Moments for L-functions for GLr GLr−1
A. Diaconu, P. Garrett, D. Goldfeld
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
[email protected]
http://www.math.umn.edu/˜garrett/
The moment expansion
Spectral expansion: reduction to GL2
Spectral expansion for GL2
Continuation and holomorphy at s0 = 0 for GL2
Spectral expansion for GLr
Continuation and holomorphy at s0 = 0 for GLr
Appendix: half-degenerate Eisenstein series
Appendix: Eisenstein series for GL2
Appendix: residues of degenerate Eisenstein series for P n−1,1
Appendix: degenerate Eisenstein series for P r−2,1,1
We exhibit elementary kernels Pé which produce sums of integral moments for cuspforms f on GLr by
Z
A
ZA GLr (k)\GLr ( )
Pé · |f |2 =
Z
X
F on GLr−1
Re (s)= 12
|L(s, f ⊗ F )|2
M (s) ds + (continuous part)
hF, F i
over number fields k, with certain weights M (s). Here F runs over an orthogonal basis for cuspforms
on GLr−1 . There are further continuous-spectrum terms analogous to the discrete-spectrum sum over
cuspforms. The kernel (Poincaré series) Pé admits a spectral decomposition, surprisingly consisting of only
three parts: a leading term, a sum arising from cuspforms on GL2 , and a continuous part from GL2 . That
is, no cuspforms on GL` with 2 < ` ≤ r contribute. This spectral decomposition makes possible the
meromorphic continuation of Pé in auxiliary parameters.
Moments of level-one holomorphic elliptic modular forms were treated in [Good 1983] and [Good 1986], the
latter using an idea that is a precursor of part of the present approach. Level-one waveforms over appear in
[Diaconu-Goldfeld 2006a], over (i) in [Diaconu-Goldfeld 2006b]. Arbitrary level, groundfield, and ∞-type
for GL2 are in [Diaconu-Garrett 2006] and [Diaconu-Garrett 2008].
Q
Q
We do have in mind application not only to cuspforms, but also to truncated Eisenstein series (with cuspidal
data) or wave packets of Eisenstein series, giving a non-trivial application of harmonic analysis on larger
groups GLr to L-functions attached to smaller groups, for example, on GL1 , giving high integral moments
of ζk (s).
Q
For context, we review the [Diaconu-Goldfeld 2006a] treatment of spherical waveforms f for GL2 ( ). In
that case, the sum of moments is a single term
Z
Q
A
ZA GL2 ( )\GL2 ( )
Pé(g) |f (g)|2 dg =
1
2πi
Z
L(s0 + s, f ) · L(s, f ) · Γ(s, s0 , s00 , f∞ ) ds
Re (s)= 12
where Γ(s, s0 , s00 , f∞ ) is a ratios of products of gammas, with arguments depending upon the archimedean
data attached to f . Here the Poincaré series Pé(g) = Pé(g, s0 , s00 ) has a spectral expansion
00
1−s00
X L( 1 + s0 , F )
π 2 Γ( s 2−1 )
0 00
1
2
0 +
Pé(s , s ) =
·
E
· G( 12 − itF , s0 , s00 ) · F
1+s
00
s
00
2
hF, F i
π − 2 Γ( s )
2
+
1
4πi
Z
Re (s)= 12
0
F on GL2
0
ζ(s + s) ζ(s + 1 − s)
G(1 − s, s0 , s00 ) · Es ds
ξ(2 − 2s)
1
(for Re (s0 ) 21 , Re (s00 ) 0)
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
where ξ(s) = π −s/2 Γ(s/2)ζ(s), where G is essentially a product of gamma function values [1]
0
−(s0 + s2 ) Γ(
00
00
G(s, s , s ) = π
0
0
00
0
00
s0 +1−s
) Γ( s 2+s ) Γ( s −s+s
) Γ( s +s−1+s
)
2
2
2
s00
0
Γ(s + 2 )
and F is summed over (an orthogonal basis for) spherical (at finite primes) cuspforms on GL2 with Laplacian
eigenvalues 41 + t2F , and Es is the usual spherical Eisenstein series
Es
y
0
0
1
= |y|s +
ξ(2 − 2s) 1−s
|y|
+ ...
