INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS Adrian Diaconu Paul Garrett 1 ...

41 Pages·2006·0.38 MB·English

INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS
Adrian Diaconu
Paul Garrett
Abstract. This paper exposes the underlying mechanism for obtaining second integral moments of
GL2 automorphic L–functions over an arbitrary number field. Here, moments for GL2 are presented
in a form enabling application of the structure of adele groups and their representation theory. To
the best of our knowledge, this is the first formulation of integral moments in adele-group-theoretic
terms,distinguishingglobalandlocalissues,andallowinguniformapplicationtonumberfields. When
specializedto the fieldof rational numbers Q, we recoverthe classicalresults.
1. Introduction
2. Poincar´e series
3. Unwinding to Euler product
4. Spectral decomposition of Poincar´e series
5. Asymptotic formula
Appendix 1: Convergence of Poincar´e series
Appendix 2: Mellin transform of Eisenstein Whittaker functions
1. Introduction
§
For ninety years, the study of mean values of families of automorphic L–functions has played a
central role in analytic number theory, for applications to classical problems. In the absence of
the Riemann Hypothesis, or the Grand Riemann Hypothesis, rather, when referring to general
L–functions, suitable mean value results often serve as a substitute. In particular, obtaining
asymptotics or sharp bounds for integral moments of automorphic L–functions is of considerable
interest. The study of integral moments was initiated in 1918 by Hardy and Littlewood (see
[Ha-Li]) who obtained the asymptotic formula for second moment of the Riemann zeta-function
T
2
(1.1) ζ(1 +it) dt T logT
| 2 | ∼
Z0
About 8 years later, Ingham in [I] obtained the fourth moment
T 1
(1.2) ζ(1 +it) 4 dt T(logT)4
| 2 | ∼ 2π2 ·
Z0
1991 Mathematics Subject Classification. 11R42,Secondary 11F66, 11F67,11F70,11M41, 11R47.
Key words and phrases. Integral moments, Poincar´e series, Eisenstein series, L–functions, spectral decomposi-
tion, meromorphiccontinuation.
Typesetby AMS-TEX
1
2 ADRIAN DIACONU PAUL GARRETT
Since then, many papers by various authors have been devoted to this subject. For instance see
[At], [H-B], [G1], [M1], [J1]. Most existing results concern integral moments of automorphic L–
functionsforGL (Q)andGL (Q). Noanalogueof(1.1)or(1.2)isknownoveranarbitrarynumber
1 2
field. The only previously known results, for fields other than Q, are in [M4], [S1], [BM1], [BM2]
and [DG2], all over quadratic fields.
Here we expose the underlying mechanism to obtain second integral moments of GL
2
automorphic L–functions over an arbitrary number field. Integral moments for GL are presented
2
in a form amenable to application of the representation theory of adele groups. To the best of our
knowledge, this is the first formulation of integral moments on adele groups, distinguishing global
and local questions, and allowing uniform application to number fields. More precisely, for f an
automorphicformon GL and χ an idele class characterof the number field, let L(s,f χ) denote
2
⊗
the twisted L–function attached to f. We obtain asymptotics for averages
∞
2
(1.3) L 21 +it, f ⊗χ Mχ(t)dt
χ Z
X −∞ (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
for suitable smooth weights M (t). The sum in (1.3) is over a certain set of idele class characters
χ
which is infinite, in general. For general number fields, it seems that (1.3) is the correct structure
of the second integral moment of GL automorphic L–functions. This was first pointed out by
2
Sarnak in [S1], where an average of the above type was studied over the Gaussian field Q(i); see
also [DG2]. From the analysis of Section 2, it will become apparent that this comes from the
Fourier transform on the idele class group of the field.
Meanwhile, in joint work [DGG] with Goldfeld, the present authors have found an extension to
treat integral moments for GL over number fields. We exhibit specific Poincar´e series P´e giving
r
identities of the form
moment expansion = P´e f 2 = spectral expansion
·| |
ZZAGLr(k)\GLr(A)
for cuspforms f on GL . The moment expansion on the left-hand side is of the form
r
1
L(s,f F) 2 M (s)ds + ...
