {3k , a l ( S ) ; 2) kp n (2.3. 18) � � ® al , {3k (S ) ® {3k , y m ( 1 - s ) = 0ayOlm , ,B= l k = l
THEOREM
where Oay is-the Kronecker symbol, 1 � a � n, 1 � I � k a ' 1 � Y � n and 1 � m � ky o
PROOF. Part 1) follows easily from (2.3. 17) and the Hermitian property of the kernel b(z, z ' ; s ) (see Theorem 2.2.5 and (2.1 .28)).
I I EX PANSION IN EIGEN F U N CTIONS OF )I ( J' ; X )
40
We prove part 2). By (2.3.17), the left side of (2.3. 1 8) has the form
2 ) oayo/m + � � 1 I I I I a 1 . The notation ( raEra) �a/3 is defined in Theorem 3 . 1 .2. ( _I
X
exp
[27Ti(
0k/3 + j
/3
C
l
Our Theorems 3. 1 .2 and 3. 1 .3 completely describe the Fourier expansion of any Eisenstein series (3. 1 . 1). As an application we now obtain the following facts.
The following asymptotic formula holds for the parameter values 1 a, /3 n and 1 I k a: E( g/3 z ; s; a; et( a) ) = �af3 e, ( a ) y S + 0( 1 ) , where Z = x + iy E IT a and Re s > 1 (s is fixed). LEMMA
::;;;;;
::;;;;;
3. 1 . 1 .
::;;;;;
::;;;;;
y ---> oo
The lemma follows from Theorems 3 . 1 .2 and 3. 1 .3 and the asymptotic behavior of KsCy) as 00 (see §2. 1). The proof is complete. PROOF.
y 3 . 1 .4. The following equalities hold for a , I and /3 as in the hypothesis of ......
Lemma 3. 1 . 1 1) ( ek ( /3 ) THEOREM
:
,
r;; r (s - ! ) r ( S rl l1o(S ; a ; I ; f3 )
) = @5 at,/3k(S ) ,
I I I . F I R ST REFI NEMENT OF THE EXPA N S I O N THEOREM
48
2) £ ( z ; s; IX; et( IX» = I/;( z; s; IX; I), where 3. 1 .2, and the functions f3 al./'Jk( s) and 1/;( z; s;
11 0( S ; IX; I; f3 ) = l1o( S ) is from Theorem IX; l) are defined in §2.3.
It is not hard to see that 2) implies 1 ), since equality of the functions implies equality of the coefficients in their asymptotic expansions. We prove 2). We note that both of the functions E( z; s; IX; et ( IX» and 1/;( z ; s; IX; I) are defined in the strip 1 < Re s < 2. From Lemma 3. 1 . 1 and Theorems 3. 1 . 1 and 2.3.2 it follows that the difference £( z; s; IX; etC IX» - 1/;( z; s; IX; I) is bounded in z and is an eigenfunc tion for the operator m ( f ; X ) with eigenvalue A = s( 1 - s) for any s, 1 < Re s < 2. Since m (f; X) is a selfadjoint operator, this is only possible if this difference is identically zero. The proof is complete. PROOF.
3. 1 .5 . 1 )
The functions E( z ; s; IX ; e{(IX» , l1/ S ), �/ s ), I/;( z; s; IX; I) and ® a ' , f3k( s) can be extended '!leromorphically in s onto all of C , 1 � a � n, 1 � I � k 1 � f3 � n, 1 � k � kf3,} E Z, z E H. 2) The kernel of the resolvent 91 (s; f; X) and the kernel of the operator iB (s; f; X) (see Chapter 2) can be extended meromorphically in s onto all of C . PROOF. 1 ) By Theorem 3 . 1 .4, it suffices to prove the assertion for E( z ; s ; IX; e,( IX» , l1)( S ) and �/s). The last two functions appear in the Fourier coefficients of the Eisenstein series; hence, it is sufficient to prove meromorphicity of E( z ; s; IX; e,(IX» . THEOREM
a'
By Theorems 2.3. 1 , 3. 1 . 1 , and 3. 1 .4, this function is meromorphic in the half-plane Re s > O. Consequently, the scattering matrix f3(s) is meromorphic in the half-plane Re s > O. Then its functional equation (2.3.2 1 ) implies that 8(s) is meromorphic on C , and the functional equation for the functions I/;( z ; s; IX; I) in Theorem 2.3.6 implies the desired meromorphicity of E(z; s; a ; et C a» . 2) This is a consequence of part 1 ) applied to the functions 1/;( z ; s; IX; I ), the functional equation (2.3. 1 4), and the formula (2.2.5). The proof is complete. §3.2.
The Maass-Selberg relation
In the scalar spectral theory of 2f(f; X ), i.e., in the special case of the theory when the dimension of the space V on which the representation X acts is equal to one, the Maass-Selberg relation is what one calls the formula expressing the inner product in :Je( f; X) of two Eisenstein series (modified in a special way) in terms of the constant terms in their Fourier expansion relative to parabolic subgroups fa C f (see [50], and also [28], Chapter II). This relation plays a vital rolejn one of Selberg's methods for proving meromorphic continuation of Eisenstein series onto C (see [52]) and in the method for proving the theorem : on expansion in eigenfunctions of m: (f; X ) which does not use the integral equation for the kernel of the resolvent 91 (s; f; X ) (see [46]; for references to other important works, see §8 of [66]). I n addition, the Maass-Selberg relation occupies an important place in the derivation of the Selberg trace formula (see [50]). In the theory developed in the present monograph, the Maass-Selberg relation (the multidimensional version) is essentially necessary only for the derivation of the Selberg trace formula (see §4.3). In the generality we need (dim V � 1 , X nontrivial) the relation was derived by Roelcke in [46] under the hypothesis of meromorphic continuation of Eisenstein series. Since we have now proved this meromorphic
§3.2. THE MAASS-SELBERG RELATION
. 49
continuation in Chapter 2 and §3. 1 , in §3.2 we shall only give the statement of the fundamental theorem of §§9- 1 1 in [46]; we shall limit ourselves to the special case of the relation which will be useful to us in §4.3. For the convenience of the reader, we mention that the derivation of the Maass-Selberg relation is based on Green's classical integral formula for the operator L (see § 1 . 1), the automorphic property of the Eisenstein series, and part 3) of Theorem 3. 1 . 1 . We introduce some notation. Recall the partition of the fundamental domain F into the sets Fj, j = 0, 1 , (see Theorem 1 .2.4). This partition depended on the number We now emphasize this dependence by letting Fj( denote the elements of the partition. For each Eisenstein s�ties E ( z; a; l a we define the function E( z; a; el a) ; by the following formula for z F:
s;
a.
. . . ,n
a)
E{z; s; a; e,( a ) ; a ) =
zE
a)
s; e )) E E { z; s; a; e,( a )) , z E Fo(a) , E{z; s; a; e,( a ) ) - y ( gplz rc5ape, ( a )
- .r;; r{ s - !)r- I ( s ) r] o( s )y ( gplz r-s,
, n,
z,
( 3 .2 . 1 )
where Fp , f3 = 1 , 2, . . . y( z ) = 1m and we extend the function by the automorphic property involving X to all of H (as was done for a; I ) In (2.3. 10)). In (3.2. 1 ) we have used the notation in Theorem 3. 1 .2.
l/t( z ; s ;
Suppose that 1 a ";;;;; n, 1 ,,;;;;; I ,,;;;;; ka and s E C, Re s 1 /2, is not a singular point for the operator �(s) on the interval ( 1 /2, 1) (see Theorem 2.3. 1). Then the following formula holds ( the Maass-Selberg relation): THEOREM
,,;;;;;
3.2. 1 .
>
... .
( 3 .2.2 )
where ® (s) is the scattering matrix (see §2.3), and the bar denotes complex conjuga tion. We note that the relation (3.2.2) can be extended by continuity onto the line Re = 1/2 as well. The Maass-Selberg relation enables us to fill in a small gap in the proof of part 3) of Theorem 2.3. 1 .
s
The point s = 1/2 is a regular point for each of the functions E( z ; s; a ; et( a)), I ,,;;;;; a ,,;;;;; n, 1 ,,;;;;; I ,,;;;;; ka• THEOREM
3.2.2.
The proof of the analogous fact in the scalar theory is given in §4.3 of [28]. If one supposes that the point = 1/2 is singular, one obtains a contradiction with (3.2.2) by means of a simple but rather lengthy argument. The proof of Theorem 3.2.2 is similar. We shall not give it here, instead referring the reader to Kubota's book.
s
50
III. FIRST REFINEMENT OF THE EXPANSION THEOREM
§3.3.
Incomplete theta-series cusp-vector-functions. The operators K( and
f; X )
The first part of this section is devoted to a certain intrinsic characterization of the subspace of the continuous spectrum-and to a lesser extent the subspace of the discrete spectrum-of the operator W (f; X ). The theory developed here uses ele ments from the spectral theory of Selberg Roelcke Godement Langlands and Kubota in which the main emphasis is on the properties of Eisenstein series and the Maass-Selberg relations, and especially the theorem on expansion in eigenfunctions of W(f; X) which we proved in Chapter 2 using Faddeev's method from the scalar theory of automorphic functions. Part 2) of Theorem is a resolvant version of a well-known theorem of Gel'fand and Pjateckii-Sapiro (see Chapter I, or Chapter I, §2). In the second part of the section we introduce a fundamental class of integral operators in the spectral theory of automorphic functions, and prove some proper ties of these operators. These operators were first introduced by Selberg (see [S I D. The idea of studying them using the resolvent f; X ) in the scalar theory is due to L. D. Faddeev, V. L. Kalinin and the author (see [72]). For every function E COOO([ 0, 00) ; C) we define the O-series
[33]
3.3.2
[52],
[28],
[13],
[46],
[14],
§6, [16],
9t(s;
l/; � l/;(y ( g� lz ) ) X* ( y )v O(z; l/;; a ; v) = y Ef,,\f
(3.1.1),
( 3.3.1)
(the incomplete theta-series), in analogy with the Eisenstein series where E Va ' = 1m and � a � Thanks to the finiteness of l/; and the discrete ness of the group r, for any fixed E H the series has only finitely many nonzero terms. We enumerate some simple properties of l/;; a ; which follow directly from the definition.
v
y( z )
1
z
n. z
(3.3.1) O(z;
v)
3.3.1. Every O-series has the following properties: l) O( yz; v) X( y )O( z; v)for any y r andz O( z; v) in the variable z.
THEOREM E E H. l/;; a'; l/; ; a ; = 2) l/;; a ; E COO( F; V; X) n :Je(f; X) We define e( f ; X) c :Je ( r ; X ) to be the closed subspace spanned by all of the O-series with E Co([ 0, 00); C ), E Va' � a � The set of O-series obviously forms a dense linear subset in e(f; X). We now introduce the orthogonal comple ment of e(f; X) in :Je(f; X):
v
l/;
1
:Je ( f ; X ) = :Jeo ( r ; X )
n.
EEl
( 3.3.2)
8(f ; X). It is not hard to describe the elements of :Jeo( f; X ). Let E :Je(r; X) be a continuous vector-valued function which is bounded on F in the norm of V. We have
f
the following equalities:
°=
{z
(f, O) F.x fF(f(z), O(z; l/; ; a ; v ) dp, (z) =
z f,
iy
2. Then
�R ( s ; f ; x )f( z ) = j r ( z , z ' ; s )f( z ') dJ.L ( z ') F
=
jIIk ( z , z ' ; s )f( z ') dJ.L ( z ') .
( 3 .3 .7 )
We have used Theorem 1 .4.5 and ( 1 .4. 1). Next, because of the invariance of the kernel k( z , z ' ; s) (i.e., k( gz, gz ' ; s) = k ( z, z ' ; s), g E G , z E H ), from (3.3.7) we obtain for any a = I , . . . , n and v E Va
1o 1 ( ffi ( s )f( ga z ) , v ) dx = jIIk ( iy , Z ' ; s ) 110 ( f( ga( z ' + x ») , v > dxdJ.L( z ') ,
(3 .3 .8)
z = X + iy, z ' = x ' + iy ' . The right side of (3.3.8) is identically zero in y, because f( z ) is a parabolic form of weight zero. This proves part I ) for Re s > 2. I t is obvious how to extend this result to any regular point s. 2) We first suppose that s E C and Re s > 2. Let f E :Je(f; X ). We consider the expression for ffi ( s ) in Theorem 2. 1 . 1 . From the definition of the operator � ( s ) we have the following relations for the a-component of the function �(s )f( z): ( :� ( s )f ) o( z ) = 0, (3 .3.9 ) where a = I , . . . , n , z = x + iy and z ' = x ' + iy ' . If f E Xo(f; X ), then the right side of (3.3.9) is identically zero, since the kernel t(y, y '; s ) does not depend on x or x ' on ITa' Consequently, by Theorem 2. 1 . 1 we obtain 1
Since � o is a bounded operator in X(f ; X ) and, by Theorem 2. 1 .2, �(s), �(s) and �(s) are Hilbert-Schmidt operators, it follows that 91 (s)�o is a Hilbert-Schmidt operator. To prove the theorem for an arbitrary regular point s ' it suffices to make use of the Hilbert identity
91 ( s ') - 91 ( s ) = ( s '( 1 - s ') - s ( 1 - s » 91 ( s ') 91 ( s ) ,
( 3 .3 . 10 )
taking into account that s is chosen with Re s > 2. Multiplying (3.3. 10) on the right by � 0 and using the fact that 91 (s) � 0 is a Hilbert-Schmidt operator, we obtain the assertion in the theorem.
"
§3 .3. INCOMPLETE THETA-SERIES AND CUSP-VECTOR-FUNCTION
53
:Jeo(f;
X) is 3) From parts 1) and 2) proved above it follows that the subspace spanned by the eigenfunctions of the discrete spectrum of Hence X) X), X) � (3 . 3 . 1 1 ) and we would like to prove equality: X ) EB ( 3 . 3 . 1 2) X) . X) = It is not hard to verify that the functions which can be represented in the form
1,l3 :Je(f;
:Jeo(f; EE)(3 1 (f;
:Jeo(f;
8 1 (f;
W.
1,l3 :Je(f;
1 s ( 3 .3 . 1 3 ) ( 8(z; 0/ ; a ; v) = 2 JRe .V. =SoLvl ) E ( z; s; a ; v) ds, are a dense set in the subspace 8(f; X ) = 81(f; X) EB 8 (f; X); here the integral is taken along the line Re s = so, so > 1 , and E( z; s; a ; 2v) is the Eisenstein series (3. 1 . 1 ). In (3.3. l 3) Lis) is the Mellin transform 1 ( s=soL.is )yS ds, Lvl s) = 1000o/(y )y-S dy� , o/(y) 2 JRe 0/ E C:([ 0, (0); C), and the function 8( z; 0/; a; v) is the incomplete theta-series (3.3. 1 ). Hence, the subspace 8(f; X) does not contain any eigenfunctions of the discrete spectrum of W except for the elements of 8 1 (f; X ), and this, along with (3.3. 1 1), gives us (3.3. 12). The proof is complete. 'TTl
.
=
.
"
WI
.
We now proceed to the second part of this section, which is devoted to certain integral operators in :Je(f ; X ) which are functions of We introduce some notation. Let be a real eigenbasis for X ) in the subspace of the be the corresponding set of discrete spectrum X) EB X), and let eigenvalues .
{w( z; A)}j :Jeo(f;
W(f; { AJ
8 1 (f;
W.
1) If the function h: [ 0, (0 ) C is measurable and bounded, then the operator h(W(f; X)): :Je (f; X) :Je(f; X) is defined, and its kernel h(z, z' ; W) as an integral operator has the spectral decomposition h(z, z'; W) = �h (Aj) w (z; Aj) ® w (z' ; AJ 1 n k a 00 ( 1 + ) ( 1 r 2 E z; + ir; a ; e,( a ) ) � � f h THEOREM
3.3.3.
�
�
j
+ 4w 0' = 1
/= 1
- 00
4
2
( 3.3 14) .
The series and integral in (3.3.14) converge in the metric of :Je (f; X)· 2) Let the function h: C C have the following properties: a) h(s) h(s(1 - s)) is analytic in the strip -E < Re s < 1 + Efor some E > 0; and b) it satisfies the estimate h(s) = o ( ( 1 m s 1 + 1 ) -2 - 8), -E < Re s < 1 + E, for some 0 > O. Then the kernel of the operator heW) K(f; X) is given by the series h(z, z ' ; 2I ) = k(z, z ' ; f; X) � X {y )k { u { z, yz ' )) , ( 3.3.15 ) =
�
1
=
=
y E T'
III . FIRST REFINEMENT OF THE EXPANSION THEOREM
54
where u(z, z') is the fundamental invariant of a pair of points, and the function k: [ 0, 00) � C is connected with h by the following transformation (the inversion formula of Selberg and Harish-Chandra): foo Vtk-{ t)w dt Q { w ), k { t) -! f oo dVWQ {-wt) , (3 .3 . 1 6) Q{ exp u + exp { -u) - 2) = g { u), g{ u) 21'TT i: (exp - iru)h ( ! + r 2 ) dr. Here the series (3.3. 1 5) converges absolutely in the norm of V, converges uniformly in any compact subregion of H X H, and gives a continuous map H X H � =
=
w
'TT
t
=
V.
Part 1) is a direct consequence of Theorems 2.3.7 and 3.1 .4. We prove part 2). Let be a fixed number, -f < < 1 + f, where f is the number in the conditions of the theorem. A simple but fairly lengthy calculation, which we shall not carry out here, shows that, if the function h( satisfies conditions 2 a) and 2 b), then the inversion formula (3.3. 16) is equivalent to
So
So
PROOF.
k ( t) = - 2 �i i
Re s = So
k(t; s) k(u(z, z'); s). s so. k(t)
s)
(1 - 2
s ) h { s ( 1 - S )) k ( t; s ) ds,
(3.3 . l7)
where was defined in Theorem 1 . 1 . 1 and is the basic ingredient in the Green's function The integral in (3.3. 17) is taken over the vertical line Re This integral clearly does not depend on the choice of within the indicated region. We suppose that lies in the region 1 < < I + €. We show that is a continuous function on the semiaxis (0, 00), has a finite limit as 0, and satisfies the estimate (3.3 . 1 8) From the definition of it follows (see Theorem 1 . 1 . 1) that the following estimate holds for in any interval [ f l ' 00): =
t
So
So
So
t E [O, oo ).
k(t; s)
I k { t; s) I « ( 1 + t rRe "I
s,
t
�
(3 .3 . 1 9)
f l > 0. Because of this and assumption 2 b) of the theorem, the integral (3.3. 17) converges absolutely and uniformly for [ f l ' 00 ) and satisfies (3.3. 1 8) for � f l .
tE
k(t) t O.
t
This implies that is continuous on (0, 00). It remains for us to prove that there is a finite limit as � This is most ' easily done using (3.3. 16). The assumptions 2 a) and 2 b) concerning h(s) imply that behaves like u l +c2 as � 0, where f '::> ° is 2 some fixed number. From this it follows that Q( w ) w l /2+E2i2, W � 0, and this, in turn, implies that has a finite limit as 0. We now substitute the fundamental invariant in place of on the left and right in (3.3.17) and average the resulting kernels (multiplied by the representation over the discrete group. Using ( 1 .4. 1), we obtain
g(u)
k( t)
X)
2:
yET
X {y )k { u {z , yz')) -�'TT l 1 =
t
�
�
Re s = ,\'o
u( z, z')
(1 -
u
t
2s)h { s {1 - s))r { z, z ' ; s) ds.
(3 .3 .20)
§3 .4.
THE SELBERG-NEUNHOF FER INTEGRAL EQUATION
55
It is permissible to interchange the summation and integration in the right side of (3.3.20), because of the properties we proved for the integral in (3.3. 1 7) and Theorem 1 .2.5. Making the change of variables A = s( 1 s ) in the integral in (3.3.20), we find that the right side of this equality is the kernel of the operator 1 (3 .3.2 1 ) - - h ( A ) 9't ( A ) dA , 2m Q where �ft ( A ) = m ( s ) and Q is a contour which encloses the spectrum of 2! in the positive direction. Hence, by a well-known theorem in functional analysis, the integral (3.3.2 1) is the operator h( 2! ). Because of the properties of k( t ) which we proved and Theorem 1 .2.5, the series on the left in (3.3.20) determines a continuous map H X H V. The proof is complete. The class of functions h(s( 1 s )) for which the corresponding kernel k( z , z ' ; f; X) is given by an absolutely convergent series over the discrete group is not exhausted by the set defined in condition 2) of Theorem 3.3.3. To illustrate this, we shall be satisfied here with the following special fact, which will turn out to be useful in Chapter 6. -
.
->
j
-
THEOREM 3.3.4. Let the function h: C C have the following properties: a) h( s ) = h (s( 1 s )) is analytic in the region bounded by the rays Q l : s = -e or - iYJr, Q 2 : s = 1 + e + or + iYJr for r > 0, Q3 : s = -e or + irYJ , Q4 : s = 1 + e + or - i YJ r for r < 0, -
->
-
-
where e , 0 and YJ are fixed positive numbers. b) In this region it satisfies the estimate , .
Then the kernel of h ( 2! ) is given by the series (3.3. 1 5), which is absolutely convergent and uniformly convergent in {z, z ' } E H X H outside of an arbitrarily small neighbor hood of the surfaces z = z ' (mod f).
The proof of the theorem is similar to the preceding one, with the integral (3.3. 1 7) replaced by 1 t � el > O . (3.3 .22) (1 h s( s )) k ( t ; s ) ds , k(t) = -2 7T l !J4 U !J , - 2 s ) ( I PROOF .
.
j
Here one must use the easily verified estimate I k ( t ; s ) 1 « ( I + t ro r
for s E Q4 U Q 2 and t � e l > O. The proof is complete. We note that, in general, a weaker restriction on the growth of the function h in Theorem 3.3.4 than that in condition 2) of Theorem 3.3.3 will lead to singularity of the kernel k(z, z ' ; f; X) on the diagonal z = z ' (mod f ). §3.4. The integral equation which we shall discuss here was first proposed by Selberg in order to prove meromorphic continuation of Eisenstein series in the scalar theory (dim V = 1 ) (see [50]). In essence, it is a Fredholm equation of the third kind, whose kernel of the integral operator is the harmonic Green's function of the automorphic
The Selberg-Neunhoffer integral equation
56
I I I . FIRST REFINEMENT O F THE EXPANSION THEOREM
Laplacian modified in a special way. To prove the existence of this kernel, Selberg used the Dirichlet principle. Twenty years later, Neunhoffer [39] proposed a some what more general integral equation than the Selberg equation for proving meromor phicity of Eisenstein series (also in the scalar case). As the integral operator in this equation he used a modified resolvent ffi(s; f ; X), whose existence can possibly be proved by a simpler theory than the Dirichlet principle on a noncompact Riemann surface with elliptic singular points. We shall call the latter equation the Selberg Neunhoffer equation. The purpose of this section is to derive an even more general version of the Selberg-Neunhoffer equation, which is suitable for studying the vector-valued Eisen stein series (dim V � 1) which depend upon a representation X E 9C sC f), X: V ..... V. We derive the equation here not only for the sake of completeness of the exposition (we have actually already proved meromorphicity of Eisenstein series in Chapter 2), but because from the Selberg-Neunhoffer equation it is relatively simple to find an upper estimate for the order of meromorphicity of the Eisenstein series and the scattering matrix. Unlike in [50] and [39], we shall derive the Selberg-Neunhoffer equation from Faddeev's equation, using the information we now have concerning the resolvent ffi (s) of the operator m . We introduce the following kernel, defining it component by component (see § 1 .2):
ro:oo(z , z '; K; a ) = ro:o(z , z'; K ) , ro:°,B ( z , z' ; K; a ) = ro:,B (z, z ' ; K )
(3 .4. 1 )
a = 0 , 1 , . . . , n , /3 = 1 , . . . , n , where z ' = x ' + iy ' E ITa' r( z , Z ' ; K ) is the kernel of the resolvent of m , K is a fixed regular point, K > 3, and E( z ; K; a; et ( a» is the Eisenstein series (3. 1 . 1). From
(2.3. 1 1) and Theorem 3. 1 .4 we have
E{z; s ; a ; et( a )) = [ s ( I - s ) - K(I - K )] !. [ r ( z, z ' ; K) - r O (z , z' ; K ; a )] F
X E{z'; s ; a ; et( a )) dp. (z') + [s( I
-
s ) - K(l - K )]
(3.4.2) which can be regarded as a definmg equation for E( z ; s; a; et ( a » , 1 ,.;;;; a ,.;;;; n , 1 ,.;;;; I ,.;;;; k o: . From the definition (3.4. 1) we see that the free term, i.e., the first term on the right in (3.4.2), is equal to k s ( I - s ) - K(I - K ) n p E{z; K; /3; e k(/3 )) � � 2K - I /3= 1 k= l
§3 .4. THE SELBERG-NEUNHOFFER INTEGRAL EQUATION
57
We now use Theorem 3 . 1 .2 to compute the integral in (3.4.3) in terms of the constant term in the Fourier expansion of the Eisenstein series:
( 3 .4.4) Substituting this value in (3.4.3), we arrive at the following expression for the free term in (3.4.2):
s ( I - s ) - ,, ( 1 2 ,, - 1
- Ie)
1 2" - 1
X E{ z ;
kp
n
[
as -
al -
K.-S K. � � 8ap8kl ,,' ._ S + ® al ,Pk( s ) ,, + s _ I
fJ = 1 k = 1
1
kp
� � [ 8ap 8kl ( " + s - 1 ) a S-K. + ( " - S ) ® al,Pk( s ) a l - K. - s] n
fJ= 1 k = 1
Ie ;
( 3 .4.5 )
/3 ; e k( /3 ) ) .
In deriving (3.4.5) we used Theorem 3 . 1 .4. We introduce the notation
,
'
Val,Pk( S ) = 8ap 8kl( " + s - 1 ) a S-K. + ( " - S ) @5 al , Pk( s ) a l - K.-S , ( 3 .4.6) taking a and " to be fixed, and for the other indices taking 1 � a � n , 1 � I � k a ' 1 � /3 � n and 1 � k � kp . Just as in the definition of the matrix ® ( s ), we introduce new indices d and b according to (2.3.20). In this notation Val, Pk ( S ) = Vd,b( S ), 1 � d � kef ; X ), 1 � b � kef ; X). In order now to derive the desired Selberg-Neunhoffer equation from (3.4.2), we need only invert the matrix {Vd,b}��r�x? By Theorem 3. 1 .5, its determinant is a meromorphic function of s. We show that it is not identically zero. In fact, by Theorem 3. 1 .2, each function l1o( s), I � a, /3 � n and 1 � I � k a , can be represented by the Dirichlet series (3. 1 .9) in the region Re s > 1 .