ξ(2s)
It is not obvious, but the continuous part (the integral of Eisenstein series) cancels the pole at s0 = 1 of the
leading term, and when evaluated at s0 = 0 is [2]
Pé(g, 0, s00 ) = (holomorphic at s0 = 0) +
+
1
4πi
Z
Re (s)= 12
1
2
X
F on GL2
L( 12 , F )
· G( 12 − itF , 0, s00 ) · F
hF, F i
ζ(s) ζ(1 − s)
G(1 − s, 0, s00 ) · Es ds
ξ(2 − 2s)
In this spectral expansion, the coefficient in front of a cuspform F includes G evaluated at s0 = 0 and
s = 21 ± itF , namely
1
G( 12
00
00
− itF , 0, s ) = π
− s2
Γ( 2
−itF
2
1
) Γ( 2
+itF
2
1
s00 − −itF
2
) Γ(
2
s00
Γ( 2 )
) Γ(
1
s00 − 2 +itF
2
)
The gamma function has poles at 0, −1, −2, . . ., so this coefficient has poles at s00 = 21 ± itF , − 32 ± itF , . . ..
Over , among spherical cuspforms (or for any fixed level) these values have no accumulation point. [3] The
continuous part of the spectral side at s0 = 0 is
Q
1
4πi
Z
Re (s)= 12
ξ(s) ξ(1 − s) Γ( s
ξ(2 − 2s)
00
−s
s00 −1+s
)
2 ) Γ(
2
s00
Γ( 2 )
· Es ds
with gamma factors grouped with corresponding zeta functions, to form the completed L-functions ξ.
Thus, the evident pole of the leading term at s00 = 1 can be exploited, using the obvious continuation
to Re (s00 ) > 1/2.
Further, a subtle contour-shifting argument [4] shows that the continuous part of this spectral decomposition
has a meromorphic continuation to
with poles at ρ/2 for zeros ρ of ζ, in addition to the obvious poles
from the gamma functions.
C
[1] This from [Diaconu-Goldfeld 2006a], the result of a direct computation with the simplest useful choice of
archimedean data in the Poincaré series.
[2] This evaluation of the meromorphic continuation, from [Diaconu-Goldfeld 2006a], is not trivial. Note that the
leading term (after continuation) is reminiscent of the Kronecker limit formula. See [Asai 1977].
[3] The discreteness of the parameters t as F ranges over cuspforms of a fixed level follows from the compactness
F
of test-function operators on cuspforms, from [Gelfand-PS 1963]. In particular, at this point we do not need any sort
of Weyl’s law, and, thus, do not need trace formula computations yet.
[4] See [Diaconu-Goldfeld 2006a]. A contour shifting argument is also necessary to meromorphically continue the
continuous part of the spectral decomposition to s0 = 0 in the first place.
2
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Already for GL2 , over general groundfields k, infinitely many Hecke characters enter [5] both the spectral
decomposition of the Poincaré series and the moment expression. This naturally complicates isolation of
literal moments, and complicates analysis of poles via the spectral expansion. Suppressing constants, the
moment expansion is a sum of twists by χ’s
Z
XZ
2
L(s0 + s, f ⊗ χ) · L(1 − s, f ⊗ χ) · Mχ (s) ds
Pé · |f | =
A
ZA GL2 (k)\GL2 ( )
χ
Re (s)= 12
And, suppressing constants, the spectral expansion is
X
Pé = (∞ − part) · E1+s0 +
(∞ − part) ·
F on GL2
+
L( 21 + s0 , F )
·F
hF, F i
L(s0 + s, χ) L(s0 + 1 − s, χ)
Gχ (s) · Es,χ ds
L(2 − 2s, χ2 )
Re (s)= 12
XZ
χ
Q
In the simplest case beyond GL2 , take f a spherical cuspform [6] on GL3 over [7]
. We construct a weight
function Γ(s, s0 , s00 , f∞ , F∞ ) depending upon complex parameters s, s0 , and s00 , and upon the archimedean