F
2πi | ⊗ |
F Z
X ℜ(s)=1
2
summed over F in an orthonormal basis for cuspforms on GL , as well as corresponding
r−1
continuous-spectrum terms. The specific choice gives a kernel with a surprisingly simple spectral
expansion, with only three parts: a leading term, a sum induced from cuspforms on GL , and a
2
continuous part again induced from GL . In particular, no cuspforms on GL with 2 < ℓ r
2 ℓ
≤
contribute. Since the discussion for GL with r > 2 depends essentially on the details of the GL
r 2
results, the GL case merits special attention. We give complete details for GL here. For GL
2 2 2
over Q and square-free level, the average of moments has a single term, recovering the classical
integral moment
∞
2
L(1 +it,f) M(t)dt
| 2 |
Z
−∞
As a non-trivial example, consider the case of a cuspform f on GL over Q. We construct
3
a weight function Γ(s,w,f ,F ) depending upon complex parameters s and w, and upon the
∞ ∞
INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 3
archimedean data for both f and cuspforms F on GL , such that Γ(s,w,f ,F ) has explicit
2 ∞ ∞
asymptotic behavior similar to those in Section 5 below, and such that the moment expansion
above becomes
1
P´e(g) f(g) 2dg = L(s,f F) 2 Γ(s,w,f ,F )ds
∞ ∞
| | 2πi | ⊗ | ·
ZZAGL3(Q)\GL3(A) F oXnGL2 ℜ(sZ)=1
2
1 1
+ L(s ,f E(k) ) 2 Γ(s ,w,f ,E(k) ) ds ds
4πi2πi | 1 ⊗ 1−s2 | · 1 ∞ 1−s2,∞ 2 1
k∈Z Z Z
X ℜ(s1)=21 ℜ(s2)=12
where
L(s ,f E(k) ) = L(s1−s2+ 12,f)·L(s1+s2− 21,f)
1 ⊗ 1−s2 ζ(2 2s )
2
−
In theabove expression,F runsover an orthonormalbasisfor alllevel-one cuspformsonGL , with
2
(k)
no restriction on the right K –type. Similarly, the Eisenstein series E run over all level-one
∞ s
Eisenstein series for GL (Q) with no restriction on K –type, denoted here by k.
2 ∞
Thecourseoftheargumentmakesseveralpointsclear. First,thesumofmomentsoftwistsofL–
functions has a natural integral representation. Second, the kernel arises from a collection of local
data, wound up into an automorphic form, and the computation proceeds by unwinding. Third,
the local data at finite primes is of a mundane sort, already familiar from other constructions.
Fourth, the only subtlety resides in choices of archimedean data. Once this is understood, it is
clearthatGood’soriginalideain[G2], seeminglylimitedtoGL (Q),exhibitsagoodchoiceoflocal
2
data for real primes. See also [DG1]. Similarly, while [DG2] explicitly addresses only GL (Z[i]),
2
the discussion there exhibits a good choice of local data for complex primes. That is, these two
examples suffice to illustrate the non-obvious choices of local data for all archimedean places.
The structure of the paper is as follows. In Section 2, a family of Poincar´e series is defined in
termsof localdata,abstractingclassicalexamplesin a form applicableto GL over a number field.
r
In Section 3, the integral of the Poincar´e series against f 2 for a cuspform f on GL is unwound
2
| |
and expanded, yielding a sum of weighted moment integrals of L-functions L(s,f χ) of twists of
⊗
f by Gro¨ßencharakterenχ. In Section 4, we find the spectraldecompositionof the Poincar´e series:
the leading term is an Eisenstein series, and there are cuspidal and continuous-spectrum parts
with explicit coefficients. In section 5, we derive an asymptotic formula for integral moments, and
observethatthelengthoftheaveragesinvolved issuitableforsubsequentapplicationstoconvexity
breaking in the t–aspect. The first appendix discusses convergence of the Poincar´e series in some
detail, proving pointwise convergence from two viewpoints, also proving L2 convergence. The
second appendix computes integral transforms necessary to understand the details in the spectral
expansion.