From this and from Theorem 3. 1 .4 it is easy to see that each of the functions a -s ® d,b( s ) vanishes in the limit as Re s � 00 (we have the right to take the number a in Theorem 1 .2.4 to be large; in this connection, see also Theorem 3.5.2). Hence, if s has a sufficiently large real part, then the matrix { Vd,b( S )} is close to a multiple identity matrix, and so its determinant is nonzero for such s. We let Vi,b( S ) = Vai, Pk ( S ) denote the entries in the inverse matrix of {Vd,b(S)}. W e introduce the further notation:
U( z ; s ; /3 ; k ) =
n
k"
� � V;I,Pk ( s )E{ z ; s ; a ; e, ( a )),
a = 1 /= 1
( 3 .4.7 )
where /3 = 1 , . . . , n and k = 1 , . . , kp . From (3.4.2), (3.4.4) and (3.4.5) we obtain the desired generalization of the Selberg-Neunhoffer equation for the function .
U( z ; s ; /3 ; k ) =
2"
1
_ 1 E ( z ; ,, ; /3 ; e k ( /3 )) + [ s ( 1 - s )
f
-
X r O ( z ; z ' ; Ie ; a ) U( z' ; s; /3 ; k ) dp,( z' ) , F
where 1 � /3 � n and 1 � k � kp .
,,
( 1 - ,, ) ] ( 3 .4.8 )
III . FIRST REFINEMENT O F THE EXPANSION THEOREM
58
We proceed to study (3.4.8). An important difference between this equation and Faddeev's equation (2.2.6) is that the kernel of the integral operator in the former case depends more explicitly upon s. If the free term in (3.4.8) were an element of X(f; X) and the integral operator in the equation were a Hilbert-Schmidt operator, then this equation would be a Fredholm equation of the third kind, i.e., an equation of the sort well understood by classical mathematicians. However, it is not hard to see from Theorems 2. 1 . 1 , 2. 1 .2, 2.3.2, and 3 . 1 .4 that neither condition is fulfilled. Nevertheless, the question of solvability of (3.4.8) reduces to an investigation of a Fredholm equation of the third kind by means of a device similar to Selberg's " trick" in the scalar theory (see [50D. We make the following identity transforma tions in (3.4.8) :
[
at y( g�'z ) ] U( z ; k ) = � [ at y( g�'z ) ] E(z; K; 1,,[ t ( y( g� 'z ) - y( g�'z')) ] r O (z, z ' ; K; a ) X[ a� /( g;'Z')] U( z'; k ) z'),
exp -
2.
X
•
s; P;
\
exp - .
exp -
exIi -
•
P ; e k( p ) ) + [ s ( I - s ) - .(\ -
.)]
•
s ; P;
dl' (
( 3 .4.9 )
where y(z) = 1m z, z = x + iy , z' = x ' + iy', e is a parameter, e > 0, and z E F. We show that for fixed e (3.4.9) is a Fredholm equation for the vector-valued function
z E F, 1
� /3 � n , 1 � k �
kf3 .
In fact, from Theorems 2.3.2 and 3 . 1 .4 we have the inclusion
Then Theorems 2. 1 . 1 and 2. 1 .2 and the definition of :t( K ) in (2. 1 .27) imply that
fA
exp - 2 .
at ( y( g;'z ) - y( g';'z')) ] 1 r O t z, z' ;
. ; a ) I i- dl' ( z ) dl' ( z ' ) < 00 .
Thus, we can use the results of rredholm theory for (3.4.9) after passing to the limit as e -7 0, and also for (3.4.8). Here we shall limit ourselves to an outline of the steps. Equation (3.4.8) is uniquely solvable for all values of 'A = s ( l - s ) with the exception of a discrete set of singular points. In the resulting solution to (3.4.8) one can pass to the limit as e -7 ° for nonsingular A , and thereby obtain the desired solution U( z; s ; /3 ; k ) of (3.4.8). This solution can be represented in the form
U( z ; s ; /3 ; k) = W\(z ; 'A ; /3 ; k)/Wz( 'A ; /3 ; k ) , 'A = s ( 1 - s ) , ( 3 .4. 10) where W\(z; 'A; /3; k ) is a map F -7 V for fixed /3, k and 'A, and Wz('A; /3; k) is a complex-valued function for fixed /3 and k. As functions of 'A, W\(z; 'A; /3; k) and Wz('A; /3; k) are entire functions of order no greater than two. Now in order to prove
the exist�nce of a formula analogous to (3.4. 1 0) for Eisenstein series, one must
§3 .5 . THE DETERMINANT OF THE SCATTERING MATRIX
59
substitute on the right in (3.4.7) the asymptotic expansion as y -> 00 for the Eisenstein series E( g/3 z ; s ; f3; e r( f3 )), f3 = 1 , . . . , n , Z = x + iy -> E ITa (see Theo rems 2.3.2 and 3. 1 .4). Comparing the coefficients in the asymptotic formulas in the left and right in (3.4.7) leads to a proof of meromorphicity of the functions 'I',B-k ,ar ( s ), and hence of the functions 'I'/3 k , ar( S ) with order of meromorphicity in s no greater than four. This, in turn, implies meromorphicity of the Eisenstein series and the scattering matrix with order no greater than four, as well as formulas for them analogous to (3.4. 10). We have thereby proved a theorem which gives a refinement of Theorem 3 . 1 .5. 3.4. 1 . Each vJ the functions i,r- condition 1 ) of Theorem 3. 1 .5 has a representation as a ratio (3.4. 10); more precisely , THEOREM
E { z ; s ; a ; e r ( a ) ) = Wi z ; s ; a ; I )/ W4( s ; a ; I ) , �/ s ) = W7 ( s ; j ) / W8( s ; j ) ,
17/ S ) = Ws ( s ; j )/ ff6 ( s ; j ) ,
® ar , /3 k ( S ) = W9( s ; a ; I ; f3 ; k ) / WI O( s ; a ; I; f3 ; k ) ,
where 1 � a � n , 1 � I � k a , j E lL , 1 � f3 � n , 1 � k � k/3 and z for the indicated values of the parameters
E F. Furthermore,
W3( z ; s ; a ; I ) E V, W4( s ; a ; I ) E C , US ( s ; j ) E V, ff6 ( s ; j ) W7 ( s ; j ) E V, W8( s ; j ) E C , W9( s ; a ; I; f3 ; k ) E C ,
E C,
WI O ( s ; a ; I; f3 ; k ) E C . Each function ( or map ) Wm , m = 3, . . . , 1 0, is an entire function ( map ) of s of order no greater than four.
§3.5. In this section we explain several properties of the determinant of the scattering matrix { ® ar , /3k(S ) } (see (2.3. 1 7)); besides their independent interest, these properties will be useful in the derivation of the Selberg trace formula and in the study of the asymptotic behavior of a spectral function of the operator 21:(f; X) (see Chapters 4 and 5). Here we are generalizing results of the scalar theory which are due to Selberg (see [50], and also [68]). We recall that, with the change of indices (2.3.20), the scattering matrix is a square matrix of order k ef; X) = "1, � k a ' equal to the total degree of singularity of the representation X relative to the group f. We let �( s ) = �(s; f; X ) denote the determinant of the matrix ® ( s ) (see §2.3).
The determinant of the scattering matrix
3.5 . 1 . Let f and X be fixed. The function �(s ) has the following properties: 1) In the half-plane Re s > 1 it can be written in the form
THEOREM
k = k( f ; X ) , where I( s ) is an absolutely convergent Dirichlet series (Re s I( s ) =
00
� P; '
s
>
1 ),
Pm E C ' P ) =I= O ' 0 < Ql < Q2 < . .
·
·
m = 1 Qm 2) It is meromorphic on C with order of meromorphicity no greater than four. 3) It is regular in the half-plane Re s > 1 /2 except for a finite number of poles on the interval of the real axis s E 0 /2 , 1 ] ; each pole has multiplicity no greater than k e f; X ).
uv
I l l . t' l K :":>
1 K t r I N tMJ:: N T O r
THE EXPANSION THEOREM
4) For any r E IR
d ( i + ir ) 7"= O . 5) It satisfies the functional equations d ( s ) d ( 1 - s ) = 1 , des) = d ( s ) , where the bar denotes complex conjugation; in particular, 1 dU + ir ) 1 = 1 , r E IR . PROOF. All the assertions are direct consequences of theorems we have proved. In fact, 1) follows from Theorems 3. 1.2 and 3. 1.4; 2) is a consequence of Theorems 3. 1.5 and 3.4. 1 ; and 3) - 5) follow from Theorems 2.3.4 and 2.3.5. The proof is complete. The following theorems describe the subtler properties of the function d( s). THEOREM 3.5.2.
1) The function des) is bounded in the region l2 � Re s � l2 , 1 1m s I > 1 . 2) In the half-plane Re s � 3/2 the function q�Sd(s) is bounded, and, moreover, q�Sd ( s ) � O. Isl � oo
PROOF. In the Maass-Selberg relation (3.2.2) we set s = a + ir and sum the right and left sides .as I goes from 1 to k a and as a goes from 1 to n. The formula obtained in this way then gives us the inequality
[
1 20- 1 a k 2a - 1 +
/I
_
al - 20
±
�
±
a = 1 /= 1 {3 = 1
ka 1
� � 2ir [ ® al , al ( a + ir ) a2 i r - ® al , al ( a + ir ) a -2 ir] > O . a= 1 /= 1
From this we obtain
1 2 ka 4 .- 2 + 01 r a' ·- l ± a= 1
1
/I
� I ® al,.l o + ir ) I
/= 1
ka
/I
kp
> � � � � 1 ® al ,Pk ( a + ir ) 1 2 . a = 1 / = 1 {3 = 1 k = 1
( 3 .5 .1 )
From (3.5.1) it follows that each function ® al, Pk( a + ir) is bounded in the region 1 /2 � a � 3/2, 1 r I > 1 . Consequently, the determinant d( s) is bounded, and part 1) i s proved. Part 2) is a simple conseqllence of part 1) of Theorem 3.5.1. The proof is complete. Before stating the next theorem, we introduce some new notation. We let a1 � O � � o� denote all of the poles of Ll( s) in the interval s E (1/2; I ], 2 counting multiplicity (see part 3) of Theorem 3.5.1). By analogy with the scalar theory, we introduce the function •
•
•
'!)R,
d n;/ s) = q�S - I II
j= 1
s - a. � Ll(s ), · S - 1 aj
where the coefficient q l was defined in the conditions of Theorem
( 3 .5 .2 ) 3.5.1.
,
§3 .5 . THE DETERMINANT OF THE SCATTERING MATRIX
61
The function A reg( s) has the following properties: 1 ) It is regular in the half-plane Re s > 1/2 and continuous in Re s � 1/2. 2) A reg(s ) 0 as Re s 00 . + 3) It does not vanish for s = 1/2 ir, r E IR . 4) It satisfies the functional equations A reg( s ) A reg( I - s) = 1 , in particular, 1 A reg(l /2 + ir) 1 = 1 , r E IR . 5) It satisfies the boundedness condition I A reg ( S ) I .;;;;; 1' , . Re s � t . THEOREM
3.5.3. -i>
-i>
Parts 1 )-4) follow from the definition of A reg and Theorems 3.5.1 and 3.5.2. Part 5) is a consequence of parts 1 ) and 4) and the maximum modulus principle for analytic functions. The proof is complete. We now study the behavior of the function A(s ) in the half-plane Re s < 1 /2. Let p = f3 + iy be an arbitrary pole of A(s ) with f3 < 1 /2. Part 5) of Theorem 3.5 . 1 implies that p = f3 - iy is also a pole of A(s), and the points 1 - p and I - p are zeros. Obviously, p and p are poles of Areg(s), and I - p and 1 - P are zeros of this function. We consider the series PROOF.
_
�
.r:
f3 - 1/2 ' I p 12
(3 .5 .3)
where the summation is over all poles p of A(s) for which f3 = Re p < 1/2. THEOREM
3.5.4.
The series (3.5.3) converges.
PROOF. Since A reg(s) is analytic in the half-plane Re s > 1/2, and is continuous and bounded in the half-plane Re s � 1/2 (see Theorem 3.5.3), it follows by Carleman's theorem (see [56], §3.7 1) that �p f3/1 p 1 2 converges, and this implies the theorem. The proof is complete. From Theorems 3.5. 1 and 3.5.3 we already know that the functions A(s ) and A reg(s) are meromorphic with order no greater than four. We now construct special canonical products for these functions which are different from the usual Weierstrass product from the theory of entire and meromorphic functions. Using these products, we shall later refine our estimate for the order of meromorphicity of A(s ) and Areg(s) . THEOREM
3.5.5.
The logarithmic derivative A�eg I + - -lr 2 A reg -
(
.)
is nonnegative if r E IR . In addition, it can be represented by the series
where the summation is over all poles p = f3 + i Y of the function A( s) in the half-plane Re s < 1/2.
III . FIRST REt'INEMl::'N l " Ul' THE EXPAN SION THEOREM
£l reg( s), 3.5.4,
starting from the properties PROOF. We construct a canonical product for of this function described above. According to Theorem the following product is absolutely convergent:
II ( 1 (2s - 1) (s _2:)(s � p) ) +
(3.5 .4)
,
p
where the product is only over half of the poles p of in the half-plane Re s < (one pole is taken from the pair p, p ). The expression can also be written in the form
£l(s) (3.5 .4)
1/2
(3.5 .5) where the product is now over all p, but here we agree to take the factors for p and for together with one another. Next, we look for the function in the form
p
(s) £l r e g (3.5 .6) £l reg( s) = £l reg ( I ) II S -S _1 + p g ( - 21 ) ' ( s) £l 3.5 .3, g( s - �) = - !) - !f, 3.5.3, C2 £l reg(s) ± II s -s - p , £l(s) = ±ql- 2S II'JlL s -S -l oj II S -S -1 + - (3.5 .7) 1-2 _ £l'reg (l. ) � £l reg 2 - 1/2)2 (l/2) 4) E ± £l r e g 3. 5 . 3 ), q\, (3.5.2). ( s) £l r e g 3.5.5. 3.5 .6. , £l' ( 1 . ) = - £l£l'rreegg ( 1 . ) ( r12 ) , 2
P
p
exp
s
where, because of the estimate for the order of meromorphicity of result in Theorem we have
and the
c\ ( s
+ c2 ( s the constants c \ and C 2 must with c \ and constants. But, again by Theorem be zero. From this we obtain l + p
=
p
j= \
+ ir =
p
( f3
+
OJ
f3
+ (r
P
P
_
y)
2 ;;. 0
P
,
'
(see part where r �, the signs correspond to the possible values of of Theorem and the numbers OJ and M are defined in the context of · The proof is complete. The following fact is an immediate consequence of the definition of and Theorem THEOREM
+ r -T 2 z
where c3 is a constant.
2 + zr + c3 + 0
r � oo , r E IR
,
CHAPTER
4 THE SELBERG TRACE FORMULA :'.
In this chapter we shall suppose that f E 9J( and X E 9C( f ) (see and shall focus our attention primarily on the more difficult situation f E 9J( 2' X E 9C ( f). The modifications needed in the formulas in the alternative cases f E 9J( I ' X E 9C( f ) or f E 9J( 2 ' X E 9C rCf) are obvious, and involve a natural trivialization. As before, we shall suppose the group and the representation to be fixed, and shall not always indicate the dependence upon f and X in the notation.
§1.2),
s
§4.1. Nuclearity of the operator §4.1
x )Il3
0
(f; X)
In we shall prove a theorem which is in the spirit of the well-known theorem of Gel'fand and PjateckiI- Sapiro in the scalar theory of automorphic functions (see Chapter at the same time bringing in ideas from the theory of perturbations of continuous spectrum (see The proof will be based on our results from Chapter and Theorem We shall keep all of the notation used here. The proof of the theorem requires three lemmas. The first lemma was proved in Theorem For convenience, we shall state it here as a separate proposition.
[13],
1, §6),
) . [ 7 2] 3.3.2. 2.3.3.
2
, .
K( f ;
The operator O( s) satisfies the Hilbert identity; more precisely, the 4.1.1. following operator identity holds in :Je(f; X ) for any points s, s ' E C with s, LEMMA
Re s'
Re
> 1- :
O ( s ) - O ( s ,) = [ s { l - s ) - s'(l - s ')] O (s ) O(s ') . LEMMA The difference 9l (s) - O(s) is a Hilbert-Schmidt operator for any s E C satisfying the conditions Re s > 1- and s rt (1-,
4.1.2.
1].
PROOF. First suppose that Re s equality
>
2.
From Theorems
2.1.1 2.1.2 and
we have the
9l (s) - 0 (s) = � (s ) + :n (s) + (if (s ) + � ( s ) - 0 ( s ) ,
where �(s), :n(x) and
(if(s)
are Hilbert-Schmidt operators. From the definitions and it follows that the difference �(s) - O(s) is also a Hilbert-Schmidt operator, and this proves the lemma for s with Re s > Now let s be as in the lemma. We choose K E IR, K > and fix it. As in Chapter we let w(s) denote the function w(s) = - s) - K(1 - K), and let g denote the identity op�rator in :Je(f; X). We have
(2.1.27), (2.1.14), (2.2.13) (2.2.10) 2, s(1
2.
3,
( 63
4.1.1
)
1 v . 1 .t1� ��Ltl�KU I KAL� r U K M U LA
o ':t
In deriving (4.1.1) we have used the Hilbert identity for the resolvent 9T(s) and the operator D(s) (see Lemma 4. 1.1). From the Hilbert identity it also follows that
( 1 - W(S ) 9T ( K ))( 1 + w( s ) 9T (s )) = 1. (4. 1 .2 ) If we invert the operator ( 1 - W ( S )9T( K » on the left side of (4. 1 . 1 ) by means of (4. 1.2), we obtain an expression for 9T(s) - D(s) as a product of a bounded operator and the Hilbert-Schmidt operator 9T( K ) - D ( K ) (we have already proved the lemma for s = K ) ; this proves the lemma. Before stating the next lemma, we introduce some notation. We let II K II 62 denote the von Neumann-Schatten norm of an arbitrary Hilbert-Schmidt operator K (see [ 19]).
LEMMA 4. 1 .3. The following estimate holds in the region {s E e l Re s > 1 /2, s
(1/2, I ] ) :
tl
where n is the number of cusps in F, K is from Lemma 4. 1 .2, and w (s) = s( l - s ) K ( I - K ). PROOF. The formulas (4. 1 . 1) and (4.1 .2) and Lemma 4. 1.2 give us the estimate 11 9T ( s ) - D(s ) 11 62 � ( I + I w ( s ) 1 11 9T (s) 1I F.J ( 4. 1 .3 ) x ( 1 + I w( s ) I II D(s )II F,x ) II 9T ( K ) - D (K ) 1 1 6 2. Furthermore, by a general theorem in function analysis, the resolvent 9T(s) has norm bounded by
(4. 1 .4 ) Recall that the operator D(s) is associated to the kernel q(y, y ' ; s ) of the resolvent of a one-dimensional selfadjoint problem (see the situation in (2.2. 14» . Hence, its norm in :Je(f; X) satisfies an estimate similar to (4. 1.4):
II O ( s ) II F, x � n/I Im s ( l - s ) l . (4. 1 .5 ) If we substitute (4.1 .4) and (4. 1.5) in the right side of (4. 1.3), we obtain the lemma. The proof is complete. To formulate the basic theorem of this section we return to Theorem the definition of D(s) it is not hard to see that the operator
3.3.3. From
I
� {e s = so( I - 2s ) h ( s ( I - s )) O ( s ) ds
h( � ) = - 2 i
(4.-1 .6)
is correctly defined as a bounded operator in :Je(f; X ) if h(s(l - s» satisfies conditions 2a) and 2b) of Theorem 3.3.3 ; So was determined in the proof of that theorem. THEOREM
4. 1.1. I ) Suppose that the function Ii(s ) = h(s(1 - s» has the following
properties: a) it is analytic in the strip - e < Re s < 1 + e for some e > 0; and
, 65
§4.2. JUSTIFICATION OF THE SPECTRAL TRACE FORMULA
b) in this strip it satisfies the estimate
Ii(s ) =
o( ( 1 + I 1m s I) -4-8 )
for some 8 > O. Then h ( � ) - h( � ) and h ( � )�o(f; X) are Hilbert-Schmidt operators ( � o is the projection of :JC( f; X) onto :JCo(f; X » . 2) Suppose that the functions h(s), h1 (s) and 1i2(s ) satisfy conditions l a) and Ib) above, and also the equality Ii(s ) = h1 ( s )his ) ; then h( � ) - h(i) and h( � )�o( f; X) afe nuclear operators in :JC(f; X ). PROOF. Obviously, 2) is a consequence of I ), since a product of Hilbert-Schmidt operators is a nuclear operator. In addition, the definition of O(s) and the proof of part 2) of Theorem 3.3.2 imply the equality
h ( i) ) �o( f ; X ) = O. Thus, all of the assertions in the theorem reduce to the claim that the von Neumann-Schatten norm of the difference h( � ) - h(i) is finite if the function h(s(1 - s» is as in part I ) of the theorem. We prove this. We have
h( � ) - h(�)
=1
Re s = + e/2
1
( 1 - 2s )h(s ( 1 - s » [ ffi ( s ) - O (s )] ds .
( 4. 1 .7)
Lemmas 4. 1 .3 and 4. 1 .2 are now easily seen to imply the estimates
Il h ( � ) - h ( � ) 11 @52 « II ffi ( K ) - O ( K ) II @52 < ... . '
00 ,
where K is from Lemma 4. 1 .2. The proof is complete. THEOREM 4. 1 .2. Let the function Ii(s)
= h(s(1 - s» be as {n part 2) of Theorem
4. 1 . 1 . In the notation of Theorem 3.3.3, the integral operator with kernel p(z, z'; f; X)
= k(z, z'; f; X)
( � + ir; a ; e ( a ) ) dr , I
® E z' ;
(4 . 1 .8)
defined on F X F, is a nuclear operator in :JC(f; X ) , where k ( z , z ' ; f ; X ) = � X ( y )k ( u ( z , yz'» , yEr
and the functions k (t) and h(lj4 + r2) are related by the transformation (3.3. 1 6). PROOF. This theorem is a direct consequence of Theorems 3.3.2, 3.3.3, and 4. 1 . 1 . The proof is complete.
§4.2. Justification of the spectral trace formula In this section we prove that the trace of the integral operator with kernel (4. 1 .8) in :JC(f; X) can be computed as the integral of the value of the kernel on the diagonal. (According to Theorem 4. 1 .2, to do this it suffices to show that the kernel
... ..
.
... .L.L
...... ...... u
��.L...o.L, __
...
.L'-L 1.'-'
......
.I. '-'.I.'-.l.VLV L..tL"l..
x ) is defined and continuous on the diagonal.) Our method generalizes the method of [72] and the Selberg-Arthur method for justifying the trace formula for arithmetic groups on the rank one case (see [ID.
p(Z, Z'; r;
The next theorem will be stated and proved only in enough generality for our purposes in the proof of the Selberg trace formula. We let tr v denote the trace of an operator V � V. THEOREM 4.2.1 . Let the function h(s(l - s» be as in part 2) of Theorem 4.1 . 1 . In
addition, suppose that h l (s( I - s» = his(l - s» and that the function h I C A ) is real for values of the argument 0 � A � 00 . Then the following assertions are true: 1) The scalar kernel tr v p(z, z'; r ; X) (see (4.1 .8» is continuous in z for every z' E F and is continuous in z' for every z E F. 2) The function tr v p(z, z; r; X) is correctly defined, and is continuous on F. PROOF. 1) By Theorem 3.3 .3, it suffices to prove part 1) for the kernel
� t h ( ! + r' l E ( z ; ; + ir ; a ; e; { a ) l
T{z, z'; r ; X ) = L ±
a = i l= i
®
00
E (z' ; ! + ir ; a ; e, ( a )) dr
( 4.2 . 1 )
or for each term in (4.2. 1): T( z, z'; a ; I; r; X ) = } f_: h ( � + r 2 ) E ( z ; ± + ir ; a ; e, ( a) ) w ® E ( z'; � + ir ; a ; e t ( a) ) dr. ( 4.2.2 ) From part 1) of Theorem 3.3.3 we know that the kernel T(z, z'; a, I ) is defined for almost all z, z' E F. The scalar kernel tr vT(z, z'; a ; I ) is also defined almost everywhere. On the other hand, it is not hard to verify that the operator he m ) for h(s(l - s» as in the theorem is selfadjoint and nonnegative definite; and, by part 2) of Theorem 3.3.3, its kernel k( z, z ' ; r; X) is defined and continuous everywhere on F X F. Hence, tr v T( z, z; a ; I; r ; X) :;;;;; tr k ( z, z; r ; X ) , (4 .2.3 ) tr v T( z, z; r ; X ) :;;;;; tr k ( z, z ; r ; X ) for every z E F. By Holder's integral inequality we have I tr v T( z, z'; a ; l ) I 1 = 4 w I f_oo h ( ! + r 2 ) < E ( a ; � + ir ; a ; e, ( a ) ) , E ( z'; � + ir; a ; et ( a ) ) > drl v
v
oo / " 4� L C h ( ! + l lE ( z; ± + ir ; a ; e (a ) l [ dr ) 1 ' ( tooh ( i + r 2 l l E ( z'; ; + ir ; a ; e, { a) l l: dr ) - ' }
r'
,
I j'
X
trl/v 2 k ( z z · r ,· x ) tr l / 2 k ( z ' z,· r· X) (4.2.4) In deriving (4.2.4) we used the equality h(i + r 2 ) hrd + r 2 ) and the estimate (4.2.3). But we verified in the proof of part 2) of Theorem 3.3.3 that the integral �
"
v
·
'
"
.