data for both f and cuspforms F on GL2 , such that Γ(s, s0 , s00 , f∞ , F∞ ) has explicit asymptotic behavior,
and such that the moment expansion is
Z
Z
X
|L(s, f ⊗ F )|2
1
0 00
2
· Γ(s, 0, s00 , f∞ , F∞ ) ds
Pé(s , s ) · |f | dg =
1
2πi
hF,
F
i
Re (s)= 2
ZA GL3 (Q)\GL3 (A)
F on GL2
+
Z
Z
(k)
|L(s1 , f ⊗ E1−s2 )|2
1 1 X
(k)
· Γ(s1 , 0, s00 , f∞ , E1−s2 ,∞ ) ds1 ds2
2
1
1
4πi 2πi
|ξ(1
−
2it
)|
2
Re (s1 )= 2 Re (s2 )= 2
k∈
Z
where F runs over (an orthogonal basis for) all level-one cuspforms on GL2 , with no restriction on the
(k)
right K∞ -type, and Es is the usual level-one Eisenstein series of K∞ -type k. Here and throughout, for
Re (s) = 1/2, write 1 − s in place of s, to maintain holomorphy in complex-conjugated parameters. In this
vein, over , it is reasonable to put
Q
(k)
(k)
L(s1 , f ⊗ E s2 ) = L(s1 , f ⊗ E1−s2 ) =
L(s1 +
1
2
− s2 , f ) · L(s1 −
ζ(2 − 2s2 )
1
2
+ s2 , f )
(finite-prime parts only)
(k)
since the natural normalization of the Eisenstein series Es2 on GL2 contributes the denominator ζ(2 − 2s2 ).
Meromorphic continuation in s0 and evaluation at s0 = 0 gives the desired specialization of the moment
expansion. There is also a meromorphic continuation in the parameter s00 in the archimedean data.
Q
More generally, for a cuspform f on GLr with r ≥ 3, whether over
or over a numberfield, the moment
expansion includes an infinite sum [8] of |L(s, f ⊗ F )|2 /hF, F i over an orthogonal basis for cuspforms F
[5] See [Sarnak 1985], [Diaconu-Goldfeld 2006b] for
Q(i), and [Diaconu-Garrett 2006] for number fields.
[6] For our purposes, a cuspform generates an irreducible unitary representation of the adele group, so has a central
character, and the representation factors over primes. This factorization follows from the admissibility of irreducible
unitaries of reductive groups over archimedean and non-archimedean local fields (the former due to Harish-Chandra
in the 1950s, the latter reduced to the supercuspidal case by Harish-Chandra, and completed by J. Bernstein in 1972).
[7] The assumption of groundfield
Q
Q
achieves the minor simplification that for GL2 ( ) there is a single (family of)
Eisenstein series participating in the spectral decomposition.
Z
[8] In fact, it is non-trivial to prove (after Selberg) that there are infinitely-many spherical waveforms for SL ( ).
2
3
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
on GLr−1 , as well as integrals of products of L-functions L(s, f ⊗ F ) for F ranging over cuspforms on
GLr1 × . . . × GLr` for all partitions (r1 , . . . , r` ) of r. Correspondingly, the natural normalization of the
cuspidal-data Eisenstein series gives products of convolution L-functions L(∗, Fi ⊗ Fj ) in the denominators
of these terms, as well as factors hFi , Fi i1/2 · hFj , Fj i1/2 . [9]
Generally, the spectral expansion for GLr is an induced-up version of that for GL2 . Suppressing constants,
using groundfield
to skirt Hecke characters,
Q
0
Pé
=
X
(∞ − part) · Esr−1,1
+
0 +1
(∞ − part) ·
F on GL2
0
Z
(∞ − part) ·
+
ζ( rs +r−2
+
2
Re (s)= 12
1
2
0
− s) · ζ( rs +r−2
+
2
ζ(2 − 2s)
1
2
+ 12 , F )
L( rs +r−2
2
· E r−2,2
s0 +1
hF, F i
2 ,F
+ s)
·
E r−2,1,1
0
s +1, s− s
0 +1
s0 +1
2 , −s− 2
ds
where the Eisenstein series are normalized naively. The continuous part has a pole that cancels the obvious
pole of the leading term at s0 = 0.