For applications, one needs to combine refined choices of archimedean data with extensions of
theestimatesin[Ho-Lo]and[S2](or[BR])tonumberfields. However,fornow,wecontentourselves
with a formulationthatlays thegroundworkfor applicationsandextensions. In subsequentpapers
we will address convexity breaking in the t–aspect, and extend this approach to GL .
r
4 ADRIAN DIACONU PAUL GARRETT
2. Poincar´e series
§
Before introducing our Poincar´e series P´e(g) on GL (r 2) mentioned in the introduction, we
r
≥
find it convenient to first fix some notation in this context. Let k be a number field, G =GL over
r
k, and define the standard subgroups:
(r 1)-by-(r 1)
P =Pr−1,1 = − − ∗
0 1-by-1
(cid:26)(cid:18) (cid:19)(cid:27)
the standard maximal proper parabolic subgroup,
I (r 1)-by-(r 1) 0
U = r−1 ∗ H = − − Z =center of G
0 1 0 1
(cid:26)(cid:18) (cid:19)(cid:27) (cid:26)(cid:18) (cid:19)(cid:27)
Let K denote the standard maximal compact in the k –valued points G of G.
ν ν ν
The Poincar´e series P´e(g) is of the form
(2.1) P´e(g) = ϕ(γg) (g G )
A
∈
γ∈ZkXHk\Gk
for suitable functions ϕ on G described as follows. For v C, let
A
∈
(2.2) ϕ = ϕ
ν
ν
O
where for ν finite
A 0
(detA)/dr−1 v for g = mk with m= Z H and k K
(2.3) ϕ (g) = ν 0 d ∈ ν ν ∈ ν
ν  (cid:18) (cid:19)
 0(cid:12)(cid:12) (cid:12)(cid:12) otherwise

and for ν archimedean require right K –invariance and left equivariance
ν
v
detA A 0
(2.4) ϕ (mg) = ϕ (g) for g G and m = Z H
ν dr−1 · ν ∈ ν 0 d ∈ ν ν
(cid:12) (cid:12)ν (cid:18) (cid:18) (cid:19) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Thus, for ν , the furt(cid:12)her da(cid:12)ta determining ϕ consists of its values on U . The simplest useful
ν ν
|∞
choice is
x
1
(2.5) ϕν Ir0−1 x1 = 1+|x1|2+···+|xr−1|2 −dν(r−1)wν/2 x = ...  and wν ∈ C
(cid:18) (cid:19) x
(cid:0) (cid:1) r−1
   
with d = [k : R]. Here the norm x 2 + + x 2 is invariant under K , that is, is the
ν ν 1 r−1 ν
| | ··· | | |·|
usual absolute value on R or C. Note that by the product formula ϕ is left Z H –invariant.
A k
We have the following
INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 5
Proposition 2.6. (Apocryphal) With the specific choice (2.5) of ϕ = ϕ , the series (2.1)
∞ ν|∞ ν
⊗
defining P´e(g) converges absolutely and locally uniformly for (v) >1 and (w ) >1 for all ν .
ν
ℜ ℜ |∞
Proof: In fact,the argumentappliesto a much broaderclass of archimedeandata. For a complete
argument when r = 2, and w =w for all ν , see Appendix 1. (cid:3)
ν
|∞
We can give a broader and more robust, though somewhat weaker, result, as follows. Again, for
simplicity, we shall assume r =2. Given ϕ , for x in k = k , let
∞ ∞ ν|∞ ν
Q
1 x
Φ (x) = ϕ
∞ ∞ 0 1
(cid:18) (cid:19)
For 0 <ℓ Z, let Ω be the collection of ϕ such that the associated Φ is absolutely integrable,
ℓ ∞ ∞
∈
and such that the Fourier transform Φ along k satisfies the bound
∞ ∞
Φ (x) (1+ x2)−ℓ
∞b ≪ | |ν
ν|∞
Q
b
For example, for ϕ to be in Ω it suffices that Φ is ℓ times continuously differentiable, with
∞ ℓ ∞
each derivative absolutely integrable. For (w ) >1, ν , the simple explicit choice of ϕ above
ν ∞
ℜ |∞
lies in Ω for every ℓ> 0.