=
§4. 3 . DERrvAnON OF THE FORMULA
67
0.3.20) for k(z, z ' ; f; X), regularized by means of (3.3.16), converges uniformly for z in a compact subregion of F. This implies uniform convergence of the integral on the left in (4.2.4) with respect to z in a compact region, for every fixed z' E F, and, in exactly the same way, its uniform convergence with respect to z' in a compact subregion of F, for any fixed z E F; this proves part 1). The proof of 2) is analogous to that of 1); it follows from (4.2.4) and part 2) of Theorem 3.3.3. The proof is complete. For convenience, we now combine the results in Theorems 4.1.1, 4.1.2, and 4.2. 1 into a single theorem. THEOREM 4.2.2. Suppose that the function h\(s) = h \(s(1 - s)) has the following properties: a) It is analytic in the strip - < Re s < 1 + for some O. b) In that strip it satisfies the estimate f
f
f >
for some 0 > O. c) h l "A) is real for 0 � "A < 00 . Then the function h (s( 1 - s)) = h �(s( I - s)) satisfies the trace formula � h ( AJ = j
,.
f tr p (z , z' ; f; x) d ( z ) , F
JL
v
(4.2.5)
where "Aj runs through the discrete spectrum of2:((f; X), and the kernel p( z, z'; f; X) is defined by (4.1.8).
§4.3 . Derivation of the Selberg trace formula
In this section we discuss transforming the spectral trace formula (4.2.5) to the Selberg trace formula. The following special cases of the Selberg trace formula for f E W1 and X E 9C(f) are well known and have been discussed several times in the literature: 1) f E W1 \ and X E 9C(f) (see [51], [13], [ 19] and others); 2) f E W1 2 and X E 9C /f), dim V = 1 (see [50], [28] and [72]); 3) f E W1 2 and X E 9C /f) (see [51)); and 4) f an arithmetic group in W1 2 and X E 9C(f) (see [23], [7], [1] and [18]) (for the notation, see 1.2). However, the general case of the Selberg trace formula for f E W1 and X E 9C /f) has not been considered before, as far as we know, either in the published literature or in Selberg' s lectures in Princeton (1952) and Gottingen (1954) (see [50)); hence, we shall focus our attention primarily on this case. As will be clear later, the Selberg trace formula we obtain for a general grollp f E W1 2 and a general representation X E 9C(f) in a certain sense includes all of the trace formulas I )-4) considered before, and its proof, modulo the results of Chapters 2 and 3, does not differ in any essential way from the proof of those formulas. We begin the transformation of the right side of (4.2.5) by observing that
§
f tr p ( z, z ; f ; X ) d ( z ) = lim f tr ( z ; f ; X ) d ( z ) lim f (tr k ( f; X ) - tr T( z, z ; f; X )) d ( z ) , F
JL
v
a -> 00
Fo( a )
a -> 00
v
z, z;
Fo( a )
vP
z,
v
JL
JL
(4.3.1)
68
IV. THE SELBERG TRACE FORMULA
where Fo(a) is a component in the partition of F (see §1.2 and (3.2.1» . We have to introduce an artificial passage to the limit in (4.3.1) because, in general, neither of the functions trv k(z, z; f; X ) or trv T(z, z; f; X) is integrable on F. The particular choice of passage to the limit is suggested by the Maass-Selberg relation (3.2.2). We now find asymptotic expansions for the integrals (4.3.2) fFo( a )trv k(z, z; f ; X) dJ.l. (z), (4.3.3)
as a � 00 . The divergent principal terms in these expansions will be the same, by Theorem 4.2.2, and so they will cancel in (4.3.l). After taking the limit as a � 00 , we will find that the other terms give the desired value for the matrix trace in (4.2.5). We now carry out this program. We first consider the integral (4.3.3). To construct its asymptotic expansion we shall need a certain consequence of the Maas-Selberg relation (3.2.2). THEOREM 4.3.1. Let r E IR . Then k f) E ( z ; s ; a ; e, ( a ) ; a ) l 2v d (z) = 2k ( f ; X)ln - 611' ( "21 + ir ) 11
�
a
a
1
_
I �I
a
J.l.
+
) ( + ir a 2 ir ( tr ® � 2�r
-
tr ® ( � + ir ) a- 2 ir )
( 4.3.4)
,
where 11'/11( s) is the logarithmic derivative of the determinant of the scattering matrix ® (s), tr ®(s) is the trace of the matrix ® (s), and k(f; X) is the total degree of singularity of the representation X relative to the group f.
The equality (4.3.4) is obtained from the Maass-Selberg relation (3.2.2) if we sum both sides over I and a, and then in the resulting relation pass to the limit as s � 1/2 + ir, r E IR. Here one must use Theorem 2.3.5 and (2.3.21). In fact, we consider the limit PROOF.
(
1
n
n
ka
a 2 a- 1 k ( f ; X ) - a l - 2 a � � � lim 17 --> 1 /2 211 - 1 a = I /= 1 {3 = 1
1 . hm (7 --> 1 /2 211 - 1
(
kp
� I ® a" ,8k ( J + ir)/
k= I
k ( r ; x ) k ( r; x )
k e f ; x )a 2 a- 1 . '--- a l - 2 a
�
d= 1
-
d 1
1 k ( r ; x) k ( r ; x)
2
d= I
b= 1
d= 1
b= 1
1 k ( r ; x ) k ( r ; x)
1
d ,b
1
b 1
® d ,b
® d ,b ,b
1
)
2 � I ® d,b ( + ir )1
2 / ) / = kef; x )ln a + (In a ) � � ( "2 + ir ( + ir ) ( + ir ) � � ® � ,b "2 "2 ) ) � ® ( 2 + ir ® � ( 2 + ir . - "2 � k ( r ; x) k(r ; x)
2
b= 1
11
)
1
1
(4.3.5)
§4.3. DERIVATION OF THE FORMULA
69
Next, we have
1 ® d,b ( 21 + ir 1 2 = tr [ ® ( 21 + ir ® ( 21 - ir = k ( r ; x ) . ) ) )] d�l b�l
k ( r ; x ) k ( r ; X)
(4 . 3 .6)
+ ir,
Differentiating the second equality in (2.3.21 ) with respect to we obtain
®'( 1 + ir ) ® (1 - ir )
,
. t5
s and setting s = 1/2
(1 + ir ) ®' (1 - ir ) .
(4 .3 .7 )
Finally, it is easy to verify the equality
tr[®'(1 + ir )®*(1 + ir )] = tr [ ®'(1 + ir )®(1 - ir )] = tr [ ®'(1 + ir )® - 1(1 + ir )] = det'( ®(1 + ir )) dec l ( ®(1 + ir )) = �' (1 + ir ) � (1 + ir ) . (4 . 3 . 8 ) -I
Substituting (4.3.6)-(4.3.8) in the right side of (4.3.5), we obtain the theorem. The proof is complete. We return to (4.3.3). We have
fFo(a)trv T(z, z; r ; X ) dp. ( z ) ,.
� � fFo( a)dp. ( z ) f_ dr 1 E ( z ; 2'1 + ir; a; e, ( a ) ) 1 2 h ( 41 + r 2 ) k" 1 1 4 = f dr h ( 4 + r 2 � � fF dp. ( z ) 1 E ( z ; 21 + ir; a; e, ( a ) 1 2 . 1 = 4'7T '7T
n
k"
a= 1
1= 1
00
00
)
- 00
00
n
V
/= 1
a= 1
)
o(a)
v
(4.3.9 )
We can interchange the order of integration in (4.3.9), because the integrand is nonnegative. We now prove a theorem which enables us to estimate the error which arises when we replace the sum under the integral sign on the right in (4.3.9) by the sum (4.3.4) as --') 00 .
a
Let h(s(l - s)) be as in Theorem 4.2.2. Then fFo( a )tr v T( z, z ; r ; X ) dp. ( z ) - -f- f drh ( ! + r 2 ) THEOREM 4.3.2.
'7T
x
�
�f
a = 1 1= 1
s; a; e,(a); a)
Fo( a )
00
- 00
i ( � + ir; a; e,(a); a ) i 2 = 0 ( 1 ) .
dp. ( z ) E z ;
V
a
-'>
00
(4.3 . 1 0 )
PROOF. According to the definition (3.2. 1) of the vector-valued function the difference on the left in (4.3. 10) is equal to E ( z;
70
IV. THE SELBERG TRACE FORMULA
. . ' 1 -\ + "S / / eJ a ) 2 2-,,;; a - / p + / 1' (ir )1' ( "2 + ir ) ( 21 + ir ) 2v] ' (4.3.11) �O
since integrating the Eisenstein series over IT a reduces to integration of the corre sponding constant term in the Fourier expansion (see Theorems 3.l.2 and 3.l.3). Now, because of (4.2 .3), which holds because the operator K(f ; X ) is . nonnegative definite, the right side of (4.3.11) is bounded above by the expression
[ k ( gp z, gp z ; 1' ; X ) - 4� t drh ( ! + r ' ) at /�1 �1 /I / I'(Z) oo P Xly l /2 +"Sape/ ( a ) + y ' /H';; r( ir )1' ( � + ir ) ( � + ir ) J - I �O
(4.3.12) as we see in our usual way, using (3.3.17),
But (4.3.12) is bounded by 0 (1) as a 00 , (3.3.20) and Theorem 2.1.1. We shall not dwell on the verification of this bound. The proof is complete. We now state a theorem which gives a final characterization of the asymptotic behavior of the integral (4.3 .3). �
.f
THEOREM 4.3.3. Let the function h(s(1 - s)) be as in Theorem 4.2.2. Then
Fo( a )
tr v T( z, z; f ; X ) dfJ. ( z ) =
1 k( f ; X)g(O)ln a - 47T
+ h(1{4) tr ® ( � )
1_0000 h ( 41 + r 2 ) Lf/1' ( "21 ir ) dr +
(4.3.13) 0(1), where the integral on the right in (4.3.13) is absolutely convergent; g( u) is the Fourier transform of the function h(1j4 + r 2 ) in (3.3.16). PROOF. By (4.3.9) and Theorems 4.3.1 and 4.3.2, we have f0)( a ) tr T( z, z; f; X ) d ( z ) 1 1_00 h ( ! + r 2 ) ( 2k( r; X ) In a - � ( � + ir ) + �. ( tr ® ( � + ir ) a 2 ir = 2r 4 7T 00 - tr ® ( � + ir ) a - 2 i r) ) dr + �� L v
J..t
+
. ' J
Q -). <X)
§4 . 3 . DERIVATION OF THE FORMULA
71
Next, from the Fourier integral formula we obtain 1 f_oo h ( ! + r2 ) � ( tr ® ( � + ir ) a2 ir - tr ® ( � + ir ) a -2 ir ) dr 4 w oo 2r =
h ( It4) tr ® ( � ) + 0 ( 1 ) . a -- 00
The integral involving the logarithmic derivative of the determinant of the scattering matrix is absolutely convergent because of the absolute convergence of (4.3.3) for each fixed a and because of Theorems 3.S.S and 3.S.6. The proof is complete. We now proceed to the asymptotic �xpansion for the integral (4.3.2). We make use of a device which is fundamental to the derivation of the analogous expansion in the scalar theory. The integral (4.3.2) is equal to the sum � � tr X ( Y ) i. k ( u (Z , y , - l yy'z ) ) d/-t (z), (4.3.14) F
'
v
{ Y } r y E r \r
y
o(a)
where { y h is the conjugacy class in f with representative y, fy is the centralizer of the element y E f in f, and the function k(u) is defined in Theorem 4.1.2. The summation in (4.3.14) is over all conjugacy classes { Y h and all cosets modulo the subgroups fy • A change in the variable of integration transforms (4.3.14) to the following form: (4.3.1S ) � tr x (y)j k(u(z, yz)) d/-t (z), {Y}r
v
B( a ; y )
where we have introduced the notation
B(a; y) =
U
Y ' E ry\ r
y'Fo (a).
It is easy to see that B(a; y ) becomes a fundamental domain for fy on the half-plane H as a 00 ; we shall denote this domain by Fy' i.e., B(oo; y ) = Fy• Just as in the scalar theory, we verify the following claim: ->
If y is the identity, is hyperbolic, or is elliptic, then the following integral is finite: lim j k(u(z, yz)) d/-t (z) = jFyk ( u(z, yz)) d/-t (z). (4.3.16) a -- 00 The structure of the centralizer fy is well known for such y, as is the procedure for computing the corresponding integrals (4.3.16) (see [SO], or [72], §3, or the Appendix to [28]). Hence we shall immediately give the result of the calculation of these integrals in terms of h (lj4 + r2) for the identity, for any hyperbolic element, and for any elliptic element. We introduce some notation. Every hyperbolic element P E G is conjugate in G to an element (transformation) z -> N(P )z , N(P) 1, z E H. Following Selberg, we shall call N(P) the norm of the hyperbolic element (or B(a; y)
>
the norm of the conjugacy class {Ph, since it is invariant under conjugation). THEOREM 4.3.4. 1) The following equality holds: t k(U (Z , z)) dl' (z) = 1;:1 f r(tanh wr) h ( ! + r ' ) dr,
oo E where E is the identity of the group f, I F I is the volume of F relative to the measure dp" and tanh is the hyperbolic tangent.
72
IV . THE SELBERG TRACE FORMULA
2) Suppose that k E 71... , k � 1 , and y = p k is a power of a hyperbolic element p E f. Then f k(u( z , yz )) d ( z ) In N(P) _ g(k ln N(P)), N(P) - N(P) 2 where g( u) is the Fourier transform of h(I/4 + r2) (see (3.3.16» , and In is the natural logarithm . 3) Let y = R k be a power of an elliptic element of order m, where k, m E k � 1, m � 2. Then f k ( u ( z , yZ )) d ( z f oo exp ( - 27Trk/m) h ( 1 + r ) dr. 1 ) - 2 m . (k 7T/m ) - f 1 - exp ( - 2 7T r ) 4 Thus, to construct the desired asymptotic expansion of the integral (4.3 .2), it remains for us to study the contribution to the sum (4.3.15) from all of the parabolic -
f.t
Fy
k /2
k/
71... ,
Fy
f.t
-
SIll
2
00
conjugacy classes in f; we now proceed to do this. We start with a definition. A parabolic element (or conjugacy class) will be called primitive if, first, it is not an integral power of any other element (conjugacy class) in f, and, second, in the group G = PSL(2, IR) it is equivalent (i.e., conjugate) to the transformation Soo: z -7 Z -+: J � Z E H. Any parabolic element in G is equivalent either to Soo or to S;;' 1 . The latter two elements are only equivalent relative to the reflection transformation. It is well known (see [SOD that the total number of primitive parabolic conjugacy classes in f is equal to n , the number of pairwise inequivalent cusps of F. For representatives of these classes we take the generators Sj ' 1 � j � n, of the groups If C f (see §1.2). The total contribution from all of the parabolic conjugacy classes to the sum (4.3.15) is equal to
tr ( Sf ) f a . k ( u ( z , SfZ )) df.t ( Z ) . (4.3.17) B( .S ) I We have used the equality B(a; 5;) = B(a; Sf), k E 71... , and the relation fSj = fj ' which follows from the discreteness of the group f (the group If was defined in §1.2),j = 1, . . . ,no We consider the integral 11
�
.j=
00
�
k = - 00 k of=. O
v
fB( a ,. S)k( u(z, 5;kZ ) ) dJL( z ) .
(4.3.18) Recall the equality t; ISfgj = S!. The. ,'change of variable of integration Z = gj Z ' reduces (4.3 .18) to the form (4.3.19) i -I B( a .,S)k{ u{z, z + k )) df.t ( z ) . We consider the domain of integration (tB(a; S). It is not hard to see that gjIB ( a; 5; ) = IT - IT a - D/ a ) , where, as in §1.2, IT a = {z E H I 0 � � 1 , a < y < oo}, ITo = IT a for a = 0, D;C a ) gj
X
is a measurable set lying strictly in ITo - ITc ' where the constant 0 depends only on the discrete group f, and, in addition, the Euclidean measure v(D;Ca)) of the set c >
§4.3 . DERlVATION OF THE FORMULA
73
approaches zero as a � 00 (see [72], §3). Now recall that the fundamental invariant of a pair of points is equal to Dia)
1 Z z ' 12 U ( Z , Z ') = -'--yy"--'-
and the function k(t) satisfies the estimate (3.3.l8): k(t) « ( 1 + t)- eo , where 1 1 + > o. From this we obtain
oo1). Now one has to consider the remaining part of the terms in (4.3.20), correspond ing to the projection (Iv - P) in (4.3.21): ak ( / 2 ) d 2 Re trv[ ( Iv - pJ x S (4.3.27 ) { f )] · y 2 ; j� l I�l 1a We use the notation in §2.1. The trace trv[(Iv - Po,)S.f] is equal to the sum of the corresponding eigenvalues (see (2.1.16) and (2.1.17» 4.3.28) tr A ( 1v - Pa)S:J = � exp 21Tip(}'a , ( I=k",+ where 0 < (}'a < 1 and h = dim We suppose that the eigenbasis of the operator X(Sa) is chosen in such a way that its first ka vectors correspond to the subspace Va' Thus, a typical term in (4.3.27) is an expression of the form (4.3.29) -- 00
n
00
h
V.
1
Similar sums were studied by Selberg in the scalar theory (see [50]). We shall give the computation of the sum (4.3.29) in terms of the function h(Ij4 + r 2 ), where we set () 21T(}/a for convenience, and we use (3.3.16). =
§4. 3 . DERIVA nON OF THE FORMULA
75
First of all, it is easy to verify that as a 00 the expression (4.3.29) has a finite limit, which is equal to 00 d = 2 1000 P00 (cos pO)k(p 2 U 2 ) duo (4.3.30) 2 1000 P� ( cos pO)k �22 ; �I ->
( )
I
We compute the right side of (4.3.30), introducing a small parameter and then passing to the limit as 00 : 00 (4.3.3 1 ) !�� 2 � 00 P� I (cos pO )k( p 2 U 2 ) duo EO
EO ->
Next,
f10 oo
:
oo � (cos pO)k(p 2 U 2 ) du = � (cos pO ) f k( p 2 U 2 ) du
p= I
p=I
10
(4.3.32)
Since 0 < Ola < 1, it follows that ei() =1= 1 . We have cos pO = In I I - exp i0 1 + 0 � I
-
P
C
V(eU
( ) Substituting (4.3.33) into (4.3.32), we obtain
� � u�
( 4.3.33 )
with a uniform constant in O. 00 00 f � (cos pO) k ( P 2 U 2 ) du = - "21 g( 0)In I I - exp i0 I + 0 ( (e) . 10
p= 1
Passing to the limit as left in (4.3.30):
EO ->
10 --> 0
0 in the last equality, we find a value for the sum on the -
g(O)ln l l - exp iO I .
Finally, the sum (4.3.27) is equal to n
-g(O) �
h
ln l l - exp 2 7T i0 a l + o(l).oo 1
�
(4.3.34)
a-+
If we now take (4.3.26) into account, we obtain the following result. THEOREM 4.3.5. The contribution to the sum (4.3.l5) from all of the parabolic
conjugacy classes is equal to k(f ; x) [ g(O) n a - g(O) In 2 + h( Y4) - 2� too h ( ! + r 2 ) i ( I + ir) dr] g (O) � � ln l l - exp 2 7T iO a l + 0( 1oo) . a-+ J
-
n
h
a = I I=k,,+ 1
l
Theorems 4.3.4 and 4.3.5 completely describe the asymptotic expansion of the integral (4.3.2). Along with Theorem 4.3.3, which gives an asymptotic expansion of the integral (4.3.3) as a 00 , they transform the spectral trace formula (4.2.5) into a Selberg trace formula. ->
r U l O. 3) In this strip h I es) satisfies the estimate THEOREM 4.3.6
for some � O. 4) h\( A ) is real for 0 � A < 00 . Then the following Selberg trace formula holds: >
�h }
( ! + 1/) = '2:' dim Vf_oooor (tanh 1Tr)h ( ! + r 2 ) dr +
�
{R}r
+2
�
d 1
k= 1
tdrsinv{xkk1T( R)/d) foo e +p exp-21Tr-21Tr k/d h .!. + r 2 ) dr (
� � � {P}r k = 1
)
(
x
- 00
I
(
(
) )
4
� trv{ xk ( P ) ) ln N(P ) g ( k ln N(P )) k/2 -k/2 - N( P )
N(P )
+ 211T f_:h ( ! + ·r � ) � ( � + ir ; f ; x ) dr - ke
f; X )
1T
fOO h ( .!. + r2 ) f'f ( 1 + ir) dr - 00
4
+ � ( k ( f ; X ) - tr ® ( � ; f ; X ) ) h ( ! )
- 2 ( k( r; X )ln 2 + at Ij: ln i l - eXP 2?TiOla l ) g(O). I
(4.3 .35)
§4.4. THE WEYL-SELBERG ASYMPTOTIC FORMULA
77
In the left side of (4.3.35) j runs through the set of all eigenvalues Aj = 1/4 + r/ of the discrete spectrum of the operator �(f; X), where we take both values of 1j which give the same Aj (i.e., the sum on the left in (4.3.35) gives twice the spectral trace). In the right side of (4.3.35) {R h runs through the set of all primitive elliptic conjugacy classes in f, d = d( R ) is the order of R, and {Ph runs through the set of all primitive hyperbolic conjugacy classes in f. PROOF. In the spectral trace formula (4.2.5) (more precisely, in the right side of it), we substitute the asymptotic expansions for the integrals (4.3.2) and (4.3.3) which are given by Theorems 4.3.3-4.3.5 and by (4.3.1 5). After passing to the limit as a � 00 and multiplying both sides of the resulting formula by two, we obtain the desired Selberg trace formula. The proof is complete. We now observe that, just as in the scalar theory, (4.3.35) remains true for a broader class of functions. We proceed to the definition of this class of functions. §4.4. The Weyl-Selberg asymptotic formula. Extension of the Selberg trace formula to a broader class of functions h
In this section we generalize the Weyl-Selberg asymptotic formula from the scalar theory (see [50]), refine our estimate of the order of meromorphicity of the determi nant �(s; f, X) of the scattering matrix, and, finally, describe a broader class of functions h (s( l s » than in the statement of Theorem 4.3.6 for which the Selberg trace formula still holds as an identity. We begin with an a priori estimate for the distribution function for the values of the norms of primitive hyperbolic conjugacy classes in f . We set -
'1T ( x ; f ) = { the number of primitive hyperbolic {P h I N( P} � x} . (4.4 . 1 ) The function '1T(x; f ) is obviously defined on the semiaxis 1 < x < 00 . Because the
series (4.3. 1 5) is absolutely convergent, we easily find the following estimate for this function (we shall later obtain an asymptotic formula). LEMMA 4.4. 1 .
'1T ( x ; f ) = O( x ) ,
x�
00 .
(4.4.2)
We now introduce the distribution function for the eigenvalues Aj of the discrete spectrum of �(f; X ) :
N( A ) = N( A ; f ; X ) = { the number of Aj for � ( f ; X ) I Aj � A} . The function N( A ) is obviously defined on the semiaxis 0 � A < 00 . By the Weyl-Selberg formula we mean the asymptotic formula, first obtained by Selberg [50] in the scalar theory (dim V = 1), for the sum of the function N(A; f; X ) and the value of the argument of the determinant of the scattering matrix at the point A of the continuous spectrum of �(f; X). In our opinion, this formula is d natural generalization of the well-known formula of Weyl in the spectral theory of the Laplace operator on a compact manifold (in particular, for � ( f; X), f E ITn I ) in the case when �(f; X) does have a continuous spectrum, i.e., f E 9)( 2 and X E 9( S < f). We note that in the theory of the operator �(f; X ) the Weyl-Selberg formula also degenerates into the usual Weyl formula on a noncompact fundamental domain F for f E ITn 2 if the representation X is regular.
IV . THE SELBERG TRACE FORMULA
78
4.4.1 (the Weyl-Selberg formula). The following asymptotic formula holds: . V · A , (4.4.3) N(A ; f; X ) - 41'IT f-TT T ( 21 + lr,. f; x ) dr A :OO I4'ITF l dlm where A = 1 /4 + T 2 , T > o. The second term on the left in (4.4.3) is absent if either f E 9)( 1 or f E 9)( 2 and X E 9C l f). PROOF. In the Selberg trace formula (4.3.35) we take for h(1 /4 + r2) the function h 0 + r 2 ; t ) = exp ( - (i + r 2 ) t ) , where the parameter t > 0 is fixed. It is easy to see that this function satisfies Theorem 4.3.6; in addition, its Fourier transform g(u; t) (see (3.3.16» can be THEOREM
!J. '
computed exactly:
g(u; t) = ( 2 F! ) l exp( t/4 - u 2/4t ) . We now successively find the asymptotic behavior as t � 0 of each term on the right in (4.3.35), except for the asymptotic behavior of the term with the logarithmic derivative !J.(1 /2 + ir; f; X). We have I F 1 dim V f oo r( tanh 'lT r)h ( 1 + r 2 ; t ) dr = I F 1 dim V 1. + 00 ), 2 'IT t ( ..... 0 4 2 'IT � tr� ( x k ( R» f oo exp( -2'ITrk/d) ( 1. � dsm(k'IT/d) 1 + exp(-2'ITr) h 4 + r 2 ; t ) dr = 0 ( 1), k (P»)ln N(P) 0 2 � � Ntrv(X (P) k/2 - N(P) _ k/2 g(k ln N(P); t) = 0 ), (4.4 .4) -
-
- 00
d
\
{Rh k= \
t
- 00
00
t
{Ph k= \
� ( k ( f ; X) - tr ® ( � ; f ; X ) ) h ( ! ; t ) = o( 1 ) ,
-2 kef; x)ln 2 +
(
n
�
dim
�
V
a = \ l= k", + \
t
)
O
.....
.....
O
O
( )
ln l l - exp 2 'ITi8la l g(O; t) = 0 r;1 .....
Vt
.
( ..... 0
The only relatively nontrivial of these formulas are the estimates for the integral with the logarithmic derivative of the gamma-function and the sum over the hyperbolic classes {Ph. We obtain the first estimate in §4 of [62], to which we refer the reader. The second estimate follows from Lemma 4.4.1 and the definition of g( u; t). Thus, we have 1 + 2 ,. t ) - 1 f oo h ( 1 + r 2 ,. t ) !J.' ( "21 + lr ) dr �h( £.J J
4
1
lj
2 'IT
_ 00
4
T
V 1. + 0 ( In t ) - I F I 2dim 'IT t vt _
r;
.