Again over
Z
Q, the most-continuous part of the moment expansion for GLn is of the form
Re (s)= 21
Z
|L(s, f ⊗ E min
)|2 Mt (s) ds dt =
1
2 +it
t∈Λ
Z Z Π1≤`≤n−1 L(s + it` , f ) 2
Π1≤j<`<n ζ(1 − itj + it` ) Mt (s) ds dt
Λ
where
Λ = {t ∈
Rn−1
: t1 + . . . + tn−1 = 0}
and where M is a weight function depending upon f and F . More generally, let n − 1 = m · b. For F on
GLm , let
F∆ = F ⊗ ... ⊗ F
on GLm × . . . × GLm . Inside the moment expansion we have (recall Langlands-Shahidi)
2
Z
Z
Z Z Π1≤`≤b L(s + it` , f ⊗ F )
M ds dt
|L(s, f ⊗ EF ∆ , 12 +it )|2 MF,t (s) ds dt =
∨
Π1≤j<`≤b L(1 − itj + it` , F ⊗ F ) Re (s)= 21 Λ
Q
If we replace the cuspform f on GLn ( ) by a (truncated) minimal-parabolic Eisenstein series Eα with
α ∈ n−1 , the most-continuous part of the moment expansion contains a term
Z Z Π1≤µ≤n, 1≤`≤n−1 ζ(αµ + s + it` ) 2
ds dt
Π1≤j<`<n |ζ(1 − itj + it` )
Λ
C
Taking α = 0 ∈
Cn−1 gives
Z Z Π1≤`≤n−1 ζ(s + it` )n 2
Π1≤j<`<n ζ(1 − itj + it` ) M ds dt
Λ
R
For example, for GL3 , where Λ = {(t, −t)} ≈ ,
Z Z ζ(s + it)3 · ζ(s − it)3 2
M ds dt
ζ(1 − 2it)
R
and for GL4
Z
(s)
2
Z ζ(s + it1 )4 · ζ(s + it2 )4 · ζ(s + it3 )4
ζ(1−it1 +it2 ) ζ(1−it1 +it3 ) ζ(1−it2 +it3 ) M ds dt
Λ
[9] The identification of these denominators in the natural normalization of the Eisenstein series is part of the
refinement in [Shahidi 1978] and [Shahidi 1983] of [Langlands 1976]’s treatment of L-functions arising in constant
terms of these Eisenstein series. Here we need to keep track of constants, due to the average over F .
4
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
1. The moment expansion
Let G = GLr over a number field k. Let P be the standard maximal proper parabolic
(r − 1)-by-(r − 1)
∗
P = P r−1,1 = {
}
0
1-by-1
Let
1
U = { r−1
0
∗
}
1
(r − 1)-by-(r − 1) 0
H={
}
0
1
and
N = {upper triangular unipotent elements in H}
= (unipotent radical of standard minimal parabolic in H)
Let Z be the center of G. Let Kv be the standard maximal compact in the kv -valued points Gv of G. Thus,
for v < ∞, Kv = GLr (ov ). For v ≈ , take Kv = Or ( ). For v ≈ take Kv = U (r).
R
R
C
The standard choice of non-degenerate character on Nk Uk \NA UA is
ψ(n · u) = ψ0 (n12 + n23 + . . . + nr−2,r−1 ) · ψ0 (ur−1,r )
where ψ0 is a fixed non-trivial character on
f has a Fourier expansion [12] along N U
Z
X
f (g) =
A/k. A cuspform [10]
Wf (ξg)
where
Wf (g) =
ξ∈Nk \Hk
Nk Uk \NA UA
ψ(nu) f (nug) dn du
The (Whittaker) function Wf (g) factors over primes. [13]
[1.1] Poincaré series For s0 ∈ C, let
ϕ=
O
ϕv
v
where for v finite
ϕv (g) =

 s0
(det A)/dr−1 v

0
(for g = mk with m =
A
0
0
d
in Zv Hv and k ∈ Kv )
(otherwise)
[10] A cuspform f satisfies the Gelfand-Fomin-Graev condition [11]
R
Nk \NA f (ng) dn = 0 for all unipotent radicals
N of (proper, rational) parabolics, generates an irreducible representation locally everywhere (hence, has a central
character ω), and is in L2 (ZA Gk \GA , ω).