ℓ
Theorem 2.7. (Apocryphal) Suppose r = 2, (v), ℓ sufficiently large, and ϕ Ω . The series
∞ ℓ
ℜ ∈
defining P´e(g) converges absolutely and locally uniformly in both g and v. Furthermore, up to an
Eisenstein series, the Poincar´e series is square integrable on Z G G .
A k A
\
Proof: See Appendix 1. (cid:3)
The precise Eisenstein series to be subtracted from the Poincar´e series to make the latter
square-integrable will be discussed in Section 4 (see formula 4.6). For our special choice (2.5)
of archimedeandata, boththeseconvergenceresultsapplywith (w ) > 1 for ν and (v) large.
ν
ℜ |∞ ℜ
For convenience, a monomial vector ϕ as in (2.2) described by (2.3) and (2.4) will be called
admissible, if ϕ Ω , with both (v) and ℓ sufficiently large.
∞ ℓ
∈ ℜ
3. Unwinding to Euler product
§
From now on, we shall assume r = 2. Recall the notation made in the previous section, which
in the present case reduces to: G = GL over the number field k together with the standard
2
subgroups
1 0
P = ∗ ∗ N =U = ∗ M =ZH = ∗
0 0 1 0
(cid:26)(cid:18) ∗(cid:19)(cid:27) (cid:26)(cid:18) (cid:19)(cid:27) (cid:26)(cid:18) ∗(cid:19)(cid:27)
Also, for any place ν of k, let K be the standard maximal compact subgroup. That is, for finite
ν
ν, we take K = GL (o ), at real places K =O(2), and at complex places K =U(2).
ν 2 ν ν ν
With the Poincar´e series defined by (2.1), our main goal is to unwind a corresponding global
integral to express it as an inverse Mellin transform of an Euler product. For convenience, recall
that
(3.1) P´e(g) = ϕ(γg) (g G )
A
∈
γ∈MXk\Gk
6 ADRIAN DIACONU PAUL GARRETT
where the monomial vector
ϕ = ϕ
ν
ν
O
is defined by
χ (m) for g = mk, m M and k K
0,ν ν ν
(3.2) ϕ (g) = ∈ ∈ (for ν finite)
ν
0 for g M K
(cid:26) 6∈ ν · ν
and for ν infinite, we do not entirely specify ϕ , only requiring the left equivariance
ν
(3.3) ϕ (mnk)= χ (m) ϕ (n) (for ν infinite, m M , n N and k K )
ν 0,ν ν ν ν ν
· ∈ ∈ ∈
Here, χ is the character of M given by
0,ν ν
a v a 0
(3.4) χ (m)= m= M , v C
0,ν d ν 0 d ∈ ν ∈
(cid:12) (cid:12) (cid:18) (cid:18) (cid:19) (cid:19)
(cid:12) (cid:12)
Then,χ = χ isM –invar(cid:12)ian(cid:12)t, andϕhastrivialcentralcharacterandisleftM –equivariant
0 ν 0,ν k A
by χ . Also, note that for ν infinite, our assumptions imply that
0
N
1 x
x ϕ
−→ ν 0 1
(cid:18) (cid:19)
is a function of x only.