,
t
--'>
o.
(4.4 .5)
§4.4. THE WEYL-SELBERG ASYMPTOTIC FORMULA
79
The left side of (4.4.5) is up to O( 1/ Ii) equal to 2 foo ( exp t ;\ )d ( N( ;\ ; f ; X ) + Q ( ;\ ; f ; X )) (see Theorem 3.5.6), where, by definition o
Q(A ;
r;
(4.4.6)
-
X) =
! 41'71" fF� -
-
-
0,
VA - 1 /4
(1 l.r) dr, 2
/:).' re g
- +
/:). r e g
1
o � A � -4 ,
and the function �reg(s) defined in §3.5. According to Theorem 3.5.5, the function Q(;\; f; X ) is monotonically nondecreasing on [ 0, 00). The sum N( ;\; f; X ) + Q(;\; f; X) obviously has the same property. Hence, the theorem follows from (4.4.5), (4.4.6) and a well-known Tauberian theorem. The modifications needed in the proof of the theorem when the spectrum of 21(f; X ) is purely discrete, i.e., either f E W( I or f E 9Jl 2 and X E �n r(f) (see §1.2), are obvious and are connected with corresponding simplifications in the Selberg trace formula (4.3.35). The proof is complete. We now refine our information on the order of meromorphicity of �(s), s E C (see Theorem 3.5.1). We return to the notation of §3. THEOREM 4.4.2. � IYI"';; T 1 « T 2 , where the summation is over all poles p f3 + i y of the function �(s) in the half-plane Re s < 1/2. PROOF. We note that all poles p of �(s) with f3 < 1/2 lie in the strip - cj Re s < 1/2 for some sufficiently large Cj > 0 which depends only on f and x . This follows from parts 1) and 5) of Theorem 3.5.1, since the Dirichlet series l(s) does not vanish for Re s > Cj if cj > 0 is sufficiently large. Furthermore, from Theorems 3.5.5 and 4.4.1 we have =
",;;;;:
j
T
'" £.J
-T p
According to Theorem 3.5.4, �
I yl > 2 T
f
T
-
T (f3
-
1 - 2f3 ( f3 - 1/2)2 + ( r
_
y )2
dr « T 2 .
(4.4.7)
1 - 2f3 = 0 ( T ) . 1 - 2f3 T � dr « 2 2 2 1/2) + ( r - y ) I y I2 I yl > 2 T
T-> 00
Consequently, the following sum gives the basic contribution to the asymptotic behavior of the left side of (4.4.7 ): �
j
T
l y l ",;; 2 T - T
1 - 2f3 dr = 2 2 2 r + 1/2 ( y) (f3 - )
This leads to the estimate
�
l y l ",;; 2 T
j
( T -- Y )/( 1 /2 -. {3 )
- ( T + y )/( J /2 - {3 )
1 dr. 1 r2 +
1 ",;;;;: � f T ( f3 1/21 � +2f3( r - ) dr ",;;;;: C3 � 1 y - ) with constants 0 < C2 ",;;;;: c3 ; this implies the theorem. The proof is complete. The following fact follows from (3.5.6) and Theorems 3.5.1 and 4.4.2. THEOREM 4.4.3. �(s) is a meromorphic function on C of order no greater than two. c2
�
lyl ,;;; 2 T
lyl ",;; 2 T
--
T
2
l yl ,;;; 2 T
ov
I V . T H E SELBERG TRACE FORMULA
Using Theorem 4.4.1 and Lemma 4.4.1, we easily verify the following theorem, which extends the class of functions h( s(l - s)) for which the Selberg trace formula (4.3.35) is valid. THEOREM 4.4.4. The Selberg trace formula holds as an identity for any function h(s(l - s)) h(l/4 -+- r 2 ) which satisfies the following conditions: I) h(I/4 -+- r 2 ) is analytic as a function of r in the strip 1 1m r 1 < 1/2 + for some E > O. 2) h(l/4 + r 2 ) = 0((1 + 1 r 1 2 )- I -f) in this strip , and all of the series and integrals =
to
in (4.3.35) converge absolutely.
In conclusion, we note that the class of functions h(s(1 - s)) for which (4.3.35) remains valid can be extended further than in Theorem 4.4.4 ; however, the series over the hyperbolic conjugacy classes in the right side of (4.3.35) may stop converg ing absolutely (for example, for nonanalytic functions h ), and one must clarify what one means by convergence in such a case. We shall not dwell any more on this question here.
j
CHAPTER 5 ELEMENTS OF THE THEORY OF THE SELBERG ZETA-FUNCTION. SPECTRAL AND GEOMETRIC APPLICATIONS OF THE THEORY
In Chapter 5 all of the assumptions listed at the beginning of Chapter 4 are still in force. §5.1. The definition and basic properties of f; X) The material in this section is a generalization of analogous results in the scalar theory (dim V = 1) which we obtained in [68]. The brief remarks at the end of Selberg's lectures [50] served as a stimulus for this. By the f ; X) we mean the product
Z(s;
Selberg zeta-Junction Z(s; Z( s) = Z(s; f ; X ) = II
00
II
{Ph k=O
...
.
det v ( I v - x ( P ) N( P fS- k ) ,
(5 . 1 . 1)
where {Ph runs through the set of all primitive hyperbolic conjugacy classes in f, N( P ) is the norm of P f (see §4.3), I v is the identity operator in V, and det v is the determinant of an operator in V (see §1 .2). The definition (5. 1 . 1 ) is correct for Re > 1 , where the product converges absolutely, by Lemma 4.4. 1 . The zeta-function Z(s; f; X ) is connected with the Selberg trace formula in the same way as the Riemann zeta-function is connected with Weil's explicit formula in analytic number theory (for f 9)( 1 see [5 1], [ 1 8] and [68]). From (4.3.35) we shall derive an important representation for the logarithmic derivative of the Selberg zeta-function from which we shall later obtain all the basic properties of We now state and prove a theorem, using the notation from Theorems 3.5.3, 3.5.4 and 4.3.6.
E
s E. C, s
E
Z(s),
Z(s).
THEOREM 5. 1 . 1 . Suppose that s E C (Re s > 1), and let a E IR (a Then Z' (s '' f . x ) = _ ( s ! I F I dim V � ( _1_ 1 Z , 2) 7: s k a k ) �
+
'IT
_
d� l
k 0 trv { X k ( R ) ) d sin 'IT/ )
_
{Rh
k= I
be Jixed.
'IT
'IT
+
1)
+
(k d X [ ( 'lTexp ( - 2 ik ( s - � ) /d ) ) ( 1 - exp ( - 2 is ) ) - I _
>
i ( S - ; ) �1 ( exp ( -2 " ik ( I - ; ) /d ) ) [ ( S - ; ) ' - ( I - ; ) 'r]
I
81
V . Ttl!:': S!:':LH!:':KU Zb l A-l' U N CTlUN
Il' f; X) + ( s - "21 ) - "21 -X-(s; 1 1 X � [ ( s - 1/2)2 - ( p - 1/2)2 ( a - 1/2f - ( p - 1/2)2 1 f ' ( -3 - s ) - ( s - -21 ) � s 1/2f 1 + kef X) · ' f 2 /= ( - - «1/ - 1/2f 1 -2 k( f ; X) ( s - -21 ) k� = l ( s - 1/2) 2 - k 2 - ( k( f; X) - tr ® ( � ; f; X ) ) 2s � 1 1 1 ) + c4 + c5 ( s - -1 ) + � (s - 1/2)1 2 1/ s 2 2 ' (a - 1/2)2 + 1/ (5 1 .2) where c4 and c5 do not depend on s. PROOF. In the Selberg trace formula (4.3.35) we choose for h(1/4 + r2) the function h ( I /4 + r 2 ; s; a) which depends on two parameters s and a: h( t + r 2 ; s; a ) = [( s - 1) 2 + r 2r l - [( a - 1) 2 + r 2] - I. In order for this function to satisfy the conditions of Theorem 4.4.4, it is sufficient to require that 1 Re s a, where a is a fixed positive number, 1 < a. The corre sponding function g(u) = g(u; s; a) in (3.3.16) is easy to compute: g(u; s; a) = (2s - 1 )- l exp( - ( s - 1 ) I u I ) ( 5 . 1 . 3) - (2 a - 1 )- I exp ( - ( a - 1) I u I ) . The Selberg trace formula for the function h(1/4 + r 2 ; s; a) has the form � h ( ! + r/ ; s; a ) = I F I:: V f : r(tanh 7Tr )h ( ! + r2; s; a ) dr \ tr ( X k ( R )) OO .l . exP(-27Trk/d) d � + � \ d sin ( k7T/d ) f h ( 4 + r 2 ." s a ) 1 + exp(-27Tr) dr k= tr ( X k ( p )) In N(P) (5 . 1 .4) +2 � � N(P) k/2 - N(P)_k/2 g(kln N(P); s; a ) (Ph k=\ '. + 27T1 t h ( 1 + r 2; s; a ) -X-Il' ( 21 + ir ) dr oo k( f · X) i ( 1 - ; oo h 4 + r 2 ; s; a ) Tf ' ( 1 + ir) dr + l2 (k( f ·' X) - tr ® ( l2 ) ) ( ( s - 11/2) 2 ( a 1_) - 1 /2) 2 1 ) - ke f ; ) ln 2 + a� /=-� �+ \ ln l l - exp 2 7T i Ola l ) ( 1 \ s - 1/2 a - 1/2 ' p , /3 < 1 j2
�
1
00
j(
+
""[ j
.
00
( -27Tik(l ; 1 /2)/d ) ( a + iT _ I2 ) � ( a exp = 0( 1 ) , + iT - 1 /2 ) - ( 1 - 1 /2 ) 2 f' ( 3 a - iT = O 1n T , ( ) r 2 ) ( a + iT - 21 ) I�00 ( a + iT - 1 /2 - I )(1 a + iT - 1 /2 + I ) = O��� ) , I 1 ) ( a + iT _ I2 ) � 2 = o( ! 2 1= 1
T-" oo
T-" 00
-
j= 1
(a +
�
iT - 1 /2 ) - ( aj - 1 /2)
T-"
We now consider the sum
( s - -21 )
�
P . f3 < �
[
1 2 ( s - 1 /2 ) - ( p
-
1 /2 ) 2 ( a - 1 /2) 2
'
1 -
1(5 .2.. 10)
( p - 1 /2 ) 2
By Theorem 4.4.2, it is bounded, like the sum (5.2.9), by a term of order 0(T 2 ) for " admissible" values of T. We shall not give the details of this estimate, since they are standard in analytic number theory. Finally, Theorem 3.5.5 gives us the estimate
( d'/d )( s; f; X ) = 0(T 2 ) for an admissible value of T, by analogy with the estimate for (5.2 . 10). Combining
these results leads us to the final estimate
(Z'/Z)( s; f ; X ) = 0( T 2 )
( 5 .2. 1 1 )
v. THE SELBERG ZETA-FUNCTION
90 a
the region (5.2.7) and an " admissible" value of T. Integrating (5.2.1 1) with respect to s along the line segment from a + iT to 2 + iT, we obtain for s =
+
iT in
r; X ) 1 = 0( T 2 ) , T � 00 , which implies the desired estimate (5.2.8). We have also used the easily verified fact I ln Z( s ;
that I ln Z ( 2 +
iT; r; X ) 1 = 0(1 ) , T � 00 , as follows from the definition (5. 1 . 1). To derive the lemma from (5.2.8) we use the Phragmen-Lindelof principle. This principle will be applied in the sector
'IT/4 � arg( s + 1 - i ) � 'IT/2. ( 5 .2 . 1 2) We study the asymptotic behavior of Z(s; r; X) on the rays which bound this sector. In the first place, it is obvious that on the segment arg( s + 1 - i) = 'IT/ 4 we have Z(s; r; X) = 0(1), since this ray extends into the region where Z(s ; r; X) is given by the absolutely convergent product (5. 1 .1). Next, we set s = - 1 + iT, T > 1 . To estimate Z(- 1 + iT; r; X ) we make use of the functional equation for this zeta-func tion (see Theorem 5.1.5). In the right side of (5. 1 .1 3) we take s = 2 - iT, and we estimate the behavior of each factor for large values of T. The following estimates are easily verified:
Z(2 - iT; r; X ) = 0( 1 ) , � (2 - iT; r; X )
(
i
=
0( 1 ) ,
)
2 iT exp -I F I dim V 3/ - t tan 'ITt dt = exp o
( 3/2 - iT
J0
[ exp( - 2'ITikt/d ) 1 (-2 'IT it ) + exp
( � 1 F I dim V · T ) + 0( 1 ) ,
exp(2'ITikt/d )
1
+ 1 dt = 0( 1 ) ' + exp (2 'IT it ) T� 00
T� oo
l r ( r( ! + iT ) r( i iTf ( ; x ) = 0( 1 ) . -
T--> 00
-
We have thereby found that the function
r; x ) exp ( 1 1 F I dim Vis ) is analytic in the interior of the sector (5.2.12) and admits the estimate there, Frs) = exp 0 ( 1 s 1 2 ) , F(s ) = Z( s ;
while on the boundary of the sector it admits the estimate F(s ) =
( 5 .2.13 ) §5.61), the
0( 1 ) .
Consequently, by the Phragmen-Lindelof theorem (see [56], Theorem estimate (5.2. 13) also holds everywhere inside the sector (5.2. 1 2); and this proves the claim Z( s ;
r ; X)
=
O ( exp 1 1m s I ) ,
and hence the lemma. The proof is complete.
-1
� Re s � 2 , 1m s
> 1,
§S . 2 . THE REMAINDER IN THE WEYL-SELBERG FORMULA
LEMMA 5.2.4.
arg Z o - + iT; f ; X )
91
= 0( 1 ) ,
where the value of arg is defined in Lemma 5.2. 1 . PROOF. The lemma follows from Lemma 5 .2.3, and also from a version of Jensen' s theorem (see [57], Chapter 9, §4) and the following lower bound, which is easily derived from (5. 1 . 1 ): Re Z( a + it ; f; X) � 1-
for any a � a o = a o (f; X) > 2 for some fixed a o . (In this connection, see also [ 1 9], Theorem 7.3.) The proof is complete. It is clear from Lemmas 5.2. 1 , 5 .2.2 and 5 .2.4 that we have already refined the Weyl-Selberg asymptotic formula (see Theorem 4.4. 1 ), by taking out another prin cipal term proportional to TIn T. However, the remainder term in this refined formula still has order O(T), by Lemma 5 .2.4. In order to increase the precision of the formula, we must find a better estimate for the order of the function arg Z(-t + iT). We now show that
l arg Z o - + iT; f ; x ) l « Tjln T. Lemmas 5.2.5-5.2. 1 1 are devoted to proving this assertion. They are formal generali zations of corresponding facts from the scalar spectral theory for a cocompact group f (see [44]) or for f E WC2 (see [68]). Here we shall focus our attention primarily on the differences in the proof, and in the parts of the proof which are the same we shall refer the reader to these articles (mainly, to the detailed paper [44]). We introduce some standard notation. Let
Se T ) =
! arg z ( � + iT; f ; X ) ,
where the value of arg is defined in Lemma 5 .2. 1 . Next,
l
S I ( T ) = TS(t) dt, qJ ( t )
=
o
max
l ';;; u';;; t
l SI ( U ) I .
We shall now let p denote an arbitrary zero of the Selberg zeta-function Z(s; f; X ). We set s = a + it, t > 1 0, 1 < v < 5 j4 is fixed. LEMMA 5.2.5.
fV In I Z( a + it ; f; X ) I da = O( t ) . 1�
t - oo
PROOF. Lemma 5 .2.3 and Lemma 1 of [57], Chapter 3, §9, imply that Z' 1 Z
(s; f ; X )
=
-
� s - p + OCt ) , p
t- 00
(5 .2 . l 4)
which hold.s for I a - v I ,,;:;; 1 . The sum on the right in (5.2. 1 4) is taken over all zeros p lying in the disc I p - v - it I ,,;:;; 2. If we integrate the left and right sides of this
v.
THE SELBERG ZETA-FUNCTION
equality over the line segment from a + it to a In Z ( a + it ; f; X ) - In Z ( v + it ; f ; X )
+ it, we obtain
(5.2.15)
= � [In( a + it - p ) - In( v + it - p )] + O ( t ) .
4.4.2, 5.1.3 (5.2.15), V
t � oo
p
From Theorems and Lemmas of zeros p over which we are summing in Integrating we obtain
f
I�
5.2.1, 5.2.2, 5.2.4 it follows that the number (5.2.14) is a number of order OCt) ( t (0) . �
I ln Z ( a + it; f; X ) I da
= O( t) , t � oo
from which the lemma follows. The proof is complete. The proofs of the next three lemmas repeat almost verbatim the proofs of Lemmas in they are based on Lemma Here we shall limit ourselves to the statements of these lemmas.
5 - 7 [44] ;
5.2.6. I S I ( T ) 1 « T. LEMMA 5.2.7. Suppose that -t
5.2.5.
LEMMA
1, the following estimate follows from (3.2.2):
( 6.3.9 ) By (6.3.6) and (6.3.9), we have
fF1 T( g) £ ( z ;
s;
a ; a ) 1 2 dJL ( z ) « a 2 Re S - I ,
= 1, . . . , n 1 T( g)E( gpz; s; a ) l « yRe s , 1m z and 1 .:;;;; a .:;;;; n.
and this shows that for every /3
=
(6 .3 . lO)
where z E ITa' y N ext, from the differential equation
-LT( g )E( z ; s ; a) = s ( l - s ) T( g )E(z ; s ; a ) and (6.3.l0), we find, just as in §3. 1 , that the following Fourier expansion holds for the values of the parameters 1 .:;;;; a, /3 .:;;;; n: T( g) E ( gp z; s ; a )
=
A ( lX ; /3; s ) y S + B ( a ; /3; s ) i - s + fY � Crn ( a ; /3; s ) Ks - 1 /2(27T 1 m 1 y ) exp 27Timx, m E Z, m*O
(6.3 . l 1 ) where z = x + iy E H, with various coefficients A( a; /3; s), B ( a ; /3; s) and Crn( a; /3; s) not depending on x or y . For a fixed value of a one can say that not all of the coefficients A( a; /3; s ) are identically zero. Otherwise, T(g)E( z; s ; a) would be a square integrable (on F ) eigenfunction for 2£ ( r; 1 ) with eigenvalue A = s(l - s ),
112
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
Re s > 1, and this would contradict the selfadjointness of difference
W(f ; 1). We consider the
n
T( g) E ( Z ; s ; a) - � A ( a ; fi ; s ) E ( z; s ; fi ) .
( 6.3 . 1 2)
{3 = 1
Since (see Lemma 3.1.1), it follows that the difference (6.3. 12) lies in the space X(f ; 1) and is an eigenfunction for the operator W(f ; 1) with eigenvalue s(l - s). Again using the selfadjointness of W(f ; 1), we find that the difference (6.3. 12) is identically zero; this proves (6.3.7). Here one should set
Ha{3( s ; g) = A( a ; fi ; s ) (see (6.3. 1 1» . Finally, (6.3.8) is an easy consequence (6.3.7). The proof is complete. We now describe some properties of the coefficients Hais ; g), where we assume that the group f satisfies the hypothesis of Theorem 6.3.3. The definitions (3. 1 . 1 ) and (6.3.3) imply that e
T( g)E(g{3 z ; s; a) = �
� y S ( g� lgYj g{3 Z ) ,
( 6 .3 . 1 3 )
j = 1 y E ra\ r
where the series converges absolutely for Re s > 1. From Theorem 6.3.3 it follows that Hais; g) is the coefficient of yS in (6.3. 13). This coefficient is already de termined by the terms of the series which correspond to transformations 0': H � H of the form ( 6 .3 . 1 4) O'Z = g� l ygYj g{3 z = p 2Z + q ,
where Z E H and p , q E R . We show that the set of such 0' in (6.3.13) is finite. In the first place, since (6.3. 13) converges absolutely, so does the series
( 6.3 . 1 5 )
a
for Re s > 1. By (6.3.2) and (6.3.3), the set of elements through f,j = 1, . . . , e, coincides with the double coset
g� l ygYj g{3'
g� l fgfg{3 .
where
y
runs
( 6.3 . 1 6)
Next, since r is a group, it follows that whenever g E r we also have g- l E r, i.e., there exists a (nontrivial) Hecke operator T(g- l ). If we consider the function which is defined by a series absolutely convergent series
T( g- I ) E ( gaz ; s ; fi ) , analogous to (6.3.1 3) for
where O" z = gp l yg- l Yj'gaz = p,2 + q ', Y E f{3\ f, j' that the set of elements gp l yg- l Yj'ga for y E f and with the double coset
Re s >
1,
we arrive at an
( 6 . 3 . 1 7) = I , . . . , e'. It is not hard to see Yj' E ( gfg - 1 n f)\f coincides ( 6 .3 . 1 8)
113
§6.3. THE SPECTRUM OF m: ( f ; I ) . FIRST CASE
However, the cosets (6.3. 1 6) and (6.3. 1 8) consist of mutually inverse elements; hence p = p '- I , and the series (6.3. l 5) and (6.3 . l 7) can only converge at the same time if one of them (and hence both) has only finitely many terms. We have proved the following theorem.
If r E ffiC 2, n and T(g) is a Hecke operator, then the coefficients HaP (s ; g) in the expansion (6.3.7) have the form Hap ( s ; g ) = � p 2 s ( a ; /3 ; I ; g) , ( 6 .3 . 1 9) THEOREM 6.3.4.
I
where p(l) = p( a; /3; I; g) E R and the . sum only has finitely many terms; a, /3 = 1, . . . ,no A s before, we shall let ® (s) denote the scattering matrix of the operator mer; 1 ), ® (s) = { ® ap(S)}:,P = 1 (see the context of (2.3.21» . Recall that in this section X = 1 and dim V = 1 . We introduce the notation for a matrix of order n: THEOREM 6.3.5. Suppose the conditions of Theorem 6.3.3 are fulfilled. H(s; g) = ® ( s )H( 1 - s; g)® (1 - s ) .
Then (6 .3 .20)
PROOF. The explicit formula (6.3. l9) implies that Hap (s; g) is an entire function of s for every a , /3 = 1, . . . , n. By Theorems 2.3.6 and 3 . 1 .4, we have the functional relation
n E ( z; s ; a) = � ® ap ( S ) E ( z ; s ; /3 ) , P= 1
a = 1, . . . ,no We have T( g )E(z ; s ; a) =
( 6 .3 .2 1 )
n � Hay (s; g )E( z ; s ; y )
y= 1
n n s; Ha g) � ® Y8(S )E(z; 1 - s; 8 ) . = � y( 8= 1 y= 1
(6 .3 .22)
The left side of (6.3.22) has the form
T( g)
[ Y� ® .y ( s )E(z ; 1 - s ; y) ] 1
=
® .y( s ) T( g)E(z; 1 - s ; y) t y
n n s ®a = � y ( ) � Hy8 ( I - s; g)E(z ; l - s; 8 ) . y= 1
8= 1
Because of the linear independence of the functions obtain
n � HaJ s ; g ) ® eP ( s )
f= 1
=
E(z; s; a), a =
n � ®a8 ( s )H8P ( 1 - s; g ) .
8= 1
1 , . . . , n, we (6 .3 .23)
From the relation ®(s)®(1 - s) In ' where In is the n X n identity matrix (see Theorem 2.3.5), and the equality (6.3.23), we obtain the desired relation (6.3.20). The proof is complete. =
1 14
VI . SECOND REFINEMENT OF THE EXPANSION THEOREM
We now prove a theorem which confirms Roe1cke's conjecture that there are infinitely many eigenvalues of the discrete spectrum of &(f ; X ) in the case of a group f E WC 2 n with a " large" commensurable and X = 1 (compare with Theorems 6.2.4, 6.2.5, 6. 1.1, and 6. 1 .2). For simplicity we shall not deal with the most general case. We introduce some notation. Let g E f for f E WC 2 n . We let 1) denote the j operator ,
,
1) =
T( gj )
+
T( gil ) ,
( 6.3.24)
and we let H ,a{J( s) denote the function
j
Hj ,a{J ( s ) = Ha{J ( s ; gJ + Ha{J(s ; gi l ) , where T(g) and T(gil) are Hecke operators, and Hap( s ; g) and Ha{J( s ; gil ) are the
corresponding coefficients in (6.3.7). We shall suppose that the group f has n 2 operators 1). Recall that n denotes the number of pairwise inequivalent cusps on the fundamental domain F for f. We make the change of indices
�1(S ) = Hj , a{J ( s ) ,
( 6.3 .25 )
where 1 = ( a l)n + /3. If a and /3 vary in the range 1 � a, /3 � n , then I runs through the set 1, . . . ,n 2 • We introduce the square n 2 X n 2 matrix made up of the functions (6.3.25), and its determinant -
� (s ) = det { �l ( s ) } ��l= . l THEOREM 6.3.6. Suppose that f E we 2 n is a group whose fundamental domain F has exactly n pairwise inequivalent cusps, and such that: 1) there exist n 2 of the operators 1) in (6.3.24), and 2) the determinant �(s) does not vanish in the region where the function h( s ) in Theorem 3.3.4 is analytic. Then the space of cusp-functions X02( f; 1) is infinite dimensional. PROOF. We consider the n 2 + 1 spectral decompositions (see (3.3.14)) 1) k/ z , z ' ; f ; 1 ) = � A ( j ; k ) h/ A k )w( z ; k ; j )w ( z ' ; k ; j ) 1 + -. 1 4 'TT l Re k n2+ I ( Z , Z ' ;
k
s=
hj (s( I - s ))
1 /2
n
n
� � � . a{J ( s ) E ( z ; I (3 = I
a=
j=
s; /3 ) E ( z ' ; s; a ) ds, 1 , . . . ,n 2 ;
( 6.3 .26 )
f ; 1 ) = � hn2+ I ( A k )w(z ; Ai Jw( z ' ; A k ) k
n 1 +4 'TT l. 1Re s = 1 /2hn2+ I ( s ( 1 - s )) L I E ( z ; s; a ) E ( z ' ; s; a ) ds. ( 6.3.27 ) {w( z ; k; j)} k is a compatible real eigenbasis for the operators the subspace Xo(f; 1) ffi 8 1 (f ; 1) of the discrete spectrum (the a=
In these formulas &(f; 1) and r;. in operators &(f ; 1) and 1) are selfadjoint and commute with one another; the operators 1) and 1), do not, in general, commute if j =1= j'). Furthermore, in (6.3.26)
§6 . 3 . THE SPECTRUM OF
9l ( r ; I ) . FIRST CASE
115
and (6.3.27) we have
l )w{ z ; k ; i ) = A k ( i ) w ( z ; k ; i ), 1jw( Z ; k ; i ) = Ak ( i ) w ( z ; k ; i ) , { W( Z; A k )} k is the standard real eigenbasis for m ( f; 1 ) in the subspace %o( f ; 1 ) E9 8 1 ( f; 1 ), and { A d is the corresponding set of eigenvalues. The functions h/s(l - s» satisfy the conditions of Theorem 3.3.4 for i = 1 , . . , n 2 • We are also supposing that the operator 1j acts on the variable z. The existence of the spectral decomposition m (f;
.