[12] Fourier-Whittaker expansions for GL(n) with n > 2 are due to [Shalika 1974] and [Piatetski-Shapiro 1975],
independently.
[13] Uniqueness of Whittaker models, by [Shalika 1974] at archimedean places, [Gelfand-Kazhdan 1975] at non-
archimedean, implies the factorization over primes. The local factors are ambiguous up to constants, naturally.
For cuspforms f at primes v where f is spherical, the spherical Whittaker function Wv◦ with the same local data as
f , normalized by Wv◦ (1) = 1, is the standard choice. However, even at good primes, for natural normalization in
construction of Eisenstein series presents us most naturally with a local Whittaker function which is an image under
the intertwining operator from principal series to the Whittaker model. This contributes an extra factor which is
a product of L-functions, as studied at length in [Langlands 1971], [Shahidi 1978], [Shahidi 1983], and elsehwere.
Luckily, our subsequent convolution L-functions will lack archimedean factors, so avoid the most acute concern about
choices of data at archimedean places, thus skirting issues addressed in [Jacquet-Shalika 1990], [Cogdell-PS 2003].
5
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
and for v archimedean require right Kv -invariance and left equivariance
0
det A s
A 0
(for g ∈ Gv , for m =
∈ Zv Hv )
ϕv (mg) = r−1 · ϕv (g)
0 d
d
v
Thus, for v|∞, the further data determining ϕv consists of its values on Uv . The simplest useful choice is
[14]
ϕv
1r−1
0
x
1

x1
.
(where x =  .. , and s00 ∈
xr−1

00
= (1 + |x1 |2 + . . . + |xr−1 |2 )−s
/2
and where the norm |x1 |2 + . . . + |xr−1 |2 is normalized to be invariant under Kv .
ZA Hk -invariant. We attach to ϕ a Poincaré series
X
Pé(g) =
ϕ(γg)
[15]
C)
Thus, ϕ is left
γ∈Zk Hk \Gk
[1.2] Two obvious unwindings Integrate the norm-squared |f |2 of a cuspform f against Pé. The typical
first unwinding is
Z
ZA Gk \GA
Pé(g) |f (g)|2 dg =
Z
ZA Hk \GA
ϕ(g) |f (g)|2 dg
Next, express f in its Fourier-Whittaker expansion, and unwind further:
Z
Z
X
ϕ(g) Wf (g) f (g) dg
ϕ(g)
Wf (ξg) f (g) dg =
ZA Hk \GA
ZA Nk \GA
ξ∈Nk \Hk
[1.3] Iwasawa decomposition, simplification of integral Suppose for simplicity that f is right KA invariant, so we can use an Iwasawa decomposition G = (HZ)U K (everywhere locally) to rewrite the whole
integral as
Z
ϕ(hu) Wf (hu) f (hu) dh du
Nk \HA ×UA
Use a spectral decomposition [16]
Z
F =
hF, ηi · η dη
[1.4] Spectral decomposition on GLr−1
inexplicitly
for F ∈ L2 (Hk \HA ),
(η)
[14] Over
Q
Q
and (i), in effect this is the choice of archimedean data in [Good 1983], [Diaconu-Goldfeld 2006a],
[Diaconu-Goldfeld 2006b].
[15] Thus, for this purpose, to be K -invariant at real places v, we use the standard norm. At complex places we also
v
use the standard norm, not the norm compatible with the product formula. That is, for this purpose, for v ≈
√
norm is not |z|v = z · z, but is |z| = z · z.
[16] Let H 1 = {h ∈ H
A
C the
: | det h| = 1}. Then Hk \HA = Hk \H 1 × (0, +∞) is the relevant decomposition. The
most obvious continuous part coming from (0, +∞) will eventually give integrals over vertical lines. Still L2 (Hk \H 1 )
has both continuous and discrete parts in its decomposition. Since some of the necessary functions η for such a
spectral decomposition are not literally in L2 (both Eisenstein series and ordinary exponentials in Fourier transforms
and Fourier inversion), the inner integral is not at all symmetric. Further, that integral, (as well as the outer)
only make literal sense for F in a suitable (dense) subspace (e.g., pseudo-Eisenstein series and Schwartz functions
in the corresponding circumstances), and the mappings indicated must be defined by isometric extension. See
[Langlands 1976] and [Moeglin-Waldspurger 1995].