| |
Let f and f be cuspforms on G . Eventually we will take f =f , but for now merely require
1 2 A 1 2
the following. At all ν, require (without loss of generality) that f and f have the same right
1 2
K –type, that this K –type is irreducible, and that f and f correspondto the same vector in the
ν ν 1 2
K–type (up to scalar multiples). Schur’s lemma assures that this makes sense, insofar as there are
no non-scalar automorphisms. Suppose that the representations of G generated by f and f are
A 1 2
irreducible, with the same central character. Last, require that each f is a special vector locally
i
everywhere in the representation it generates, in the following sense. Let
(3.5) f (g) = W (ξg)
i i
ξ∈ZXk\Mk
be the Fourier expansion of f , and let
i
W = W
i i,ν
ν≤∞
O
be the factorization of the Whittaker function W into local data. By [JL], we may require that
i
for all ν < the Hecke type local integrals
∞
a 0 s−1
W a 2 da
i,ν 0 1 | |ν
Z (cid:18) (cid:19)
a∈kν×
differ by at most an exponentialfunction from the local L–factorsfor the representationgenerated
by f . Eventually we will take f =f , compatible with these requirements.
i 1 2
INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 7
The integral under consideration is (with notation suppressing details)
(3.6) I(χ )= P´e(g)f (g)f¯(g)dg
0 1 2
ZZAGk\GA
For χ (and archimedean data) in the range of absolute convergence, the integral unwinds (via
0
the definition of the Poincar´e series) to
ϕ(g)f (g)f¯(g)dg
1 2
ZZAMk\GA
Using the Fourier expansion
f (g) = W (ξg)
1 1
ξ∈ZXk\Mk
this further unwinds to
(3.7) ϕ(g)W (g)f¯(g)dg
1 2
ZZA\GA
Let C be the idele class group GL (A)/GL (k), and C its dual. More explicitly, by Fujisaki’s
1 1
Lemma(seeWeil[W1], page32,Lemma 3.1.1),theideleclassgroupC isa productofacopyofR+
and a compact group C . By Pontryagin duality, C Rb C with C discrete. It is well-known
0 0 0
≈ ×
that, for any compact open subgroup U of the finite-prime part in C , the dual of C /U is
fin 0 0 fin
finitely generated with rank [k : Q] 1. The generabl Mellin tbransformband inversion are
−
(3.8) f(x) = f(y)χ(y)dyχ−1(x)dχ
b
ZCZC
1
= f(y)χ′(y) y sdyχ′−1(x) x−sds
2πi | | | |
χX′∈Cb0 ℜ(sZ)=σ ZC
for a suitable Haar measure on C.
To formulate the main result of this section, we need one more piece of notation. For ν infinite
and s C, let
∈
(s, χ , χ ) = ϕ (m n )W (m n )
ν 0,ν ν ν ν ν 1,ν ν ν
K
ZZν\MνNν ZZν\Mν
(3.9) W (m′n )χ (m′) m′ s−21 χ (m )−1 m 21−sdm′ dn dm
· 2,ν ν ν ν ν | ν|ν ν ν | ν|ν ν ν ν
and set
(3.10) (s, χ , χ) = (s, χ , χ )
∞ 0 ν 0,ν ν
K K
ν|∞
Y
Here χ = χ is the character defining the monomial vector ϕ, and χ = χ C . When
0 ν 0,ν ν ν ∈ 0
the monomial vector ϕ is admissible, the integral (3.9) defining K converges absolutely for (s)
ν
N N ℜ
sufficiently large. We are especially interested in the choice b
1+x2 −w2 for ν real, and n = 1 x N
ν
|∞ 0 1 ∈
(3.11) ϕ (n) =  (cid:18) (cid:19) (v, w C)
ν  (cid:0)1+ x(cid:1)2 −w for ν complex, and n = 1 x N ∈
ν
| | |∞ 0 1 ∈
(cid:18) (cid:19)
 (cid:0) (cid:1)


8 ADRIAN DIACONU PAUL GARRETT
The monomial vector ϕ generated by this choice is admissible for (w) > 1 and (v) sufficiently
ℜ ℜ
large. This choice will be used in Section 5 to derive an asymptotic formula for the GL integral
2
moment over the number field k. The main result of this section is
Theorem 3.12. For ϕ an admissible monomial vector as above, for suitable σ > 0,
1
I(χ ) = L(χ χ−1 1−s, f ) L(χ s, f¯) (s, χ , χ)ds
0 0 1 2 ∞ 0
2πi · |·| · |·| K
b Z
χX∈C0 ℜ(s)=σ
Let S be a finite set of places including archimedean places, all absolutely ramified primes, and all
finite bad places for f and f . Then the sum is over a set C of characters unramified outside
1 2 0,S
S, with bounded ramification at finite places, depending only upon f and f .