(6.3.26), (6.3.27) is a consequence of Theorems 3.3.4 and 6.3. 1 -6.3.5. The basis idea of the proof is as follows. We regard the set of spectral decomposi tions (6.3.26), (6.3.27) as a system of: linear equations for the integrals with the Eisenstein series, and we regard the set of functions hj as a set of free parameters. Starting from condition 2) of the theorem, we select these free parameters in such a way as to eliminate the unknown integrals from (6.3.27) and obtain a spectral decomposition containing only the eigenfunctions of the discrete spectrum of m ( f; 1); then our investigation of this discrete spectrum leads to the proof of the theorem. We introduce the notation
Xt(Z , z ' ; s ) = E(z; s ; 13) E( z'; s ; a ) ,
I= a
-
n
+ f3 n .
We consider the system of equations
n2 L Hjl ( s )Xt(z, z' ; s ) = I/ z, Z'; s),
J.
/= 1
- 1 , . . . ,n2 , -
( 6.3 .28 )
for the functions x t Cz, z '; s) with certain known kernels £ (z, z'; s). By assumption 2) of the theorem, the system (6.3.28) has a unique solution if s is not a zero of �(s). We have
n2 Xt( Z , Z'; S ) = L Qt/ s )£(Z, Z'; S ),
(6 .3 .29 )
j= 1
where
n2 L Hj / s ) Q r/ ( S ) = 13jt , r= 1
1
' ';:;;; J
,
I .;:;;;
n2,
and 13jt is the Kronecker symbol. From (6.3.27) we formally obtain
k n2 + I (Z, z' ; f ; 1 ) - L h n2 + I ( A k )W(Z; A k )W(Z' ; A k ) k n 1 ( h n 2 + I (s( 1 - s )) L xa n + a --n(z , z'; s ) ds = 4 WI J Re s = I /2 a= 1 .
n n L L Hj , yfl ( s )E(z ; s ; 13 ) E(z'; s ; y ) ds. X· fl = 1 y = 1
(6 .3 .30)
116
V I . SECOND REFINEMENT OF THE EXPANSION THEOREM
From Theorem 6.3.5 and the functional equation (6.3.2 1) we see that the function II
II
� � Hi, y/3 ( s ) E ( z; s; f3 ) E ( z' ; s; y ) /3= I y = I is invariant under the change of variable s 1 - s. Hence, the function --->
"
h n 2 + I ( S( 1 - s )) � Qan+a - n( s ) , ( 6.3.3 1 ) a= l 2 has the same property; we denote this function by hJ (s( l - s» , j 1 , . . , n • Using Theorem 6.3.4, we see that each of the functions hJ(s( l - s» satisfies the conditions of Theorem 3.3.4, provided that h,,2 + / S(l - s» satisfies these conditions, and we =
.
are assuming that this is the case. Thus, there exist kernels
kJ(z , z'; f ; 1 ) = � kJ ( u(z, yz')) , yEr
o
}
. -- 1 , . . . , n 2 ,
where the function kj (t) is determined from hJ(s(1 - s» , s 1- + ir, r E �, by (3.3. 16). Taking this into account, we obtain the desired spectral decomposition from (6.3.26) and (6.3.30): =
,,2 k,, 2 + I (Z , z' ; f; 1 ) - � Tj kJ(z , z' ; f; 1 ) j= l � h n 2 + 1 ( A k )W(Z ; A k )W(z' ; A k ) k =
� � A ( j; k ) hJ{A k ( j ))W(z; k; j)w(z'; k; j ) . (6 .3.32) = l k j We study (6.3.32) with the proof of nontriviality of Xoi f; 1) in mind. The proof
that this subspace is infinite dimensional will use proof by contradiction. Suppose that dim X02 (f; 1) < 00 , and hence the sum on the right in (6.3.32) is finite. All of the generalized eigenfunctions w(z; A k ) and w(z; k; j) may be assumed to be continuous on F, and thus the right side of (6.3.32) is a function on F X F which is continuous in z and z' for any functions hJ(s(1 - s » , j = 1, . . . , n 2 , satisfying the conditions of Theorem 3.3.4. For kn 2 + I ( Z, z'; f; 1) we now choose the kernel of the resolvent 91 ( K; f; I) (see §1 .4) of the operator 2f (f; 1). It is not hard to see that the corresponding function h n 2 + I (s(1 - s » has the form
h n 2 + I (S{1 - s ) ; K ) = 1/ (s{1 - s ) - K( 1 - X )) . ( 6.3.33 ) We suppose that K is a sufficiently large fixed positive number. From the definition (6.3.33) it is clear that the function h n2+ l (s(1 - s) ; K ), and hence also all of the functions hJ(s(1 - s) ; K ), which are connected with it by (6.3.3 1), satisfy the conditions of Theorem 3.3.4. Therefore, the right side of (6.3.32) is a continuous function of z and z ' for our h n 2 + ! and hJ. On the other hand, we now show that the kernel in the left side of (6.3.32), i.e., the �um n2 (6.3.34) r (z, z'; K ; f ; 1 ) - � Tj kJ(z, z'; f ; 1 ; K), ,
j= !
§6.3. THE SPECTRUM OF � (r ; 1 ) . FIRST CASE
117
is not a continuous function of z and z' in a neighborhood of the surface z = z '. In this manner we obtain a contradiction with the supposition that dim X02 ( f; 1) < 00 . I t suffices to prove that each of the kernels 1jkJ ( z , z'; f; 1 ; Ie) in (6.3.34) has a finite limit as z � z'. (The kernel r(z, z'; Ie; f; 1) obviously has a weak logarithmic singularity as z � z' (mod f).) This, in turn, follows from the definition (6.3.3), since
u( z , g/Ytz ) > 0, u ( z , g/Yfz ) > 0,
Yt E ( g;l fgj n f ) \ f, gj t£ f, yf E ( gjfg;l n f ) \ f, g/ t£ f,
where u(z, z') is the fundamental invariailt of a pair of points. The proof is complete. We now give a simple example to illustrate Theorem 6.3.6. Let f be any of the groups in Fricke's family. The Fricke groups play an important role in Markov's theory of the minima of quadratic forms and its generalizations (see [47]). By definition, f is a group in 9)1 2 with signature f = f( 1 ; 0; 1) (see § 1 .2). The set of all such groups up to conjugation in G forms a family which depends continuously on one complex parameter (the groups have a nontrivial deformation). As shown in [47], f is a subgroup of index two in a group fo having signature fo = f(O; 2, 2, 2; 1 ), and so it has a nontrivial commensurable. It is not hard to see that the commensura ble contains the parabolic transformation g; z � z + 1 , g t£ f, which induces a Hecke operator T( g). The determinant a(s ) is easily computed: a(s ) = 1 , since the group f only has one primitive parabolic conjugacy class. Consequently, Theorem 6.3.6 holds for f. We end this section by giving other examples of groups f E 9)1 n. We shall focus Ollr attention primarily on groups f E 9)1 2,n' although examples include cocompact groups as welL Later, in §6.4, we shall prove Roe1cke's conjecture for each group in this class, by carrying further the idea of the proof of Theorem 6.3.6. All of the examples are constructed from a single point of view: we work with groups which are commensurable with groups generated by reflections. In order to give a detailed description, we must first extend somewhat our definition of the commensurable of a group f. Before we assumed that f E G = PSL(2, IR). However, as we shall see later, it is useful to allow the commensurable to contain reflections relative to geodesics on H, i.e., motions of the second kind. [In §6.5 a) we shall give a group theoretic description of the motions of the second kind.] An example is reflection relative to the geodesic {z E H I Re z x = O } - this is the map ED : H � H, z � -z, where the bar denotes complex conjugation. Obviously, &; fl G. The condition for &; to belong to the commensurable is the same as before: the groups &; f&; and f must be commensurable in the narrow sense. We now describe the set of Fuchsian g£oups of the first kind which are connected with the groups generated by reflections relative to the sides of geodesic polygons in the Lobachevsky plane H. Let M be a regular polygon in H, i.e., it has the properties that 1) M is a closed polygon bounded by a finite number of geodesics in H, and 2) the interior angles of M are of the form 'TT/k, where k E 1, k > I or k = 00 , if the corresponding angle is zero. Let m = m(M) be the number of zero interior angles in M. If m =t= 0, then M is noncompact. However, in all cases we consider M has finite volume I M I relative to -
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
1 18
the measure dp.. Let ft be the group generated by reflections relative to all of the sides of M. We define fM to be the subgroup of index two in ft consisting of words of even length in the generators of ft (see [54]). We have fM E IDe . Its fundamental domain (in general, not canonical) can be chosen as follows: FM = M U SM, where S is reflection relative to some side of M. We note that the fM are classical groups. They were known to Klein in connection with the problem of existence of an analytic automorphic form for a given Fuchsian group (see [26b], and also [ 1 5]). For the groups fM the problem was solved in a particularly simple and elegant manner using Riemann's conformal mapping theo rem and the Schwartz symmetry principle. In our view, there is a curious analogy between this problem of Klein and Poincare and the problem of nontriviality of the subspace 8 \(f; 1 ) EB 5(o(f; I) which we stated in §6. 1 . As we shall show later, Roelcke's conjecture can also be proved in a relatively simple manner for the groups fM We now give a simple classical example of a family of groups fM ' This is the Hecke family. For every natural number k > 1 we define the group f(k ) to be the group of transformations H � H with the two generators z � -z - \ Z � Z + 2 cos 7T/k. It is well known that f(k) E IDe 2 ; in addition, the Hecke group f(k) can be obtained up to conjugation in G by the above procedure using a regular triangle with interior angles (0, 7T/2, 7T/k). In particular, the modular group fz is the Hecke group f(3). Hence, the set of all groups commensurable with the groups fM (in the case m(M) =1= 0) includes the set IDe 2 of all arithmetic groups, but it is much larger. Even the Hecke group f( k ) is arithmetic only in the cases when k = 3, 4, 6 or 00 (see [24]). In the next section we show that each of the groups fM ' and hence every group commensurable with it in G, has nontrivial commensurable. But from the standpoint of size of the com mensurable one must especially distinguish the arithmetic groups. It is well known that the modular group fz has GL(2, C) for its commensurable. Having such a gigantic commensurable is a characteristic property of an arithmetic group. ·
,
a
§6.4.
The spectrum of
�(f; I)
with
for a group commensurable the group f
fM
In this section we shall specialize the theory in §6.3 to the case of a group f which is commensurable with an arbitrary group fM (see §6.3 for the definition of fM ) ' The basic results are 1) establishment of a connection between the spectral theory of automorphic functions for fM and the Dirichlet and Neumann boundary value problem on M, and 2) proof of Roelcke's conjecture (and even a somewhat stronger conjecture) for the group f, with X = 1 . These results were first published in [6 1 ] and [70]. A special case of the conjecture which we are now calling Roelcke's conjecture was proved in Roelcke's dissertation [45] for the Hecke groups with X 1 . We begin by examining the theory for the groups rM (see §6.3). =
THEOREM 6.4. 1 .
Every group rM has a nontrivial commensurable.
PROOF. Suppose that the group rt is generated by the reflections R I , . . . ,Rk, with RJ = E, j = I , . . . , k, where E is the identity of the group. Any element 0 E rt is
represented by a word
§6 .4. THE SPECTRUM OF � ( f ;
I ). SECOND CASE
119
in the generators Rj' The group fM is identified in fZ. by the condition that the length of the words it contains must be even (the length of 0 is the number of letters in the word, i.e., the number of generators). If 0 E fM ' then RjoRj E fM' since the length of the word RjoRj is either the same, or two greater, or two less than the length of o. (We do not count the letters RjRj = E .) This gives us the inclusion RjfMRj C fM for any j = 1, . . . ,k, and we also have the reverse inclusion because of the equality R; = E; hence,
(6 . 4 . 1 ) Consequently, every reflection Rj is in,the commensurable fM' and since Rj fl fM' it follows that the commensurable is nontrivial. The proof is complete. In what follows we shall let &; = &;(M) denote the reflection relative to a side of the polygon. THEOREM 6.4.2.
Each reflection &; relative to a side of the regular polygon M induces a Hecke operator T(&;) connected with the group fM. For f E C( FM; C ; 1) T( &; )f( z ) = f( &; z ) . ( 6 .4.2 ) PROOF. The theorem is a direct consequence of the definition (6.3.3) of the Hecke operator and the equality (6.4. 1). 6.4.3. The operator T( E9) in Theorem 6.4.2 has the following properties: 1) It is bounded and selfadjoint in X ( fM; 1). 2) It commutes with m ( fM; 1). 3) The operators PI ( &; ) = 1/2( g - T(&;)) and P2 ( &; ) = 1/2( g + T(&; )), where g is the identity operator in X(fM; 1), are orthogonal projections of X(fM; 1) onto the mutually orthogonal subspaces VI ( fM ) and Vi fM) respectively; X ( fM; 1) = VI ( fM ) THEOREM
,
.
E9
V2 ( fM ) .
PROOF. Parts 1) and 2) are proved in a similar way to Theorems 6.3.1 and 6.3.2. The operator T( &; ) is selfadjoint because &; = E9 - I . To prove 3) it suffices to show that P I ( E9 ) and P2( E9 ) are selfadjoint and have the properties
P I ( E9 ) 2 = PI ( E9 ) , P2 ( E9 ) 2 = P2 ( E9 ) , ( 6.4.3 ) P I ( &; ) P2 ( E9 ) = P2 ( &; ) PI ( E9 ) = 0. The selfadjointness is obvious from part 1). Next, we have PI (&; ) 2 = 1/4 ( g - T( &; ) - T( &; ) + T( &; ) 2) = 1 /2 ( &; - T( E9 )) = P I ( E9 ) , P2 (E9 ) 2 = 1/4 ( g + T( &; ) + T( &; ) + T( E9 ) 2) = 1/2 ( g + T( &; )) = Pi &; ) , PI ( E9 ) P2 ( E9 ) = 1/4 ( g - T( &; ) 2) = 0, P2 ( E9 ) P I ( E9 ) = 0.
The proof is complete. As before, we let E (z; s; a) = E(z; s; a ; fM ) denote the Eisenstein series (more precisely, the corresponding meromorphic function of s E C ) for the group fM and the cusp z a of the fundamental domain EM ' a = 1 , . . . , n , n being the number of pairwise inequivalent cusps on FM (see §§1 .2 and 3.1). In Theorems 6.4.4 and 6.4.5 we shall assume that the regular polygon M has the property that m ( M) -=1= 0, where
12 0
V I . SECOND REFINEMENT OF THE EXPANSION THEOREM
m is the number of zero interior angles. This property is necessary and sufficient for
rM E 9J( 2 ·
Theorem 6.4.2 and/or every a = 1 , . . . , n , T( (9 ) E ( Z ; s; a ) E ( z ; s; a ) .
THEOREM 6.4.4. For every (9 as in
=
PROOF. We fix (9 as in the hypothesis. As we already pointed out in §6.3, the fundamental domain FM for can be chosen as the union FM = M U (9M. We fix such an F In general, it is not canonical, in the sense that it can have cusps which are equivalent under This is always the case if the number m(M) of zero angles is greater than one. We give a drawing to help visualize the situation. In it the arrows indicate the order in which the sides of FM are identified. We have chosen as our example a polygon having two zero angles and a line along the y-axis relative to which the reflection (9 is taken. The cusps Zil and Zi 2 with the stars in the diagram are equivalent. We now continue the proof of the theorem in the general situation. From the definition of the group it follows that the cusp Zi of the canonical fundamen tal domain corresponds to two equivalent cusps Zil and Zi2 of FM , which may coincide, if they border on the side relative to which the reflection (9 is taken. On the other hand, (9Zil Zi 2 • We draw two congruent oricycles W I and through the vertices Z i and Z /" respectively (see the figure). The reflection 0 takes the two oricycles into one another. We shall call this property a) of the transformation 0. We consider the Eisenstein series E ( z; s; a; Re s > 1. We define the function
rM
M.
rM .
rM
W2
=
I
2
rM )' tY I
M
By property a) of the reflection 0, the functions
E( giz ; s ; a ; rM ), E{ gi(9 Z ; s; a ; rM ) have the same asymptotic behavior with respect to y for Z E ITa' since 0 only acts on the variable x = Re z. In other words, E{gi0z ; s ; a ; rM ) �ia Y s + 0( 1 ) . =
y - oo
12 1
§6.4. THE SPECTRUM OF � ( r ; 1 ) . SECOND CASE
On the other hand, we know the asymptotic formula (see Lemma 3. 1 . 1 ) E ( giZ; s ; a ; fM ) = 8iay s + 0( 1 ) . y � oo
Because of this and the selfadjointness of � ( fM; 1), just as in the proof of Theorem 6.3.3 we obtain the desired equality E(z; s ; a ) = E ( f9z ; s ; a ) , a = 1 , . . . ,no The proof is complete.
The subspace Xoi fM ; 1) C X(fM; 1 ) is infinite dimensional. PROOF. The spectral decomposition for t4e kernel r(z, z ' ; K; fM; 1 ) of the re
THEOREM 6.4.5.
solvent of � ( fM; 1 ) (see (3.3. 1 4» and Theorem 6.4.4 give us the decomposition
r(z, z' ; K ; fM ; 1 ) - r { z, f9z' ; fM ; 1 )
= 2� k
1
Ak - K( 1
- K
)
w ( z ; A k )W(Z ' ; A k ) , ( 6 .4.4)
where {w( z; A k ) h is the part of the real eigenbasis of the subspace of the discrete spectrum of � ( fM ; 1 ) in X(fM ; 1 ) whose elements satisfy the additional condition -w(z, A k ) = w { f9z; A k ) . ( 6 .4.5 )
Just as in the proof of Theorem 6.3.6, the supposition that dim Xoi fM ; 1) < 00 would imply continuity of the kernel on the right in (6.4.4) as a function of z, z ' E F. The left side of (6.4.4) can be represented as a series
� [ k( u(z, yz ') ; K ) - k( u(z, yf9z') ; K ) ] ,
yEr
( 6.4.6)
which is absolutely convergent for K , Re K > 1 and z =1= z ' (mod fM ). Hence, the kernel on the left in (6.4.4) has a singularity for Z = z ' (mod f), but this contradicts the assumption that the subspace X02 ( fM ; 1) is finite dimensional. The proof is complete. In parts 1) and 2) of the next theorem, the regular polygon M can be either compact (m(M) = 0) or noncompact (m(M) =1= 0). Part 3) only has content for noncompact M.
The restriction of the operator -� ( fM ; 1) to the subspace U1(fM ) is isomorphic to the operator of the Dirichlet boundary value problem on M for the Laplace differential operator L of the metric ds. 2) The restriction of -� ( fM; 1) to the subspace U2 (fM ) is isomorphic to the operator of the Neumann boundary value problem on M for the Laplace differential operator L of the metric ds. 3) The operator of the Dirichlet problem in part 1) has only a discrete spectrum. THEOREM 6.4.6. 1)
The subspaces U1(fM ) and U2( fM ) were defined in Theorem 6.4.3. PROOF. 1) We fix a fundamental domain FM for fM which is connected with a reflection f9, FM = M U f9M. It is sufficient to show that the elements of the intersection U1(fM ) n C( FM; C ; 1) vanish on the boundary of the polygon M. The proof that the indicated Dirichlet problem leads to the restriction of � (fM; 1 ) to the subspace U1(fM) is done by reversing the process, and we shall not carry this out here. Thus, suppose that f E U1( fM) n C( FM; C ; 1). The boundary of FM consists of
12 2
VI . SECOND REFINEMENT OF THE EXPANSION THEOREM
an even number of segments of geodesics on H. It can naturally be divided into pairs of segments. The elements of each pair are equivalent under a transformation in fM (i.e., one segment is mapped to the other by this transformation). If the point Zo is on the boundary of FM , then there is an element y E fM such that yZo is also on the boundary of FM . By the definition of f( z), we have
fe z ) = 1f(z ) - 1 T( ED )f( z ) , which gives us the equality
( 6.4.7 )
f( z ) = �f( EDz ) . The function f(z) satisfies the automorphicity condition; in particular,
f( z ) = f( yz ) . ( 6.4.8 ) The equalities (6.4.7) and (6.4.8) also hold for z = zoo Moreover, yZo = EDzo, and so f(yzo ) = f( EDzo ) . ( 6.4.9) Comparing (6.4.7)�(6.4.9), we arrive at the equality f(zo) = O. We have thereby proved that f( z) = 0 if z is on the part of the boundary of M which is also a part of
the boundary of FM . The rest of the boundary of M consists of the side of M relative to which the reflection ED is taken. On this side we have z = EDz, and so f( z ) = f( EDz); this, together with (6.4.7), leads to the equality fez ) = 0, thus completing the proof of part 1). 2) The proof is similar to that of 1), but for functions f E Ui fM ) n COO( F; C ; 1). The condition f E Ui fM ) leads to the equality
f( z ) = f( EDz ) .
( 6.4 . 10)
The automorphicity condition gives
af( z ) jan
=
( 6.4.1 1 )
-af( z )jan ,
where z and yz are equivalent boundary points on FM , and a jan is the exterior normal derivative to FM . For a boundary point z E FM we have yz EDz, and hence
f( yz ) = f( EDz ) .
=
( 6.4.12 )
Comparing (6.4. 10)�(6.4. 12), we arrive at the desired equality
af( z ) jan = 0,
(6.4. 1 3 )
if z is a common boundary point of FM and M. If z is on the other part of the boundary of M, we have z = EDz. Considering this equality in a neighborhood of z and taking (6.4.10) into account, we ll�rive at (6.4.1 3). This proves part 2). 3) The entire continuous spectrum of �(fM ; 1) is generated by the functions E(z; 1r; a ) , r E IR , a 1 , . . , n (see Theorems 2.3.7 and 3.1.4). According to Theo rems 6.4.3 and 6.4.4, each function E(z; s; a ) is annihilated by the projection PI( ED), regarded as a linear operator acting on the argument of the function; this proves part 3). The proof is complete. We now prove Roelcke's conjecture for an arbitrary group f which is com mensurable in G with one of the groups fM (m(M) 7'= 0; X = 1). Moreover, we prove that N( A ; f; 1) > c A starting from some A, where c > 0 is an effective constant. Recall that N( A ; f; 1) is the distribution function for the eigenvalues of the discrete spectrum of � (f; 1). =
.
§6 .4. THE SPECTRUM OF �l ( f ; I ) . SECOND CASE
123
Without loss of generality we may assume that the groups f and fM are commensurable in the narrow sense. Let f ' denote the intersection f ' f n fM ' It has finite index in f and in fM . Let {w(z; A k )h be a real eigenbasis for 2l( fM ; 1) in the subspace VI( fM ) (see Theorem 6.4.3), and let {A d be the set of eigenvalues. =
LEMMA 6.4.1.
Let the function h( s(1 - s)) satisfy condition 2) of Theorem 3.3.3, and let k(z, z'; fM ; 1) be the corresponding kernel of the operator h(2l"(fM ; 1)), given by the absolutely convergent series k(z, z'; fM ; 1 ) = � k( u(z, YZ')) . y ETM
Then one has the spectral decomposition 1 2 a E�fIf' al E�fIf' [k( O"z, 0" \ z ; fM ; I) - k( O"z, 0 0"\z; fM ; I )] =
� h ( Ak ) k
[
]
2 � w (O"z ; A k ) , ( 6.4.14)
a E fIf'
where &; is a fixed reflection relative to some side of the polygon M. PROOF. Theorems 3.3.3 and 6.4.4 imply the decomposition
1
"2 [ k (z,
z , ; fM ; 1 ) - k(z, 0Z'; fM ; 1 ) ]
=
� h ( A; ) W( Z ; A; ) w ( z'; Aj ) , ( 6.4. 15 ) .I
from which the desired equality (6.4. 14) follows. The proof is complete. We consider the equation O"Z Y0 0"\ Z in z E H, for fixed 0", 0" \ E f/f ' and y E fM' The set of its solutions obviously lies on a geodesic (in the Lobachevsky geometry of H) which depends on 0", 0" 1 and y. We call all such geodesics for 0", 0"\ and y as indicated " special". The following lemma follows immediately from the discreteness of the group fM and the finiteness of the set of cosets f/f ' . =
6.4.2. Any compact set on the Lobachevsky plane intersects only finitely many special geodesics. LEMMA
We now prove a theorem.
6.4.7. The inequality N(A; f; I ) > c r holds beginning with some A and c r , rM > 0 independent of A . ( The constant is computed in the course of the proof.) PROOF. In (6.4. 14) we choose h(s(1 - s)) to be the function THEOREM
h ( s (I - s ) ; t ) = exp [ -ts ( I - s ) ] , which depends on a parameter t > 0 (see the proof of Theorem 4.4.1). We use this function to construct k( u) = k(u ; t) froin the transformation (3.3. 16). If we take into account the explicit formula for the Fourier transform g(u; t) of the function h(t + r 2 ; t), s ! + ir, in (3.3. 16), namely =
124
VI . SECOND REFINEMENT OF THE EXPANSION THEOREM
we easily obtain the following two properties of k(u; t ) : l ) k(O; t )
� ;? � o > 0:
(4 17 t t l ; 2) for
=
where a and b are any positive numbers. We now integrate (6.4. 1 4) with the functions h and k fixed, over a compact set Q which does not intersect the special geodesics (see Lemma 6.4.2). As t 0 we obtain the equality �
p i Q I ( 8 17 t r
l
+ O{t b ) = � [ exp ( j
-
t AJ] Cj { Q ) ,
(6.4. 16)
where p is the number of triples a, a i ' Y such that yal = a, with y E fM and a, a I E fIf' (the definition of the groups f and fM implies that 1 � P < (0 ) , I Q I is the volume of Q relative to the measure dJL, and the function c/ Q) has the form
c/ Q ) =
1 dJL { Z ) g
[
�
o E r/r'
k
w ( az ; A j )
]
2
Let k be the minimum number of elements y, E fM such that a - I Q = U y, FM , ,= I
where FM is the fundamental domain for fM in H. We introduce the number q = max k . o E r/r'
We have c/ Q) � [ f : f,] 2q. We define a monotonic function Ng{ A ) =
However,
�
>...I ,,;;; >..
ci Q ) ·
(6.4 . 1 7) On the other hand, applying the Tauberian theorem to Nri A), from (6.4. 1 6) we obtain A � 00 ,
which, combined with (6.4. 1 7), gives the theorem with any constant cr r for which 1 cr.rM = c < p I Q I ( 817q [ f : f '] 2 ) - . .