6
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
where each η generates an irreducible representation of HA .
[1.5] Expand f (hu) Since f is left Hk -invariant, it decomposes along Hk \HA as
Z
Z
f (hu) =
η(h)
(η)
Hk \HA
Unwind the Fourier-Whittaker expansion of f
Z
Z
f (hu) =
η(h)
Hk \HA
(η)
Z
η(m) f (mu) dm dη
X
η(m)
W f (ξmu) dm dη
ξ∈Nk \Hk
Z
=
η(h)
Nk \HA
(η)
η(m) W f (mu) dm dη
Then the whole integral is
Z
ZA Gk \GA
Z
Pé(g) |f (g)|2 dg
Z
=
(η)
Z
Nk \HA ×UA
ϕ(hu) η(h) Wf (hu)
Nk \HA
W f (mu) η(m) dm dh du dη
[1.6] Decoupling at finite primes The part of the integrand that depends upon u ∈ U is
Z
Z
UA
ϕ(hu) Wf (hu) W f (mu) du = ϕ(h) Wf (h) W f (m) ·
UA
ϕ(u) ψ(huh−1 ) ψ(mum−1 ) du
The latter integrand visibly factors over primes.
[1.6.1] Lemma: Let v be a finite prime. For h, m ∈ Hv such that Wf,v (h) 6= 0 and W f,v (m) 6= 0,
Z
ϕv (h) ψv (huh−1 ) ψ v (mum−1 ) du =
Z
1 du
Uv ∩Kv
Uv
Proof: At a finite place v, ϕv (u) 6= 0 if and only if u ∈ Uv ∩ Kv , and for such u
ψv (huh−1 ) · Wf,v (h) = Wf,v (huh−1 · h) = Wf,v (hu) = Wf,v (h) · 1
by the right Kv -invariance. Thus, for Wf,v (h) 6= 0, ψv (huh−1 ) = 1, and similarly for ψv (mum−1 ). Thus,
the finite-prime part of the integral over Uv is just the integral of 1 over Uv ∩ Kv , as indicated.
///
[1.7] Archimedean kernel
The archimedean part of the integral does not necessarily decouple. Thus,
with subscripts ∞ denoting the infinite-adele part of various objects, for h, m ∈ H∞ , define
Z
K(h, m) =
ϕ∞ (u) ψ∞ (huh−1 ) ψ ∞ (mum−1 ) du
U∞
The whole integral is
Z
Z
Pé(g) |f (g)|2 dg =
ZA Gk \GA
(η)
Z
Nk \HA
Z
Nk \HA
K(h, m) ϕ(h) Wf (h) η(h) W f (m) η(m) dm dh dη
7
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
[1.8] Fourier expansion of η Normalize the volume of Nk \NA to 1. Thus, for a left Nk -invariant function
F on HA
Z
Z
Nk \HA
F (h) dh =
Z
NA \HA
Nk \NA
F (nh) dn dh
Using the left NA -equivariance of W by ψ, and the left NA -invariance of ϕ,
Z
Z
ϕ(nh) η(nh) Wf (nh) dn = ϕ(h) Wf (h)
ψ(n) η(nh) dn = ϕ(h) Wf (h) Wη (h)
Nk \NA
Nk \NA
where
Z
Wη (h) =
Nk \NA
ψ(n) η(nh) dn
(The integral is not against ψ(n), but ψ(n).) That is, the integral over Nk \HA is equal to an integral against
(up to an alteration of the character) the Whittaker function Wη of η, which factors [17] over primes. The
whole integral is
Z
Z Z
Z
Pé(g) |f (g)|2 dg =
K(h, m) ϕ(h) Wf (h) Wη (h) W f (m) W η (m) dm dh dη
ZA Gk \GA
(η)
NA \HA
NA \HA
And the η th part is a product of two Euler products. It is evident that for f right Kfin -invariant only
right (Kfin ∩ Hfin )-invariant η’s will appear, due to the decoupling. However, at archimedean places v right
Kv -invariance of f does not allows us to restrict our attention to right (Kv ∩ Hv )-invariant η.