1 2
b
Proof: Applying (3.8) to f¯ via the identification
2
a′ 0
:a′ C C
0 1 ∈ ≈
(cid:26)(cid:18) (cid:19) (cid:27)
and using the Fourier expansion
f (g) = W (ξg)
2 2
ξ∈ZXk\Mk
the integral (3.7) is
ϕ(g)W (g) f¯(m′g)χ(m′)dm′dχ dg
1 2
b
ZZA\GA (cid:18)ZCZC (cid:19)
= ϕ(g)W (g) W (ξm′g)χ(m′)dm′dg dχ
1 2
b 
ZC ZZA\GA ZC ξ∈ZXk\Mk
 
= ϕ(g)W (g) W (m′g)χ(m′)dm′dg dχ
1 2
ZCb ZZA\GA ZJ !
whereJistheideles. Theinterchangeoforderofintegrationisjustifiedbytheabsoluteconvergence
of the outer two integrals. (The innermost integral cannot be moved outside.) This follows from
the rapid decay of cuspforms along the split torus.
For fixed f and f , the finite-prime ramification of the characters χ C is bounded, so there
1 2
∈
are onlyfinitely manybad finite primesfor all theχ which appear. In particular,all thecharacters
χ which appear are unramified outside S and with bounded ramification,bdepending only on f
1
and f , at finite placesin S. Thus, for ν S finite, there exists a compactopen subgroupU of o×
2 ∈ ν ν
such that the kernel of the νth component χ of χ contains U for all characters χ which appear.
ν ν
Since f and f generate irreducibles locally everywhere, the Whittaker functions W factor
1 2 i
W ( g : ν ) =Π W (g )
i ν ν i,ν ν
{ ≤∞}
Therefore, the inner integral over Z G and J factors over primes, and
A A
\
I(χ ) = Π ϕ (g )W (g )W (m′g )χ (m′)dm′ dg dχ
0 ν ν ν 1,ν ν 2,ν ν ν ν ν ν ν
ZCb ZZν\Gν Zkν× !
INTEGRAL MOMENTS OF AUTOMORPHIC L–FUNCTIONS 9
Let ω be the νth component of the central character ω of f . Define a character of M by
ν 2 ν
a 0 d 0 a/d 0
ω χ
0 d −→ ν 0 d ν 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Still denotethis characterby χ , without dangerof confusion. In this notation, the last expression
ν
of I(χ ) is
0
I(χ ) = Π ϕ (g )W (g )W (m′g )χ (m′)dm′ dg dχ
0 ν ν ν 1,ν ν 2,ν ν ν ν ν ν ν
ZCb ZZν\Gν ZZν\Mν !