The proof is complete. §6.5.
M
Selberg trace formula the Dirichlet boundary value problem on a regular polygon M for
This section is devoted to the derivation of a trace formula for a function of the operator in Dirichlet's problem in Theorem 6.4.6, a formula which seems to us to be a natural variant of the classical Selberg trace formula (see Chapter 4). The basic theorems proved here were published before in [65] (see also [62] and [66]). The plan of exposition is as follows. We first prove a theorem which applies to any regular polygon M. Then we separately consider the theory for compact M (§6.5 a» and for noncompact M (§6.5 b» .
12 5
§6 .S. SELBERG TRACE FORMULA FOR THE DIRICHLET PROBLEM
LEMMA 6.5.1. Let M be an arbitrary regular polygon, and let the function h(s(1
satisfy condition 2) of Theorem 3.3.3. Then 1
'2
� (k( u(z, yz' » - k( u (z, & yz'» )
y E rM
=
- s»
� h ( AJ w ( z ; AJW(Z'; AJ, (6.5 . 1 ) j
where Aj and w(z: A) respectively run through a complete set of eigenvalues and (real) eigenfunctions of the operator for Dirichlet 's problem in Theorem 6.4.6 ; k(t) is connected with h(s(l - s» by (3.3.16), and 0 is a fixed reflection relative to a side of M. PROOF. If M is noncompact, then the desired formula coincides with (6.4. 1 5). For a compact polygon M, (6.5.1) follows from the theorem on expansion in eigenfunc tions of 9f(fM ; 1) and Theorem 6.4.6. The proof is complete.
6.5.1 . Suppose that the conditions in Lemma 6.5.1 are fulfilled. In addition, let h(s(1 - s» = h i(s( l - s» , where the function h 1 (s(1 - s » satisfies the conditions of Theorem 4.2.2. Then the kernel on the left in (6.5. 1) is continuous on FM X FM , and determines a nuclear operator in the space :JC(fM ; 1). The spectral trace formula THEOREM
�f
� ( k( u (z, yz » - k { u(z, 0YZ » ) dp. { z ) = � h ( AJ j
FM y E rM
( 6.5.2 )
holds, where FM is the fundamental domain for fM . A ll of the series and the integral converge absolutely. PROOF. The continuity of the kernel follows from Theorem 3.3.3. The nuclearity of the corresponding operator is proved in a way analogous to Theorem 4. 1 .2. Finally, the spectral trace formula is obtained by integrating (6.5. 1). The proof is complete.
a) Trace formula for a compact polygon(6.5.2)M.
Here we shall be concerned with transforming the spectral trace formula to the Selberg trace formula for Dirichlet's problem on a compact regular polygon M. We begin the derivation of our formula by considering the integral
(6.5 .3 ) The integral (6.5.3) is obviously equal to the number
where Tr denotes the matrix trace of the operator, the function h(s(1 - s » is connected with k(t) by (3.3.16). Hence, the derivation of the classical Selberg trace formula consists of computing (6.5.3) in terms of the function h and the conj ugacy
126
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
classes in fM . We have (see §4.3)
�f
� k ( u( z , yz ) dJL( z))
FM -y E fM
=
=
� �
f k ( u(z, y'- l yy'Z )) dJL( z ) � f k( u (z, yz)) dJL( z ) = ' : ' f oo r ( tanh 7Tr)h ( ! + r 2 ) dr exp(-27Tkr/d ) h ( ! + r 2 ) dr + !4 � � d 1 d f oo 1 + exp( -27Tr ) 4 sin (k7T/ ) �
�
{ y hM Y ' E fM { y } \ fM FM
- 00
{y hM FM { y } d
\
- 00
{ R } rM k = \
1
00
+ "2 � �
(P hM k = \
In N( P )
N( P )
k/2
- N(P ) -k/2
g( k ln N( P )) .
(6.5 .4)
If we take into account Selberg's general remarks on computing the trace of the product T(g)&(f; X) for a cocompact group f with nontrivial commensurable (see [5 1 ], §2), where T(g) is an arbitrary Hecke operator, then we find an expression for the integral (6.5 .5) which is analogous to (6.5.4). However, in order to do this, we first make some preliminary transformations, as a result of which we shall be able to regard the reflection E9 as a group action. We translate the polygon M by an element in G = PSL(2, IR) in such a way that the side relative to which E9 is a reflection runs along the y-axis, y = 1m z, in the upper half-plane H. Here E9 takes the form E9 : Z � -z, (6 .5 .6) where the bar denotes complex conjugation. In what follows we shall consider M to be chosen in this way. (This choice of M in no way reduces the generality of the theory, and is only adopted for convenience.) We now construct an isomorphic model of H. This model is well known in the theory of symmetric spaces (see [5 1 D. We consider the set of positive definite symmetric matrices
Z ( x ,· y ) =
(
y
+ X 2Y - I -I .x'y ,
,
xy - 1
Y
-
I
where x E IR and y > o. If g E G, we denote g by
g=
( � �)
(mod ± 1 )
)
'
(6 .5 .7)
,
thereby expressing the fact that g E PSL(2, IR). We now define an action of G on the set { z ( x; y ) } . By definition, for g E G we have
(6.5 .8) gz( x ; y ) g [ z( x ; y )] g , where gt is the transpose of g, and the product on the right in (6.5.8) is the usual =
t
product of matrices. It is not hard to verify that the set fl = { z ( x; y)} with this
§6 . S . SELBERG TRACE FORMULA FOR THE DIRICHLET PROBLEM
12 7
action has the structure of a symmetric space and is isomorphic to the upper half-plane H. The isomorphism is given by the map z(x; y ) -) Z x + iy. This model ii of the upper half-plane has the following useful property which is not so obvious in the case of H. The reflection ED (6.5.6) for the model ii is given by the " matrix" =
( mod
± 1),
(6 .5 .9)
i.e., it is an element in GL(2, IR )/( ±E), where E is the identity matrix. It therefore makes sense to consider the products ED g and gED for g E G, regarding them as the corresponding cosets in GL(2, IR ). In the sequel we shall not distinguish between H and ii; we shall keep in mind the possibility of giving the reflection ED (6.5.6) by formula (6.5.9). We now return to the integral (6.5.5). We have
- � {0,,}�r y ' E TM �{ 0y} M fFMk( u(Z, y ,-I EDyy'z )) dll { z ) M � { 0�Y }r fFM( f9 y )k { u{ z , EDyz )) dll { Z ) . \r
(6 .5 . 1 0)
M
We have denoted by {EDy hM the relative conjugacy class in ED fM with representative EDy by fM(EDy) the relative centralizer in fM of the element EDy E ED fM and FM(EDy) the fundamental domain for fM(EDy) in H. Recall that EDy and EDy' are in the class {EDyhM if and only if there exists an element Yo E fM such that YoEDyyol = EDy'. Furthermore, fM(EDy) = {y' E fM I y'EDy = EDyy'}. The relative centralizer fM(EDy) is a group. The sum on the right in (6.5. 10) is taken over all relative classes {EDy hM ' In order to make the next transformation of the integral (6.5.5), we study the set of classes {EDyhM in some detaiL Let y E fM. We have det EDy = - 1 and tr EDy = a , with the usual notation for the determinant and trace of a matrix. We let II I and II I denote the eigenvalues of EDy . A simple computation shows that the eigenvalues can be chosen as follows: II I =
al2 - va ll4 +
1.
From this it is clear that the eigenvalues of EDy are always real and unequal. Hence, there is a transformation g E G which takes EDy to diagonal form:
gEDyg - 1
( Ill
-Ill ( Ill I )
=
0
_\
o
)
(mod
-+
1).
We shall denote the element EDy as follows:
EDy =
0
0
-Ill
( mod
-+
1).
(6 .5 . 1 1 )
We now find the relative centralizer of EDy in fM for y E fM' Here one should distinguish two cases. We first suppose that tr EDy = a =1= O. In this case the relative
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
centralizer in G of the diagonal matrix in (6.5. 1 1 ) (under the assumption (mod ± 1» consists only of diagonal matrices:
( �: _Z-I )
( mod
IL l =1=
1
± 1) ;
and this implies that the relative centralizer fM ( E9 y) c fM consists only of the hyperbolic elements and the identity. From the discreteness of fM one can conclude that the group fM ( E9y) is cyclic hyperbolic, i.e., it is generated by a single hyperbolic generator P( E9y). We now consider the alternative case tr E9y O. It is not hard to verify that the relative centralizer in G of the diagonal matrix (6.5. 1 1) with IL l = 1 (mod ± 1) consists of elements of the form =
( 6 .5 . 12 )
Obviously, P2 is an elliptic element of order two. Thus, in this case fM ( E9y) generally consists of hyperbolic, elliptic, and the identity elements. However, if we consider all possible products of the form P�P2 ' n E lL, we discover that fM ( E9y ) contains infinitely many elliptic conjugacy classes, and this contradicts the relation fM E we. Since fM is discrete, it follows that the centralizer fM ( E9y) for E9y with tr E9-y = 0 either consists of hyperbolic elements and the identity, in which case the group fM ( E9y) is cyclic hyperbolic and is generated by P( E9y) E fM ' or else it is generated by a single elliptic generator of order two. Later on, using analytic considerations, we shall show that the second case is impossible for cocompact fM ' We now proceed to compute the sum on the right in (6.5. 10). In it we consider the terms corresponding to classes {E9yhM with tr(E9y) =F O. For any y E fM we have E9yE9y E fM ' For any y under consideration, tr( E9y) =1= 0, the element E9yf9y is hyperbolic. By definition, by the norm of the class {E9yhM (or of the element E9-y) we shall mean the number N( f9y ) = 1 N( E9yE9y ) 1 1 / 2 , where the norm of a hyperbolic element was defined in §4.3, tr( E9y) =F O. In addition, we shall call a relative conjugacy class {E9yhM primitive if it is not an odd power of any other relative class {f9y ' hM • Part of the terms in the sum on the right in (6.5.10) we transform as follows: -
�
f
�
{ 6:i Y } rM FM( 6:i y ) tr 6:i y # O
� = -� = -
1
2
k{ u { z , (f9yz ))) dIL ( z )
�'
�
f
�'
�
f
�
f
{ 6:i Y } rM k = ! tr ( 6:i y ) 7"' 0
FM( 6:i y )
k k ( u ( z , ( E9 y ) 2 - I Z ) ) dIL ( z ) '
{ 6:i y hM k = ! g( 6:i y ) FM( 6:i y ) tr ( 6:i y ) #O �, £.J
£.J
{ 6:i Y } rM k = ! tr ( t9 y ) # O
!
N( P( t9 y »
k k( u(z, g( E9y )( E9 Y f - l g - l ( E9y )z)) dp, ( z )
(
I z + N( E9y )2 k - l jz I dr (" dcp . 2 k ) 2k- ! r J,0 sm cp y 2N ( E9y
)
'
(6.5 . 1 3 )
§6.S. SELBERG TRACE FORMULA FOR THE DIRICHLET PROBLEM
129
where the summation in �' is only taken over primitive relative classes, and P( f9y) is the generator of fM (f9y). The element g(f9y) E G is chosen so that g( f9y)f9yg- I (f9y) is diagonal. The region g( f9y) FM ( f9y) is a fundamental domain for the group of " diagonal matrices" g(f9y)fM (f9y)g- I (f9y). In polar coordinates x = r cos cp, Y r sin cp on H, this region is chosen as follows:
=
g(f9y) FM (f9y)
=
{z E H I I � r < N( P(f9y)) , O < cp < '1T} .
Recall that rM ( f9y) is generated by a single hyperbolic generator P( f9y). Next, the first part of (6.5. 1 3) is equal to
+
(N( f9 y ) k - I /2 - N( f9y fk + I /2) 2 ) dt 1
2
k k Q ( N( f9 y f - l + N( f9 y ) -2 + 1 � In N( P( f9y)) k N(f9y) - I/2 + N( f9y fk + 1/2 k= 1 00
�'
{ t9 y hM tr( t9 y ) 'i"' O
2)
, 00 1 In N( P(f9y)) - "2 � � k - I/2 + N( f9y) -k + I/2 g « 2 k - 1 )ln N( f9y)) ; { t9 Y } rM k = 1 N( f9y)
tr( t9 y ) 'i"' O
(6.5 . 14) the sum with the prime has the same meaning as in (6.5. 1 3), and the functions Q (w) and g(u) are determined from k(t) by (3.3.1 6). We now make a small digression from the immediate question of studying the integral (6.5.5). We shall later have need of more detailed information concerning P( f9y), the generator of rM (f9y) for f9y a primitive element, in our case tr(f9y) =1= O. For this purpose we consider the full discrete group rz. ::J fM (see the proof of Theorem 6.4.1 ) and the usual centralizer fZ.( f9y). It consists only of the powers of f9y, and perhaps the element f9. It contains f9 if and only if f9y = yf9. Correspond ingly, the relative centralizer fM (f9y) = rM n rz.(f9y) is generated by:
( 6.5 . 1 5 ) (6 .5 . 16)
1 ) either p ( f9y) = f9yf9y, or p ( f9y) y.
2)
=
We have the following alternatives for the norms:
N( P(f9y))
=
{
N 2 ( f9y ) N( f9y )
for 1 ) , for
2).
( 6 .5 . 1 7)
In order to finish the computation of the integral (6.5.5), it remains to consider in the sum on the right in (6.5. 1 0) the terms which correspond to relative classes {f9y h satisfying the condition tr( f9y) = O. Recall that an f9y for which tr( f9y) = 0 can have a centralizer fM ( f9y) of either of two types. The group fM (f9y) is M
..
V .1 . j,:)J: \...... V l�.LJ �..cr .11"11 r..1V.1.cl"11 1 v.C' I nn CArl"\.l"'f ':>.1Vl� J. n.c.v�..c.1VJ.
... ..,v
generated either by a single hyperbolic generator P ( f9y) or else by a single elliptic generator R(f9y) of order two. In the first case, just as in (6.5. l3) and (6.5.14), we have
f
FM( f9 y )
k( u (z, f9yz )) dJt ( z ) = j I
=
d q; k (cot 2 ) l q; . r sm2 q;
N( P ( f9Y » dr
1
'TT
0
2 In N(P( f9y )) g (0) .
In the second case we have
( )
4x 2 dxdy = 1 k(u(z, f9yz » dp. ( z ) = 2 J( k H , Y2 Y2 FM( f9 y )
f
(6.5 . 1 8)
00 .
(6.5.19)
The last integral diverges for all functions k(t) for which g(O) =ft 0 (see (3.3. 16)). Now, from the well-known absolute convergence of the integral (6.5.5) and the series (6.5.14), we conclude from (6.5.10) that the series and integral
�
f
{ f9 Y } rM FM ( f9 y ) tr( f9 y ) = O
k( u(z, f9yz» dp. ( z )
must converge absolutely. Taking into account (6.5.18) and (6.5. 1 9), we hence find that rM (for compact M) does not contain classes {f9y hM with tr( f9y) = 0 for which the relative centralizer fM ( f9y) consists of two elements. In addition, the series over the remaining {f9yhM with hyperbolic centralizer fM (f9y)
� In N( p ( f9y »
{ f9 Y } rM tr( f9 y ) = 0
( 6.5 .20)
is absolutely convergent. The discreteness of rM implies that the values of the norms N( P(f9y)) are isolated away from one; hence, the series (6.5.20) consists of only finitely many terms. We have completed the construction of the Selberg trace formula for Dirichlet's problem on a compact polygon M. We gather together our results in the form of a theorem, where we use all of the notation us�d in the context of (6.5.4), (6.5. 14) and 1
(6.5.20).
THEOREM 6.5.2. Suppose that M is an arbitrary regular compact polygon,
h (s( 1 - s »
=
h f { s { l - s»
and the function h )(s(1 - s)) satisfies the conditions of Theorem 4.2.2. Then the following Selberg trace formula holds for the operator of Dirichlet 's problem in Theorem 6.4.6:
§6 . S . SELBERG TRACE FORMULA FOR THE DIRICHLET PROBLEM
� h ( Aj ) .J
=
131
� I i: r (tanh wr ) h ( ! + r 2 ) dr d� 1 1 j oo exp+ (- 2 w rk/d ) h ( ! + r 2 ) dr +! I
4 ( R�}r k = I d sin kw/d 1 M In N( P ) 1
+ '2 �
- 00
00
exp (- 2 wr )
4
� -k g ( k In N( p )) k (P} rM k = 1 N( P ) /2 - N( P ) /2
1 l: � : . . In N( P ( E9 y ) ) 2 ( �� Y}r k = N( E9 y ) k - I/2 + N( E9 y rk+ I/2 M 1 tr( ��y)*o
1 X g ( ( 2k - I )In N( E9 y ) ) - 4
�
{ ��Y } rM tr( &y)=o
In N( P( E9 y ) ) · g(O) , (6 .5 .21 )
where Aj on the left runs through the set of all eigenvalues of this Dirichlet problem, and the summation on the right is taken over all primitive classes {R hM ' { P hM and { E9 y hM , tr( E9 y ) =1= O. All the series and integrals in (6.5.21) are absolutely convergent, and the sum over {E9yhM , tr( E9 y ) 0, contains only finitely many terms. Now, just as in §4.4, we extend the class of functions h for which (6.5.21) remains true. We give two definitions (here M is any regular compact polygon): w ( x ; E9 fM ) =
{number of all primitive relative conjugacy classes {E9yhM I N( E9y) ..;;; x} , NM ( A ) {number of all eigenvalues Aj of Dirichlet's problem for the operator -L on M I Aj ";;; A}. The following lemma follows from Theorem 6.5. 1 . LEMMA 6.5.2. w ( x ; E9 fM ) = O( x ) as x -> 00 . LEMMA 6.5.3 (Weyl' s formula). NM ( A ) I M I A/rw as A -> 00 . PROOF. The method of proof is analogous to the method of proving Theorem 4.4. 1 ; it is based on the trace formula (6.5.21 ) which we derived. We have = =
�
(6.5 .22) ( -> 0 , (>0
The lemma follows from the Tauberian theorem, because of the monotonicity of the function NM ( A). The proof is complete. From Lemmas 6.5.2 and 6.5.3 we obtain a theorem analogous to Theorem 4.4.4. THEOREM 6.5.3. The Selberg trace formula for the Dirichlet problem (6.5.21) is true as an identity for any function h( A) satisfying the conditions of Theorem 4.4.4.
Trace formul a for a noncompact pol y gon M. Here we shall keep the basic b) notation of the preceding subsection. Recall that the classical Selberg trace formula for a Fuchsian group of the first kind f with noncompact fundamental domain differs from the cocompact version of the trace formula because of the presence of additional terms coming from the parabolic conjugacy classes in f and the terms which are connected with the continuous spectrum of the operator m(f; X) (see
132
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
§4.3). The Selberg trace formula for Dirichlet' s problem, which we shall now derive from the spectral trace formula (6.5.2) for a noncompact regular polygon M, also differs from (6.5.21) because of the presence of new terms. In the preceding subsection we gave a classification of the relative conjugacy classes {51' hM for a cocompact group fM . If fM is constructed from a noncompact regular polygon M, then the classification of the {5yhM is analogous, except that there are new classes {5yhM for which tr(5y) = ° and the centralizer fM (5y) is generated by a single elliptic generator of order two. In other words, for each such I' E fM there exists an element g E G having the properties
{ ( � _°1 ) (mod 1 ) } , g fM ( 5y ) g I { ( � � ) ' ( _�_ � ) (mod ± I ) } , I g5yg- 1 -
=
5=
+
(6.5 .23)
=
where a ;;:. 1. These classes play the role of the parabolic classes in the classical Selberg trace formula. We proceed now to derive the desired formula. By analogy with (4.3. 14) - (4.3. 16), (6.5.4) and (6.5 . 1 0), the left side of (6.5.2) is equal to
� fFMk( u(z, z » dJL ( z ) + � ( R�}r fFM( R )k ( u( z , Rz » dJL ( z ) M
.
-� 1 2
�
fF ( �)Y)k ( u ( z , 5yz » _
{ [, Y } ru tr( �; y ) "", 0
M
dJL( z )
�
{ fD y hu tr( fD y ) = O rM ( [, y ) hyperb.
�
�
{ fD y hu y' E rM( fD y )\rM tr( fD y ) = O rM( fD y)
k ( u ( Z , y , 1 5yy z )) dJL( z ) , (6.5 .24) -
'
ellip .
where the sum is taken over all elliptic conjugacy classes {R hM ' hyperbolic con jugacy classes {Ph , and parabolic conjugacy classes {Sh in fM with representa M M tives R, P and S, respectively, over all relative nondegenerate classes {51' hM (i.e. {5yhM , tr( & y ) =F 0), and over all relative degenerate classes {5yh , tr(5y) = O. M Here we consider separately the sum over all {5yhM for which the centralizer
§6.5. SELBERG TRACE FORMULA FOR THE DIRICHLET PROBLEM
133
is a cyclic hyperbolic group, and the sum over {0 Y h with an elliptic M centralizer fM(0y). Some of the other notation in (6.5.24) is as follows: FJ = Fo = Fo( a) is from Theorem 1.2.4, where a = Y; f = fM; FJ is the compact part of the fundamental domain FM, which we choose depending on a large parameter Y > 0; and FM( R ), FM( P ) and FM( 0 y ) are the fundamental domains in H for the centralizer fM( R), the centralizer fM( P ), and the relative centralizer fM( 0 Y ), respectively, of the elements R, P and 0 Y (all of these centralizers are taken in the group fM). If the function h(s(1 - s» satisfies the condition in Theorem 6.5.1 , then the terms in the sum (6.5.24) corresponding to the identity, the elliptic classes, the hyperbolic classes, the nondegenerate {0yh , and the degenerate {0 Y h with hyperbolic M M centralizer, are defined by absolutely convergent series and .integrals, and are computed by analogy with the corresponding contributions to the classical Selberg trace formula (see §4.3) and the trace formula in the last subsection. We shall not carry out these computations here, but shall immediately formulate the result in Theorem 6.5.4 ; first we focus our attention on the last term in the sum (6.5.24) with the limit. By Theorem 4.3.5, for f = fM' X = 1 and dim V = 1 we have fM( 0 Y )
n
- 47T
f oo h ( "41 + r 2 ) r f' ( 1 + ir) dr + 0 ( 1 ) , - 00
y- oo
( 6.5.25 )
where n is the number of pairwise inequivalent cusps on the fundamental domain FM' or the number of primitive parabolic conjugacy classes in fM' and g( u) is the function in (3.3.16). From the definition of fM it is not hard to show that n is equal to m(M), the number of zero interior angles in the polygon M. We now find the asymptotic behavior as Y 00 of the remainder term in the integral under the limit in (6.5.24):
-
�
y' E rM( f9 y )\ rM
k ( u ( z , y , - 1 0 YY'Z)) dfL( Z ) .
(6.5.26 )
It can be shown that in the class of functions k(t) considered in Theorem 6.5.1, the asymptotic behavior of (6.5.26) does not depend on the order of the summation and integration. In addition,
134
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
where I fM ( E9y ) I is the order of the group fM ( E9y ). Hence, the integral (6.5.26) is equal to the sum
1
4
( 0 Y } rM tr( 0 y ) = O rM( 0y) ellip.
f
k ( u( z , E9yz )) dp.( z ) .
U y'Fj; , y ' E rM
( 6.5 .27 )
We now find the asymptotic behavior as Y � 00 of each term in (6.5.27). The technicalities of the derivation of this asymptotic behavior are reminiscent of the computation of the contribution of the degenerate elliptic elements to the Selberg trace formula for three-dimensional Lobachevsky space (see [58], and also [59], §5.4). Let E9y be as in (6.5.23). For simplicity, we first suppose that g = E, the identity of G, in (6.5.23). In addition, let E9 be a reflection relative to a side of M which is adjacent to a zero interior angle of M; let E9 be given by (6.5.6). We must find the asymptotic behavior of the integral
f
k(u(z , E9 z )) dp.( z ) .
U y Fj; y E rM
( 6.5 .28 )
We have the obvious equality ( 6 .5 .29 )
where z = x + iy. The domain of integration in (6.5.28) is all of the half-plane H except for a certain set concentrated in neighborhoods of the fM'parabolic points of the absolute of H whose Lebesgue measure vanishes in the limit as Y � 00 . On the other hand, if we study the behavior of the function k (4x2Iy2) for z = x + iy on H, we see that the set of all points at which (6.5.28) may diverge is concentrated in the region x � Y and in a neighborhood of the point x = 0, y = 0. By our assumption, the centralizer fM ( (9 ) contains an elliptic element of order two (mod ± 1 ) .
We let B( Y ) denote the set obtained from H by removing from it the two regions B \ ( Y ) = { z E H I x � Y} and B2 ( Y ) = yoB\(Y), i.e., B ( Y ) = H B\ ( Y ) - B2 ( Y ) ' ( 6.5 . 30 ) We fix a sufficiently large Yo > 0, and we consider the intersection B ( Yo ) : n U yFi; . ( 6 5 . 31 ) -
.
From the definition of B( Yo ) it follows that the difference
vanishes in the limit as Y �
f
-
This gives us
k( U ( Z , 0Z )) dp.( z ) =
U y Fit
y E TM
00 .
f
( )
4x 2 dxdy + 0( 1 ) . k - B( Y ) y2 y2 Y � oo
§6 . S . SELBERG TRACE FORMULA FOR THE DIRICHLET PROBLEM
) dxdy
We now find an asymptotic expansion for the integral 4X 2 y2 y2 ·
o
00
k(t) x
B(Y), the integral (6.5.32). is equal to the sum
dx fa Y2/ y ydy2 k ( 4yX22 ) la2!r ydy2 fyvaoo 2!I'Y- dxk ( 4yx22 ) · 1
By the definition of 1
f8(Y)k ( +
0
135
(6.5 .32)
(6.5 .33)
Recall that was chosen in accordance with Theorem 6.5. 1 , and it is connected with h(s( l - s » by the transformation (3.3. 1 6). We make a change of the variable of integration in (6.5.33): 4
Y
dx dt/ {i . =
The sum (6.5.33) is equal to dy 1 00 4 y 0
!fY a2/ y
+
k(t) dt ! la 2/ Y dy f oo k(t) dt . 4 Y 4(a2/yY-I) {i {i 0
The first term in (6.5.34) is equal to I 1 2" g(O)ln Y - 2" g(O)ln
( 6 .5 .34)
a,
a 2(yYtl T.
where g( u ) is from (3.3. 1 6). In the second term in (6.5.34) we make a change of the variable of integration y : = This term is equal to
! f oo dT f oo k( t) dt = ! f ood(In T ) f oo k({it) dt 4 I T 4(T-1) {i 4 I 4(T- I) k(t) dT d f oo --dt - 1 f oo (In T)dT 4(T-I) {i k( t) In 2 1 [ 00 - -g(O) T + In( t + 4) {i dt. 4
=
-
I
4
_
Thus, the integral (6.3.32) is equal to 1 1 In 2 1 2" g ( O ln - 2" g(O)ln a T g(O) + 4
) Y
-
Jo
l OO In(t + 4) k{i( t ) dt . 0
( 6 .5 .35)
We now transform the integral
lo°O ln(t + 4) ky�t ) dt.
( 6 .5 .36)
The integral (6.3.36) equals (see (3.3. 1 6» - � 1 00 'TT
0
dQ ( w )lw In(Vwt + 4)t dt
=
=
0
{i
-
1 1 1 {i {l=t dt � l °O Q (w ) l l (w 4/dTdw T)Fb - T 1 00 dT 2 1n 2g (O ) - 1 Q (w ) d w 1 1 (w + 4/T ) yT y� l -T
� g(O ) 2 1n 2 'TT
+
0
'TT
0
+
'TT
0
0
0
c
+
.
V I . SECON D REFIN EMENT OF THE EXPANSION THEOREM
136
00 1 1 dT 1- 1 Q(w) dw 1 1 7T (w + 4/T ) F f1=-; = -7T Q(W)7T W + 4 + 2/w 4 dw 00 x 1 1 - exp( -u) 1 fX 1 = I g( u )tanh ( -4U ) du = -7T - x h ( 4 + r 2 ) exp (-2iru) 1 + exp ( -u ) dudr ( l + 2ir)) 2 1 x g(2u) du !7T f Xx h ( �4 + r 2 ) l OO exp(I +-uexp( dudr u) 2:I h ( "4I ) 7T2 fX_ x h ( 4I + r 2 ) { 2:I ( rf' (I + ir) - rf' ( 21 + ir ) ) } dr I x f' = 2:I h ( "4I ) + 7T i h ( 41 + r 2 ) r ( 2:I + ir ) dr x - � i: h ( 1 + r 2 ) � ( 1 + ir) dr.
We continue the computation as follows : 00 0
0
o
=
+
0
0
o
0
_
=
-
Consequently, the integral (6.5.36) is equal to
2g(O)ln 2 + � h ( ! ) + � i:h ( ! + r 2 ) � ( � + ir ) dr oo ( f ! 7T - 00 h �4 + r 2 ) .cf ( I + ir) dr,
1 h ( -1 ) + -1 foo00 h ( -I + r ) -f' ( -1 + ir ) dr 1 1 O)ln Y - -g(O)In + a -g( 4 f 2 2 8 4 4 7T 2 - 417T i:h ( ! + r 2 ) � ( I + ir) dr. (6.5 .37)
and the integral (6.5.32) is equal to
2
Finally, the integral (6.5.28) differs from (6.5.37) by 0(1) as Y 00 . We now consider an arbitrary term in the sum (6.5.27) which corresponds to a having the property (6.5.23). The integral (6.5.28) is replaced by class �
{S y hM
f
k(u(z, Sz)) dp,(z).
U gy Fi; y E rM
We easily observe that the asymptotic behavior of this integral as Y 00 is the same as that of (6.5.28), except that it is computed with respect to the variable vY, where v is a fixed positive number which only depends on the transformation in (6.5.23), and hence on We set v = We sh,;tJ� also let denote the number a in (6.5.23). Taking (6.5.37) into account, we ob tain the following value for the sum (6.5.27) as Y 00 : �
S y.
v(Sy).
�
-
1 ( M )g(O)ln Y + "81
"8 q
�
{ & Y } rM tr( & y ) = O rM ( & y ) eHip.
a(Sy)
g
) ) g(O) Y In( a(S v( S y)
1 f oo h ( ! + r ) f ' ( ! + ir ) dr - �4 q ( M ) [�8 h ( �4 ) + _ 4 f 2 47T 1 h ( ! + r 2 ) � ( 1 + ir) dr] + 0(1 ), . 4 7T i: 2
- 00
Y � 00 ,
(6.5 .38)
137
§6 .5 . SELBERG TRACE FORMULA FOR THE DIRICHLET PROBLEM
where q(M ) is the number of classes {£yhM having the properties that tr( £ y ) = 0 and fM ( £ y ) is elliptic. Since the limit in the sum (6.5.24) is finite as Y � 00 , from (6.5.25) we conclude that the number q(M) is finite, and q(M ) = 4m(M ), where m( M ) is the number of zero interior angles in M. We finally obtain the following expression for the limit in (6.5.24):
�
lim
f
(
�
�
Y--+ oo FJ, {S hM y ' E rM(S ) \ rM
k( u ( z , y ,- ISy 'z ))
�
�
y ' E rM ( & y ) \ rM { & Y} rM tr( & y ) = O rM( & y ) ellip .
�
{ & Y } rM tr( & y ) = o rM( & y ) ellip.
In
k(�(Z, y'-I£YY'Z))
)
dp. ( z )
a ( £y ) - m f oo h I r 2 f' 1 r "2 + lr dr. 4'17' 00 4 + p( £y) -
(
) (
.
)
( 6.5 .39)
We note that finiteness of the number of classes {£yhM for which tr( £ y ) = 0 but the centralizer fM ( £y) is a hyperbolic group can be proved directly from the finiteness of (6.5.24), just as was done for the analogous assertion in the preceding subsection. We have proved the following theorem, which is analogous to Theorem 6.5.2. We shall state it using the notation adopted in this section. THEOREM
6.5.4. Suppose that M is an arbitrary noncompact regular polygon,
h(s(1 - s» = h�(s(1 - s» , and the function hl(s(1 - s» satisfies the conditions of Theorem 4.2.2. Then the following Selberg trace formula holds for the operator of the Dirichlet problem in Theorem 6.4.6: � h ( AJ £ r (tanh 'lT r ) h + r 2 dr )
= 1 ::;1 : 1 +4
{
�
d� )
rM k = )
00 1 + "2 � � { P }r k = )
M
1
2 +
)
(!
1
d sin( k 'lTjd )
f oo exp ( -2 'lTrkjd ) h ( 1 + r 2 ) dr - 00
-
1 + exp( 2 'lT r )
4
In N( P ) g( k ln N( P » k/2 - N( P ) -k/2 N( P )
�
In N( P ( £ y » g (( 2 k - I )ln N( £y » k 1/2 /2 N( £ y ) k - I + N( £ y r + { & Y } rM k = l
�
tr( & y ) * O
g(O) 1 2
4
�
{ & Y hM tr( & y ) = O I'M( & Y ) ellip .
a( 0Y )
In p
( £y )
- m In 2
- "21
m f OO h 1 r -4'17' 4+ -. 00
�
{ & Y}rM tr( & y ) = O I'M ( 0 Y ) hyperb.
In N( P( £y »
( - 2 ) -[[' ( -21 + lr. ) dr,
( 6 .5 .40 )
138
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
where 'Aj on the left runs through the set of all eigenvalues of this Dirichlet problem, and the summation on the right is taken over all primitive classes {RhM , {PhM and {t9yh , tr(t9y) =1= O. All the series and integrals in (6.5.40) converge absolutely, and both ofMthe sums over the classes {t9yhM , tr(t9y) = 0, contain only finitely many terms. We extend the definitions of the distribution functions 7T (X; t9 fM) ' NM('A) (see the previous subsection) word for word to the case of a noncompact regular polygon M. LEMMA 6.5.4. l) 7T ( x ; t9fM) = O( x ), x 2) NM('A) � I M I 'A/4 7T , 'A � 00 .
� 00 ;
The proof of these facts is similar to the proof of Lemmas 6.5.2 and 6.5.3. In part 2), instead of (6.5.22), from (6.5.40) we have the estimate
t ---> O,t>O
which, by the Tauberian theorem, also leads to the desired result. We conclude the section with a theorem analogous to Theorem 6.5.3. THEOREM 6.5.5. The Selberg trace formula for the Dirichlet problem (6.5.40) is true as an identity for any function h('A) satisfying the conditions of Theorem 4.4.4. It remains for us to note here that the Selberg trace formula for the Neumann boundary value problem on a regular polygon M now follows trivially by Theorem 6.4.6, the classical Selberg trace formula (4.3.35), and the Selberg trace formulas (6.5.21) and (6.5.40) for the Dirichlet problem. §6.6.
Elements of the theory of the Seibert zeta-function for the Dirichlet boundary value problem on a regular polygon
In this section we shall define a zeta-function Z M (s), which we shall call the Selberg zeta-function for the Dirichlet boundary value problem on the regular polygon M C H; then we shall prove several of its basic properties. The basic results of the section were first published in [65] (see also [66]). We shall carry out the proofs in the context of an arbitrary noncompact regular polygon. However, since all of the results we obtain will be consequences of the Selberg trace formula for the Dirichlet problem, and since (6.5.2 1 ) is formally a special case of (6.5.40) (in (6.5.40) it sUffices to take m = 0, and to take the set of classes {t9yh for which tr(t9y) = 0 and fM (t9y) is elliptic to be the empty set), it M follows that, in particular, we shall obtain the results for an arbitrary compact regular polygon as well. We proceed to the definition of the zeta-function. In accordance with (6.5. 1 5) (6.5. l 7), we introduce the following numerical function on the classes {t9Y}r and M tr( t9y) =1= 0: a( t9y)
=
{
2 p(t9y) ' 1 , p( t9y )
( t9 y ) 2 , = y. =
(6.6 . 1 )
§6.6. THEORY OF THE SELBERG ZETA-FUNCTION
139
We introduce the following product over the primitive classes {PhM and { 0Y hM , tr( &� y ) =1=
0:
Z M (S) = II II (I _ N(PfS- k ) 2 00
( P } rM k = O
S 1, Z M (s)' is
4.4.1, 6.5.2
6.5.4.
(6.6.2)
which converges absolutely for Re > by Lemmas and From the definition oLa(0Y) it follows that an analytic function of in this half-plane. We now derive a representation for the logarithmic derivative of which is analogous to the formula for the classical Selberg zeta-function; all of the basic properties of and the spectral applications will follow from this formula.
Z M (S)
(5.1.2) Z M (S)
s
6.6.1. Suppose that s E C, b E IR , Re s > 1 and b > 1 is fixed. Then ZM (s) 4 1 M 1 ( s _ 1. ) � ( _1_ _ 1 ) S+k b+k 2 ZM 7T - 2 { R�} rM kd�'= 1 d Sin( !,,/d l [ 7T ( exP ( -21Tik ( S - � ) /d ) ) (I - exp(- 21TiS)) - 1 THEOREM =
_
k =O
+4 ( s - -21 ) � [ ( s - 1/2)I 2 + r/ - (b 1/2)l 2 + r/ l + + 1 6 ( s - -21 ) ' j ( 6.6.3) where C l 5 and C I6 do not depend on s, and Aj = 1/4 + r/- (The rest of the notation is from Theorem 6.5.4.) PROOF. In the trace formula (6.5.40) we choose for h(l/4 + r2) a function h(l/4 + r 2 ; s; b) which depends on two parameters: h ( I/4 + r 2 ; s; b ) = (s - 1/2)1 2 + r 2 ( b - 1/21 f + r 2 . c1 5
_
5.1.1,
Just as in Theorem we first suppose that in has the form
1
1QT Z'Z MM ( s ) ds = 2 jQT( n ) Zz'MM (s ) ds + 1QT( n ) i'MM ( s ) ds (6.7 .3)
where Q T( n ) is the half of the contour QT to the right of the line Re s = 1 /2. The values of the arguments of the functions are obtained by continously moving from the point s = A along the broken line consisting of the two segments s A A + iT 1/2 + iT. N ow Theorems 6.6.3 and 6.6.4 imply that =
--+
Nn T - PQT = 8NM
--+
{ � + T 2 ) + 0( 1 ) . T .... oo
Comparing this with (6.7.2) and (6.7.3), we obtain the lemma. The proof is complete. LEMMA 6.7.2.
{
)
! arg 'I'M 1 + iT = 2 / M / T 2 - TIn T . 4m + i ( cM + m ) T + 0( 1 ) 2 7T 7T 7T 7T � "" oo (see the notation in (6.6.4» . PROOF. We make use of (6.6. 1 3), setting s = 1 /2 + iT. We have already consid ered the contributions from the identity and the elliptic classes in the course of proving Lemma 5.2.2; hence, it suffices to estimate the contribution from the factor with the gamma-function. Stirling's formula gives us
(
)
2m 4m 4m arg f 1 - iT = -1 arg f ( 1/2 - iT) = (-Tln T + T) + O( l ) . 2 7T 7T 7T T .... 00 f 2 m ( 1/2 + iT ) The proof is complete.
145
§6.7. THE REMAINDER IN WEYL'S FORMULA
LEMMA 6.7.3.
estimate
For any
C1 9 '
c20 E IR,
ZM( S )
C 1 9 :S;;;;
c20, the function ZM(S) satisfies the
= O(exp 1 1m s I)
in the region C l 9 :s;;;; Re s :s;;;; c20 ' 1 1m s I > 1 .
The proof of this lemma is close to the proof of Lemma 5.2.3, so we shall limit ourselves to an outline of the basic steps. It suffices to prove the lemma in the region - 1 :s;;;; Re s :S;;;; 2, (6.7.4) Im s > 1 . We set s =
a
+ i T and first prove the estimate
(6.7.5) We do this by estimating the logarithmic derivative of ZM( S ) in the region (6 . 7 .4) for " admissible" values of T. We use (6.6.3), estimating each term on the right in that formula. As shown in the proof of Lemma 5.2.3, the contributions from the identity and elliptic classes are bounded, respectively, by O(Tln T) and 0(1 ), T � 00 . From the Weyl formula for NM( "A ) (see Lemmas 6.5.3 and 6.5.4) we derive the estimate
[
(a + iT - 1.)2 � (a + iT - 11/2)2 + r/ - ( b - 1 /12) 2 + r/ j
In addition, it is not hard to prove that ( r'/r )( 1
- - iT ) a
=
1
=
O( T 2 ) . T-> oo
O(ln T ) .
This leads us to the formula T� Integrating over the line segment from a +
00 .
(6.7.6)
iT to 2 + iT, we obtain T�
00 ,
which proves (6.7.5). To prove the lemma we now use the Phragmen-Lindelof principle for the sector ?T/4 :S;;;;
arg( s + 1
- i)
:s;;;;
?T/2.
(6.7.7)
On part of the boundary of the sector, i.e., on the segment arg(s + 1 i ) = ?T/4 , we have Z M (S) = 0(1) by the definition of ZM(S) in (6.6.2). Next, we set s = - 1 + iT, T > 1 . To estimate Z M ( - 1 + i T) for large T we use the functional equation for Z M (S). On the right in (6.6. 1 2) we take s = 2 - iT and estimate the behavior of each factor. It follows from the proof of Lemma 5.2.3 that
-
Z M (- l +
iT ) = exp(6 1 M I T ) + 0( 1 ) . T-> 00
Thus, the function
F( s ) = Z M ( S )exp(6 1 M 1 is ) is analytic inside the sector (6.7.7) and satisfies the following estimate there:
F(s ) = exp 0(1 s 2 1) ,
.L<J:O
VI. SECOND REFINEMENT OF THE EXPANSION THEOREM
and the following estimate on its boundary:
F( s )
=
(6 .7 .8)
0(1 ) .
Consequently, (6.7.8) holds everywhere inside the sector (6.7.7). This gives us the lemma. The proof is complete. The proof of the next lemma is analogous to that of Lemma 5.2.4. LEMMA 6.7.4.
T � 00 , where the choice of the argument is described in Lemmas 6.7. 1 and 5.2. 1. Lemmas 6.7.1 - 6.7.4 lead us to a formula of the type (6.7.1), but with a worse order for the remainder term, namely OCT). THEOREM 6.7.1 . The formula (6.7.1) holds. PROOF. For the reasons mentioned at the beginning of the section, we shall limit ourselves to an outline of the basic steps in the proof. It suffices to prove the estimate
T�
(6.7.9)
00 .
By analogy with §5.2, we introduce the notation
)
(
SeT ) = 'TTl arg z M 21 + iT ,
S (X) I , 'J(t ) = max I x tI I We let p denote an arbitrary zero of Z M (S). We set s = a + it, t > 10, and 1 < a < 5/4, a fixed. ";;'
,,;;,
We successively verify the estimates
fa In I Z M ( a + it ) I d a = 0 ( t ) , 1 /2
For a value of the variable a, In Z M( s ) =
i
S I ( T ) = OCT ) . t < a � a, we have
!,�; ( s - ; - iy ( S( y ) dy + 0( 1 ) ,
t � 00 .
For values of the variables a and � , a > 1. 0 < � < t/2, we have In Z M( s ) =
i
f
t
+ t�
�
(s
-
1/2 - jy r I S( y ) d Y + 0 ( �- I 'J(2t )) 0( 1 ) . 1
Next, setting N(T) = 8NM (T 2 + t), we have
t� 00
N( T ) = c2 l T2 + cnTln T + c 23 T + R ( T ), R ( T ) = 2S(T ) + 0 ( 1) ,
t ---> 00
where c2l ' cn. and cn do not depend on s. By definition, N(T) is a monotonic nondecreasing function, i.e., N(T + x) - N(T) ;;;. 0 for x ;;;. O. Let x E (0, T). We have
R ( T + x ) - R ( T ) ;;;. c24 Tx,
§6 .7. THE REMAINDER IN WEYL'S FORMULA
where
C24 does not depend on T or
x.
.
147
Next, we obtain
S( T ) = ( ( T§, 2 T ) 1/ ) , which implies another estimate: +
O
( ) 2
jl/2+ln -llln I Z M ( o it) I do O {t1/4'?f3/4 (4t)ln-1/2 t ) . IP
=
1 - 00
Finally, from the last estimate and the well-known Hadamard three circle theorem we obtain the desired estimate (6.7.9) and the theorem. The proof is complete.
CHAPTER 7
THE SPECTRAL THEORY OF PERTURBATIONS OF THE SPECTRUM OF 1lJE OPERATOR � ( f ; X ) . SOME PERSPECTIVES ON THE DEVELOPMENT OF THE SPECTRAL THEORY OF AUTOMORPIDC FUNCTIONS §7.1. Deformations of the group f, the spectrum of � ( f ; X ) ,
and singular points of the resolvent 9t ( s ; f ; X )
In this section we shall prove a theorem on continuity of the singular points of !R (s; f; X ) as a function of a regular deformation of the discrete group f E WC 2 . For groups f E WC I such a result is well known (see [ 1 1 ]) and lies within the framework of the classical theory of perturbations of a selfadjoint operator with purely discrete spectrum. This is by no means the case for perturbations of � (f; X ) for f E WC 2 and X E 91 if). We begin the section with a short introduction. The perturbation theory for the spectrum of an abstract selfadjoint operator in Hilbert space lies at the base of the proof of the theorem on expansion in eigenfunctions of the automorphic Laplacian � ( f; X) (see Chapter 2). The principle involved can be stated as follows. Operators which are close to one another in some sense must, in general, have spectral properties which are not too sharply different. The quasiresolvent D(s; f; X ) with which one compares the resolvent !R (s; f; X) of �(f; X ) is induced by a simple operator with the same continuous spectrum as �(f; X ). Although D(s; f; X) and !R (s; f; X) are not near to one another in the usual norm in X(f; X ), their proximity in this situation means that the difference 9t (s; f; X) - D(s; f; X ) is a compact operator. Once the theorem on expansion in eigenfunctions of �(f; X ) has been proved, one can ask what are the spectral properties of a selfadjoint operator � e in :Je(f; X) if its resolvent !R eCs) is close to !R (s; f; X ) in the norm. (We are comparing the resolvents rather than the operators themselves, because the former are defined on the entire space :Je(f; X).) Specialists in operator theory will say that there is no clear-cut answer to this question. However, it becomes a well-posed question if it is formulated in a more specialized manner, within the framework of the spectral theory of automorphic functions. We consider a family of groups fe E WC which depends continuously on a parameter E E [0, 1 ] and has the following properties: 1 ) All the fE ' E E [0, 1 ], are algebraically isomorphic to one another, i.e., they have the same signature fE = f( g; m l , . . . , m , ; n ) (see § 1 .2). 2) Let 8E : fo fs be the isomorphism in condition 1). Then for every E E [0, 1 ] there exists a continuous differentiable one-to-one map E E : H H, such that for any y E fo we have EeC yz ) = 8eC y ) EeC z ) for any z E H. �
�
1A Q
150
VII. SPECTRAL THEORY OF AUTOMORPHIC FUNCTIONS
3) The restriction of Ee to Fo, the fundamental domain for fo in H, induces a shift transformation which is a bounded linear operator X( fo ; fe) : X(fo; X) X(fe; X ), which depends continuously on e and is strongly convergent to the identity operator § in X(fo; X) as e 0. The map X( fo; fe) defined above will be called a regular deformation of the space X ( fo; X). The regular deformation X(fo; fe) is said to be trivial if it is associated with the trivial deformation of the group fo, i.e., fE = g(e)fog( etl , where g(e) belongs to G and depends continuously on e. It can be verified that nontrivial regular deformations exist. It is especially simple to verify this for groups fo with signature fo = f( g; 0; 1), but we shall not dwell on this here. We now prove the basic theorem of this section. -,>
-,>
THEOREM 7. 1 . 1 . Suppose that there exists a regular deformation X(fo; fe). Further,
suppose that So is an arbitrary singular point with multiplicity ko of the resolvent 9l (s ; fo; X) as an integral operator, and the disc of radius ro around So has no other singular points of 9l(s ; fo; X) (see §2.2). Then there is an eo > ° such that, for every e os:;; eo and all r os:;; ro , the disc of radius r around So has exactly ko singular points of the resolvent 9l(s ; fe ; X). PROOF. We first prove that the following assertion holds under the conditions of the theorem: For every fixed s with Re s > 1 , the resolvent 9l(s ; fe ; X) depends continuously on e E [0, 1]. In fact, for such s the kernel of 9l(s ; fe; X) is represented by the absolutely convergent series ( 1 04. 1)
r ( z, z'; s ; fe; K ) = � X ( y)k ( z, yz' ; s ),
(7. 1 . 1 )
y E r,
where z *" z' (mod fE). We show that (7.1 .1) converges uniformly for e E [0, 1 ]. It is not hard to verify (see [59], Lemma 4. 1) that the kernel k(z, z', s) is majorized by a function k\(z, z') with the properties a) fHk l (Z, z') dp, (z') < 00 and b) k 1 (z, z') has regular growth, in Selberg's terminology (see [5 1]), i.e., there exist positive constants a, o and c such that for all z, z' E H with u(z, z') ;;>- a we have
where u(z, z') is the fundamental invariant of a pair of points. From the discreteness of fE it follows that the set {z E H I u( yz', z) < o} for fixed y E fE and z' E H can only intersect with finitely many sets of the same form but with y' instead of y, y' E fE. We let N(z'; y) denote this nurp�er of intersections. It is not hard to see that N(z' ; y ) does not depend on z' or y, 'but only on the group fE' and it depends on the latter continuously in e E [0, 1]. We set Nmax = max N. f E [O, I]
We have the estimate �
y ET,
I k ( u ( z, yz') ; s ) 1 os:;; cNmaxf I k \ ( z, z ") I dp,( z") , H
where u(z, z') ;;>- a (mod fJ, and this implies that the series (7. 1 . 1) is uniformly convergent in e E [0, 1] and that 9l(s; fE; X) is continuous as desired.