Suppressing the index ν, the νth local integral is
ϕ(g)W (g)W (m′g)χ(m′)dm′dg
1 2
ZZ\GZZ\M
Take ν finite such that both f and f are right K –invariant. Use a ν–adic Iwasawa
1 2 ν
decompositiong =mnk withm M,n N,and k K.The Haarmeasureis d(mnk)=dmdndk
∈ ∈ ∈
with Haar measures on the factors. The integral becomes
ϕ(mn)W (mn)W (m′mn)χ(m′)dm′dndm
1 2
ZZ\MN ZZ\M
To symmetrize the integral, replace m′ by m′m−1 to obtain
ϕ(mn)W (mn)W (m′n)χ(m′)χ(m)−1dm′dndm
1 2
ZZ\MN ZZ\M
The Whittaker functions W have left N–equivariance
i
W (ng)= ψ(n)W (g) (fixed non-trivial ψ)
i i
so
W (mn)= W (mnm−1m) =ψ(mnm−1)W (m)
1 1 1
and similarly for W . Thus, letting
2
X(m,m′)= ϕ(n)ψ(mnm−1)ψ(m′nm′−1)dn
ZN
the local integral is
χ (m)W (m)W (m′)χ(m′)χ−1(m)X(m,m′)dm′dm
0 1 2
ZZ\M ZZ\M
Weclaimthatformandm′ inthesupportsoftheWhittakerfunctions,theinnerintegralX(m,m′)
is constant, independentof m, m′, and it is 1 for almost all finite primes. First, ϕ(mn) is 0, unless
n M K N, that is, unless n N K. On the other hand,
∈ · ∩ ∈ ∩
ψ(mnm−1) W (mk)=ψ(mnm−1) W (m) =W (mn)=W (m) (for n N K)
1 1 1 1
· · ∈ ∩
10 ADRIAN DIACONU PAUL GARRETT
Thus, for W (m) = 0, necessarily ψ(mnm−1) = 1. A similar discussion applies to W . So, up to
1 2
6
normalization, the inner integral is 1 for m, m′ in the supports of W and W . Then
1 2
χ (m)W (m)W (m′)χ(m′)χ−1(m)dmdm′
0 1 2
ZZ\M ZZ\M
= (χ χ−1)(m)W (m)dm χ(m′)W (m′)dm′
0 1 2
· ·
ZZ\M ZZ\M
=L (χ χ−1 1/2, f ) L (χ 1/2, f¯)
ν 0,ν · ν |·|ν 1 · ν ν|·|ν 2
i.e., the product of local factors of the standard L–functions in the theorem (up to exponential
functions at finitely many finite primes) by our assumptions on f and f .
1 2
For non-trivial right K–type σ, the argument is similar but a little more complicated. The key
point is that the inner integral over N (as above) should not depend on mk and m′k, for mk
and m′k in the support of the Whittaker functions. Changing conventions for a moment, look at
V –valued Whittaker functions, and consider any W in the νth Whittaker space for f having right
σ i
K–isotype σ. Thus,
W(gk)=σ(k) W(g) (for g G and k K)
· ∈ ∈
For ϕ(mn)=0, again n N K. Then
6 ∈ ∩
σ(k) ψ(mnm−1) W(m) =W(mnk)=σ(k) W(mn) =σ(k) σ(n) W(m)
· · · · ·
where in the last expression n comes out on the right by the right σ–equivariance of W. For m
in the support of W, σ(n) acts by the scalar ψ(mnm−1) on W(mk), for all k K. Thus, σ(n)
∈
is scalar on that copy of V . At the same time, this scalar is σ(n), so is independent of m if
σ
W(m) = 0. Thus, except for a common integral over K, the local integral falls into two pieces,
6
each yielding the local factor of the L–function. The common integral over K is a constant (from
Schur orthogonality), non-zero since the two vectors are collinear in the K–type. (cid:3)
At this point the archimedean local factors of the Euler product are not specified. The option
to vary the choices is essential for applications.
4. Spectral decomposition of Poincar´e series
§
The objective now is to spectrally decompose the Poincar´e series defined in (3.1). Throughout
this section, we assume that ϕ is admissible, in the sense given at the end of Section 2. As we
shall see, in generalP´e(g) is not square-integrable. However, choosing the archimedean part of the
monomial vector ϕ to have enough decay, and after an obvious Eisenstein series is subtracted, the
Poincar´eseriesisnotonlyin L2 butalso hassufficientdecayso thatitsintegralsagainstEisenstein
series converge absolutely, by explicit computation. In particular, if the archimedean data is
specialized to (3.11), the Poincar´e series P´e(g) has meromorphic continuation in the variables v
and w. This is achieved via spectral decomposition and meromorphic continuation of the spectral
fragments. See [DG1], [DG2] when k =Q, Q(i).
Let k be a number field, G = GL over k, and ω a unitary character of Z Z . Recall the
2 k A
\
decomposition
L2(Z G G , ω) = L2 (Z G G , ω) L2 (Z G G , ω)⊥
A k\ A cusp A k\ A ⊕ cusp A k\ A