§7 . 1 . DEFORMAnONS OF THE GROUP r
151
We now proceed directly to the proof of the theorem. We introduce the operator � i s ): �X( fo ; X ) X( fo; X ) by the formula � Js) = X-I( fo ; fe) � (s; fe; K ) X( fo ; fe) . ( 7.1 . 2) The operator X( fo; fJ has the inverse X -I(fo; fe ) , by condition 2), which is a bounded operator for any e, at least in a neighborhood of the point e = 0, by condition 3) in the definition of a regular deformation. Furthermore, since the resolvent has been proved to be continuous, it follows that the difference �( s; fo; X ) - � e< s ) is an operator which is bounded on X( fo; X ), depends continuously on e, and vanishes in the limit as e O. Recallthat s is fixed, Re s > l . We now consider the Faddeev equation (2.2.6) for the operator ?B ( s; fo; X ) and the analogous equa tion for ?B ( s; fe; X ), where in both equations we take s in the half-plane Re s > O. We have ( 7. 1 .3 ) ?B ( s ; fo ; X ) = m ( K ; fo ; X ) + w &J ( s ; fo ; X ) ?B (s; fo ; X ) , ( 7. 1 .4) 58 (s; fe ; X ) = m ( K ; fe; X ) + w&J(s; fe ; X ) ?B (s; fe; X ) , where K is fixed, K > 3 (see §2.2). We transform (7.l .4) as follows: ( 7. 1 .5 ) ?B e(s) = m J X ) + w &J e(s ) m e(s ) , where, by definition, ?Be(s) = X-I( fo ; fe) ?B ( s ; fe; X ) X( fo ; fJ , m e( x ) = X-I( fo ; fe) m ( K ; fe; X ) X( fo ; fJ , &)e(s ) = X - I ( fo ; fJ&J (s; fe ; X ) X( fo ; fJ . We note that the operators in (7. l .5) have already been defined as operators in X( fo; X) for Re s > 1/2, s f1. (1/2 , 1] , and in 0[,_ ( fo; X ) for s with Re s > O. To I for the difference ?B ( s; fo; X ) prove the theorem we compose a new integral equation - ?B i s ) from (7. l .3) and (7. l .4). We have ?B (s; fo ; X ) - ?B e( s ) = m ( K ; fo ; X ) - mee K ) + w [ &J ( s ; fo ; X ) - &J e( s ) ] ?B e( s ) + w&J(s; fo ; x ) [ ?B (s ; fo ; X ) - ?B e(s ) ] , ( 7.1 .6 ) and so this difference is determined by the formula ?B ( s ; fo ; X ) - ?B e( x ) = ( 10 - w&J (s; fo ; X ))- I �
�
X
[ m ( K ; fo ; X ) - m e( K ) + w ( &J ( s; fo ; X ) - &J e( s ) ) ?B e( s ) ] ,
( 7 . 1 .7 )
where 10 is the identity operator in X( fo; X ). Recall that (see §2.2) &J (s; fo ; X ) = m ( K ; fo ; x ) ( 10 + w O es ; fo ; X ) ) , 91 (s; fo ; X ) = � ( K; fo ; X ) + m ( x ; fo ; X ) (and analogous formulas hold for fe ) , where � ( K; fo; X ) and O( s; fo; X ) are defined by (2. 1 .27) and (2.2.13), respectively. One can prove that the differences �(K ; fo ; X ) - X - I ( fo ; fJ � ( K ; fe; X ) X( fo ; fe) , O (s; fo ; X ) - X - I ( fo ; fe) O (s; fe; X ) X( fo ; fe)
152
VII. SPECTRAL THEORY OF AUTOMORPHIC FUNCTIONS
are bounded operators in :K (fo ; X) for Re s > 1/2 and in Cj� _ I (fo; X ) for Re s > 0, which depend continuously on f E [0, 0], 0 ';;;: 1 , and vanish in the limit as f O. By this, together with the continuity property above for m ( K; fF ; X) , it follows that the difference --->
have the same properties. We proceed to the consideration of (7. 1.7). We rewrite this formula in terms of the kernels of the integral operators which appear in it, and we take the free variables z, z' E Fo in general position; Fo is a fundamental domain for the group fo. Let So be the singular point in the theorem; it is a pole of multiplicity ko for the operator 'R(s; fo ; X) and ( 10 - w&j(s; fo; X » - I (see Theorems 2.2.3, 2.2.5 and 2.2.6). Let {Sj }7�:i e) be the set of all poles (counting multiplicity) of the operator 'R is) in the disc of radius r around so. If we compare the coefficients in the Laurent expansions at all singular points in the disc of radius ro around So on the left and right sides of (7. 1 .7) (written in terms of kernels, as indicated above), we arrive at the following relation : a s
( ) k ( s - so )
I)
-
l1 ( r ; E ) o j=
�
C/f)
I S - Sj ( f )
1 .
[
n ( ro ; e) ci f) b(s ) . ( f) f, s + 12 ( ) j�= l s - Sj ( f + h ( f , s ) (7.1 .8) II k ) ( s - so ) where a ( s ) , b( s ) , 12 ( f; s ) , h( f; s) are analytic functions of S in the disc of radius ro around so ' for all f E [0, 0 ]. The functions /1(f), /:z(f; s ) vanish in the limit as f 0, c.I.( f) =1= 0, f E [0, 0]. Now suppose that the theorem is false. Then there exist infinite sequences fm and rm, m = 1, 2, 3, . . . , such that fm 0 and rm 0 as m ---> 00 and the numbers n(rm ; fm) fail to be equal to ko for all m . But (7.1.8) easily leads to a -
-
()
--->
--->
__
--->
contradiction with this supposition. To see this, it is enough to pass to the limit zero over the sequence fm in (7. 1.8). The proof is complete. This theorem essentially says that the singular points of the resolvent m(s; f ; X ) depend continuously on a regular deformation of the discrete group. Recall (see §2.2) that the set of singular points in question is situated on the interval s E (1/2 , 1] and the half-plane Re s � 1 /2. Precisely the ones which lie in the interval s E (1/2 , 1] and on the line Re s = 1/2 correspond to eigenvalues X of the discrete spectrum of W(f; X), X = s(l - s) (Re s ;;::' 1/2). In principle, Theorem 7. 1 . 1 even gives new information concerning the discrete spectrum of W (fe ; X) for any group fe which regularly deforms to fo (and lies in a neighborhood of the latter) if we already have information concerning the discrete spectrum of W(fo; X). It is first of all natural to take for fo some arithmetic group fo E 9JC 2, a which has a nontrivial regular deformation; by Theorems 6.1 . 1 , 6. 1.2 and 6.4.7, for such a group the spectrum of W(fo ; X) is very rich. (Here we are restricting ourselves to the case of the trivial representation X, dim V = 1 .) As an example of such a group fo we can take the commutator [fz ' f�], of the modular group fz , i.e., the group generated by all the commutators of fz . It is well known that [fz , fz ] is contained in a one-parameter family of Fricke groups (see §6.3) and has a nontrivial regular deformation.
§7 .2. ZEROS OF ZETA- AND L-FUNCTIONS
153
In [53] for number-theoretic purposes (more precisely, in order to construct counterexamples to the general Petersson conjecture estimating the Fourier coeffi cients of cusp form), Selberg constructed a special subgroup of finite index fq C [ fz , fz ] depending on a parameter q E lL. He proved that there is a nontrivial eigenvalue A = s(l - s) of � ( fq; 1) lying between ° and 1/4 (this corresponds to 1 /2 < s � 1 ) and arbitrarily close to 0, depending on q E 71.. Theorem 7. 1 . 1 allows one to infer that there exist nonarithmetic groups f which can be deformed to fq, also having eigenvalues of � ( f ; 1 ) close to A = 0. We shall not dwell on this here, except to note that our " method of contin:q.ity" is especially suitable for studying precisely the eigenvalues Aj (e) = sj (e)(l - si e)) of � ( fe' X ) for which sin) lies in a neighborhood of s = 1 . In fact, the selfadjointness of all the operators � ( fe ; X ) "only permits" the deformation X( fo; fe) to move si e) as a function of e along the interval Sj E ( l /2 , 1 ], and a small deformation cannot throw si e) into the half-plane Re s < 1 /2. On the other hand, Theorem 7. 1 . 1 does not prevent an arbitrarily large discrete spectrum { AiO) = siO)(l - sin))} of � ( fo ; X } for which the corresponding sin) lie on the line Re s = 1/2 from vanishing completely for an arbitrarily small nontrivial regular deformation, because of the displacement of all the si e) into the half-plane Re s < 1/2. This circumstance also shows how difficult Roelcke's conjecture is (see §6. I). It seems to us that it is an interesting (if difficult) problem to study the directions of displacement of the singular points of 9t(s; fe; X ) as a function of a nontrivial regular deformation X( fo; fe)' A predominantly vertical displacement of the singular points (parallel to the 1m s axis) would support the Roelcke conjecture. A good model of deformation in analytic number theory is the behavior of the zeros of the Hurwitz zeta-function ns; a) as it degenerates into the Riemann zeta-func tion, i.e., for small values of the parameter a. Recall that the Hurwitz zeta-function is the meromorphic function on C which is given in the half-plane Re s > 1 by the Dirichlet series res; a ) §7.2.
=
CXJ
�
n= ]
1
(n + af '
a E �.
Zeros of zeta-and L-functions of imaginary quadratic fields and the eigenValues of ( 1 ) � fz ;
As we mentioned in the survey [66] (see also Chapter 6), the basic stimuli for the modern development of the spectral theory of automorphic forms for Fuchsian groups f E me 2 are the open problems in the theory of the discrete spectrum of the operator�(f; X ), X E � sef), in particular, the Roelcke problem. If one speaks of analogies and historical roots, then these problems have above all a function-theo retic meaning. For example, as already mentioned in §6.3, the problem of the existence of cusp-functions (i.e., the problem of nontriviality of the space Xo( f; X)) is related to the classical problem of Klein and Poincare in function theory at the end of the nineteenth century on the existence of a meromorphic automorphic function for a given Fuchsian group. However, there are other important directions for the development of the modern spectral theory of automorphic functions which arise because of the needs of number-theoretic applications. We now know some number-theoretic directions in spectral theory, such as the study of the connection
154
VII. SPECTRAL THEORY OF AUTOMORPHIC FUNCTIONS
and analogy between the Selberg trace formula and the Voronoi-Hardy summation formulas for the circle problem (see [ 1 8] and [66]), the connection between the spectral theory and estimates for Kloosterman sums in number theory (see [3]), and also its connection with the refined Kummer conjecture for cubic characters (see [41 ], and also [ 1 7]), and so on. Here we shall only discuss one aspect, which we did not discuss in our earlier survey [66]. This is the interrelation between the spectral theory and the theory of the Riemann zeta-function from the point of view of the prospects for the development of the former theory. We indicate some facts which by themselves stimulate the development of the spectral theory. 1 . The analogy between Weirs explicit formulas in number theory and the Selberg trace formula led Selberg to the definition of his zeta-function. 2. The explicit Selberg trace formula for the modular group rz made it possible to derive a new formula for the Tchebycheff psi-function, expressing l{;( x) in terms of the distribution function 71'( x ; rz ) for the norms of primitive hyperbolic conjugacy classes in rz and in terms of the eigenvalues of m-(rz ; 1 ) (see [63]). Formulas of this type can be obtained from the Selberg trace formula with Hecke operators for rz. Although at the present time these formulas do not have practical value for number theory, in the first place they point to the existence of an indirect connection between the spectral theory of automorphic forms for rz and the theory of the Riemann zeta-function; in the second place, they stimulate the study of the function 71' ( x; rz ) and the distribution of eigenvalues of m-(rz; 1). 3. The asymptotic formulas obtained by Hejhal by considering the kernel of the resolvent of 2l(r; 1) for special quaternionic discrete arithmetic groups r E WC \,a (see [20]), for example,
� N{n )N{5n + 1 ) = � ( 1n c) 2 X + � a k x sk + -.0 (X 2 / 3 ) , (7 .2 . 1 ) x oo 71' k n �x where N(n ) is the number of ideals in the field 0(/2) with norm n. The numbers Sk
on the right in (7.2. 1) have a spectral origin. Analogous formulas can be obtained by studying the kernels of the resolvents at special points for other arithmetic groups r E WC 2 • All of these formulas will then have practical value for number theory, because of the extraordinary good (from the point of view of analytic number theory) remainder term. 4. The proof that the Eisenstein series E( z ; s; rz ) has no poles on the line Re S = 1/2 by methods of operator theory (Theorem 2.3. 1 , X = 1 , dim V = 1) led us to an independent proof of the asymptotic prime number theorem (n2s) =1= 0, Re s = 1/2, �(s) the Riemann zeta-function). Thus, the prime number theorem is a consequence of the selfadjointness of the operator U(rz; 1). Finally, we cannot avoid mentioning the connection between sums of Klooster man sums and the so-called density theorems in number theory. Progress in estimating these sums based on spectral theory (see [3]) is extremely important for the theory of the distribution of prime numbers and the theory of the Riemann zeta-function. The number of facts of this sort can be multiplied, but this would carry us too far afield. The basic purpose of this section is to prove a theorem which enables one to compare the spectral singularities of the resolvants of automorphic Laplacians
§7 . 2 . ZEROS OF ZETA- AND L-FUNCTIONS
155
defined in two-dimensional and three-dimensional Lobachevsky space. Based on this theorem, we state a conjecture on a possible intersection of the set of zeros of the Selberg zeta-function Z( s; fz ; 1 ) for Re s = 1 /2 and the set of zeros of the Dede kind zeta-functions of imaginary quadratic fields. At the end of the section we indicate a new direction in the spectral theory of automorphic functions, whose development is naturally stimulated by this conjecture. We published the basic results before in [69]. We introduce some notation and definitions (compatible with Chapter 1) from the spectral theory of automorphic functions for m-dimensional Lobachevsky space. In what follows it is dimensions m 2 and m, . 3 which will be important to us. Let Hm be m-dimensional Lobachevsky space; Hm = Gm/Km' where Gm is the corre sponding Lie group and Km is its maximal compact subgroup. Next, let dS m be the Riemannian metric on Hm which is invariant relative to Gm; let Lm be the Laplace differential operator for the metric dsm; and let dP,m be the Riemannian measure on Hm determined by the metric dsm. We introduce the notation :JCm = Lz{Hm; C ; dP,m ). In addition, m m is the selfadjoint operator on :JCm induced by the operator -Lm on a suitable dense set. We have the resolvent 1Rm(s) = ( m m - sCm - s - l)t i , and the integral operator has kernel =
( 7.2.2 )
where um(z, z') is the fundamental invariant of a pair of points. The following fact follows from Lemma l .2 of [59]. LEMMA 7.2. 1 . The function k m( u; s) in (7.2.2) has the integral representation
where Re s > m/2 - 1 , f( s) is Euler 's gamma-function, and c( m) is a constant depending only on m. Let fm C Gm be a discrete subgroup acting on Hm, and let Fm be a fundamental domain for fm in Hm. We introduce the Hilbert space of automorphic functions on Hm: :JCm(fm ) = Lz{Fm; C ; dp, ; 1 ) (see § 1 .3) and the nonnegative selfadjoint operator m m(fm): :JCm(fm) � :JCm(fm), induced by -Lm. We denote the kernel of its resolvent by rm(zm' z:n; s; fm). The following lemma is proved by analogy with Lemma 1 .4 of [59]. LEMMA 7.2.2. The following expansion in an absolutely convergent series holds in the region Re s > m - 1 : rm ( zm , z:n ; s ; fm) = � km ( um ( z m , yz:n) ; s ) , y Efm
where zm ' z:n E Hm, zm =i= z :n (mod fm), and the series converges uniformly with respect to Zm and z:n in any compact subregion of Hm X Hm which does not intersect the surfaces Zm = z:n (mod fm)·
V l l . Sl'l:.C I KAL I Hl:.U K Y U.t" A U I UMUKl'H I L t· U N L. I I V N :>
As mentioned above, we shall only be interested in the dimensions m = 2 and m = 3. We set where d is an arbitrary squarefree natural number and 7L(�J) is the ring of integers in the imaginary quadratic field OCr-d). LEMMA 7.2.3. 1) Let z 2 ' z 2 ' E H2 be fixed points in general position. The function ri z 2 ' z ;; s; f2 ) has the following poles in the region Re s > 0 which will be of special importance to us: a) a simple pole at the point s = 1 ; b) simple poles Sj on the line Re s = 1 /2 , s =1= 1 /2 , where each pole is connected with an eigenvalue Aj ,2 of the discrete spectrum of m i f2 ) by the formula s/1 - s) = Aj .2 ; and c) poles p at the zeros of the Riemann zetajunction �(2s) having the same multiplicity Re s < 1/2. 2) Let d correspond to a field O(N) of class number one, and let Z3' z � E H3 be
fixed points in general position. The function ri z 3' z�; s; fid » has the following poles in the region Re s > 0 which will be of special importance to us: a) a simple pole at the point s = 2; b) simple poles Sj on the line Re s = 1 , s =1= 1 , where each pole Sj is connected with an eigenvalue Aj,3 ofm l fid» by the formula sj (2 - Sj ) = Aj,3 ; and c) poles p at the zeros of the Dedekind zetajunction �-A s) of O(N) having the same multiplicity, Re s < 1 . PROOF. The simple poles in la) and 2a) correspond to constant eigenfunctions of the operators m i f2 ) and m 3( f3 )' The simple poles in 1 b) and 2b) are guaranteed by our Theorem 2.2.6 and by Theorem 3. 1 of [59]. We proceed to consider the poles in c). From our Theorem 2.3.4 and from Theorem 3 . 1 and Lemma 3.5 of [59] it follows that at least part of the poles of the functions ri z 2 ' z ;; s; f2 ) and ri z 3 ' z�; s; f3 ) are induced by the poles of the determinants of the scattering matrices �is) and �is), respectively. The functions �is) and �is) can be computed explicitly: �
2( s ) =
r;; f ( s - 1/2) �( 2s - 1 ) f ( s ) �( 2s )
(see (6. 1 .6» , and
s 1) � 3 ( s ) = S -7T_1 LA r_d ( S ) _
-
(see [29]); and this implies l c) and 2c). The proof i s complete. In the sum in Lemma 7.2.2 for m . 3 we look at the terms corresponding to summation over f2 C f3( d). In our notation we have
r3 ( z3 ' z� ; s ; f2 ) = � k3 ( U 3 ( Z3 ' 'Yz� ) ; s ) .
(7.2.3)
y E f2
The kernel (7.2.3) is the kernel of the resolvent of m i r2 ) in the space Xif2 )' (Here f2 is considered as a group of motions of the three-dimensional space H3, PSL(2, 7L) C PSL(2, 7L(N» ). Consequently, one would expect the functions ri z 3, z�; s; fid » and ri z 3' z�; s ; f2 ) to have some poles in common i n the region Re s > O. On the other hand, we have the following theorem.
§7.2. ZEROS OF ZETA- AND L-FUNCTIONS
157
There exists a map p: H2 � H3 such that the difference f - I (2s ) r3 ( p ( z2 ) ' P ( Z� ) ; s ; fJ , r2 ( z 2 ' z � ; s; f2 ) - cf( s )f(2s - l ) f- l s
THEOREM 7.2. 1 .
( - �)
where c is a constant and z 2 and z� are fixed points in general position, is a regular function in the region Re s > 0, except at the point s = 1 /2, where it has a pole of order no greater than two. PROOF. We realize the space H3 as the quotient SL(2, C)/SU(2). We have local coordinates {y, x, v}, y > 0; x, v E IR, of H3• In these coordinates the action of SL(2, C) is given as follows (see [59], §5,.�):
z' = gz, z' = {y', x', v'} , i ' = x' + iv', i = x + iv E C , E SL(2, C ) , g=
( � �)
The fundamental invariants of a pair of points are
z -z ' u 2 ( Z 2 ' z 2' ) - I 2 � f Y2 Y2 y - y(,» ) 2 + ( x - X (,» ) 2 + ( v - V (,» ) 2 ( U3 ( Z3 ' z�'» ) = yy ( ,) If the upper half-plane H2 is given by the coordinates x, y, z = x + iy (see § 1 . 1) , then the map p : H2 � H3 is defined as follows: p : x + iy � { y, x, O } . Let Re s > 2. From Lemma 7.2.2 we have r2 ( z2 , z� ; s ; f2 ) = � k 2 ( u i z 2 ' YZ� ) ; s ) , (7.2.4) _
"
r3 ( p ( z 2 ) ' p ( z� ) ; s ; f2 ) = � k3 ( U 2 ( Z 2 ' yz� ) ; s ) . y E r2
We consider the difference
r2 ( z 2 ' z � ; s; f2 ) - g(s)r3 ( p (z2 ) P (Z� ) ; s; f2 ) ,
(7.2.5) where g(s) is specially chosen so that (7.2.5) is given by an absolutely convergent series of the type (7.2.4) in the region Re s > 0 as well. The proof of the theorem amounts essentially to proving the existence of such a function g(s). For Re s > 2, by Lemma 7.2. 1 , the difference (7.2.5) is equal to �
y E r2
{
I I [t ( 1 - t )] S - I ( t + ! Ui Z2 ' yz� ) rS dt 3 g ( s ) ( 3 ) f ( s ) 11 [ t ( 1 t ) ] S 2 ( t + 1 u 2 ( Z 2 ' YZ 2, ) ) -S dt} . ( 7.2.6)
C(2) _
0
c
f( s
_
1 /2)
0
-
--
/
4
•
... _
. ...... ... ....... ........ __ .&
_
.&
_ ... ..... ...
_
.... ...... ... ,I.
"" ... .. ... "".:"... ... ... L '-' J." U J."I '-- J. J.Ul"'�
N ext, integrating by parts, we obtain l 1 1 [t ( 1 - t ) ] S - I ( t + u rs dt ( 1 + u rS l [t ( I - t ) ] S- 1 dt o t + u ) -S - I l t [ T ( 1 - T ) ] S - I d T dt , +s o ( 7.2.7) =
11(
Ia
0
0
Ia
+s \ t + u rS - 1 t [ T( 1 - T ) ] s-3 /2 d T dt .
Using (7.2.7), we choose g(s) in (7.2.6) from the condition l c (2)( 1 + u rSl [t ( 1 - t ) ] S - 1 dt o
= g( s ) c (3 )
f(S ) ( 1 + u rs l [t ( 1 - t ) ] s-3/2 dt , f ( s - 1 /2 )
01
from which it follows that g( s )
=
c (2) f ( s ) f ( s ) f (2s - I ) f ( s - 1 /2) - c (2) f ( s ) f(2s - 1 ) . c ( 3 ) f (2s ) f ( s - I /2)f ( s - I/2)f ( s ) c ( 3 ) f(s - I /2) f ( 2s )
After this choice of g(s), the difference (7.2.5) takes the form
-
X
c
f 2 ( s ) f (2s - I ) ( 2) s 2 f ( S I /2) f( 2s ) _
f.' ( t + ! u , ( z" yz; ) r-' f.' [ T( I - T ) ] , - 3/' dT dt} .
(7.2.8)
We now prove that for u 2 ;;;;. e > 0 the general term in (7.2.8) is bounded from above by O( UiZ2 ' yz� ) -Re s - I ), where the constant in the 0 in general depends on s. By Theorem 1 .2.5, this estimate ensures the absolute convergence of (7.2.8) for Re s > 0 and its meromorphicity in this half-plane. We consider the integral
11( 1 + u ) -S - I lt [ T( I -. T )] S - I d T dt . o
0
(7.2.9)
If u > 0, then obviously (7.2.9) is a regular function in the half-plane Re s > 0; hence, for fixed Z 2 and z� with z 2 =1= z� (mod f2 ) the first series in (7.2.8) with an integral of the type (7.2.9) is a regular function in the half-plane Re s > o. We now consider the second integral ( 7.2. 1 0)
§7.:?. ZEROS OF ZETA· AND L·FUNCTIONS
where we suppose that u > O. If we represent (7. 2 . 1 0) in the form
l \ t + ufs- I l t { ( [ ,. ( 1 - ,. )] S -3/2 - ,. s - 3/ 2 o
0
-
(1
159
- ,.r - 3/2 ) ,. s- 3/ 2 + ( 1 - ,.r - 3 /2 } d T dt , +
it is not hard to see that this integral is equal to a sum of the form
I I ( s; u ) + 12 ( s; u ) / (s - 1/2) + 13 ( s; u) / ( s - 1/2 ) , where for each fixed u > 0 the function 1/ s; u ) is regular in the half-plane Re s > 0, } = 1 , 2, 3, and 1z{1/2 ; u) =F 0, /3 ( 1/2 ; u) =F O. In addition, for each } = 1 , 2, 3 we have
I l'J.(s :
I
and hence the series
Z2
'- - ,
") « - /
u-Res - I .-
,
(mod f2 ), converges absolutely in the half-plane Re s > 0, where it repre sents a regular function, except at the point z = 1/2, where it has pole of order no greater than one. To complete the proof of the theorem, it remains for us to observe that the factor =F
z;
f 2 (S) f (2s - 1 ) f 2 (s - 1/2)f(2s) in the second integral in (7.2.8) contributes only one more pole to the function (7.2.5) (with our choice of g(s)) in the half-plane Re s > 0, namely, a simple pole at the point s = 1/2. The proof is complete. Roughly speaking, Theorem 7.2.1 says that the functions have the same spectral poles in the half-plane Re s > O. It seems to us that the combination of all these results gives support for the following conjecture. CONJECTURE I. The set of numbers Sj in part 1 b) of Lemma 7.2.3 has nonempty
intersection with the set of zeros in the Dedekind zeta-function �_As) of any class one field Q(V-d). Recall that all of these numbers Sj lie on the line Re S = 1/2, and they are connected with eigenvalues Aj 2 of the discrete spectrum of � z{f2 ) by the formula
Aj,2 = s/1 - s).
,
It seems that an analogous conjecture can be formulated for a field Q(../-d ) of class number greater than one, but we shall not discuss this here. A less general conjecture than Conjecture I (for the Rieman zeta-function) was stated eariler by L. D. Faddeev, based on other heuristic considerations. Faddeev's conjecture �timulated P. Cartier to undertake computer calculations of approximate values for the eigenvalues of the operator � 2(f2 ) (see [4] and [5]); however, it is still too soon to claim any confirmation of the conjectures, because of the high degree of
.J. U V
V 11.
::IrCL
I lV"\.L I nevI'\. 1 v r t'l. U 1
V1VIVKrniL r- U N L I IUN ::'
impreclSlon in these computations. This work is primarily concerned with the eigenvalues which are connected with the Neumann boundary value problem (see Theorem 6.4.6, rM = r z). These approximate computations undoubtedly are of theoretical interest, and may turn out to be useful for supporting (or, to some extent, refuting) Faddeev's conjecture and Conjecture I. However, they cannot furnish a proof, nor even shed much light on a more general conjecture concerning the numbers Sj and the zeros of all the Dedekind zeta-functions t_is) for O(N), since the set of zeros here is too large, and one needs purely theoretical investigations on this question. What type of theory would be useful here? In §6.2, when discussing the Artin theory for the Selberg zeta-function, we derived the formula (6.2.2), which connects the kernels of the resolvents of � ( r ; X) and � ( r l ; 1), where rl is a normal subgroup of finite index in an arbitrary Fuchsian group r of the first kind. In our view, a generalization of this formula to the case of a nonnormal subgroup of infinite index in a discrete group acting in H3, in particular, to the case rl = PSL(2, Z) c r = PSL(2, Z(N» , combined with Theorem 7.2. 1, might lead to an explanation of Conjecture I. In the present state of the art, it is too difficult to derive such a formula. As a first step in this direction one might generalize the spectral theory of automorphic functions for the group r E Wl considered in this paper to infinite-di mensional representations of the groups r, and carry over the entire theory to discrete subgroups acting in Lobachevsky space.
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