[1.9] Appearance of the parameter s In fact, as usual,
A
Hk \HA ≈ GLr−1 (k)\GLr−1 ( ) ≈
where
R+ × Hk \H 1
R+ is positive real numbers, and
A
a 0
H1 = {
: a ∈ GLr−1 ( ), | det a| = 1}
0 1
The quotient Hk \H 1 has finite volume. Thus, the spectral decomposition uses functions
a 0
η
= | det a|s · F (a) with F ∈ L2 (Hk \H 1 ), s ∈ i
0 1
R
The real part of the parameter s will necessarily be shifted in the subsequent discussion. Thus, the functions
η above are of the form | det |s ⊗ F , and the Whittaker function Wη of η = | det |s ⊗ F is
a
Wη
= | det a|s · WF (a)
1
where WF is the Whittaker function of F , normalized here by
Z
ψ(n) F (ng) dn
WF (g) =
Nk \NA
[17] When η is either a cuspform (in a strong sense) or is an Eisenstein series attached to suitable (e.g., cuspidal,
in a strong sense) data, this Whittaker function Wη factors over primes. The usual normalization for the spherical
Whittaker function Wv◦ at finite places v is Wv◦ (1) = 1. This is the normalization we take for cuspforms, but this
P
is incompatible with a standard normalization of Eisenstein series E(g) = γ ϕ(γg) by requiring ϕ(1) = 1. This
produces a normalizing denominator which (as in simple cases of Langlands-Shahidi) is a product of convolution
L-functions.
8
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
where N is the unipotent radical of the standard minimal parabolic in GLr−1 .
[1.10] Non-archimedean local factors
In terms of s and F , the non-archimedean local factors are [18]
Z
Lv (s + s0 + 21 , f ⊗ F )
a
s+s0
| det a|
Wf,v
WF,v (a) da =
1
hF, F i1/2
Nv \Hv
(for Re (s + s0 ) 0)
The second Euler product is the complex conjugate of this, but lacking the shift by s0 , namely, the complex
conjugate of
0
Z
Lv (s + 12 , f ⊗ F )
a
0 s
(for Re (s + s0 ) 0 and Re (s) 0)
| det a | Wf,v
WF,v (a0 ) da0 =
1
hF, F i1/2
Nv \Hv
When η = | det |s ⊗ F is not cuspidal, but, instead, is an Eisenstein series with cuspidal data, it still does
generate an irreducible representation of GA . At a place v where η generates a spherical representation, the
Euler product expansion of degree r · (r − 1) falls apart into smaller factors, and has a denominator arising
from the (natural) normalization of the cuspidal-data Eisenstein series entering. Discussion of these terms
and their normalizations is postponed.
[1.11] Replace s by 1 − s on Re(s) = 1/2
The global integrals for the L-functions L(s0 +s+ 21 , f ⊗F ) and L(s+ 21 , f ⊗F ) only converge for Re (s0 +s) 0
and Re (s) 0, so we will need to meromorphically continue. To this end, it is most convenient for the
whole integral to be holomorphic in s, rather than having both s and s appear.
To these ends, first absorb the 1/2 into s by replacing s by s + 12 , so we have
L(s0 + s, f ⊗ F ) · L(s, f ⊗ F )
and want to eventually move to the line Re (s) = 1/2. To avoid the anti-holomorphy in the second factor,
since s = 1 − s on the line Re (s) = 1/2, we can rewrite this as
L(s0 + s, f ⊗ F ) · L(1 − s, f ⊗ F )
(for Re (1 − s) 0 and Re (s0 + s) 0)
[1.12] The vertical integral(s) Keep in mind that we have absorbed a 1/2 into s, and have replaced s
by 1 − s. The archimedean part of the whole integral is the function Γϕ∞ (s, s0 , f, F ) defined by
Γϕ∞ (s, s0 , f, F ) =
Z
Z
s0 +s− 12
K(h, m) | det a|
N∞ \H∞
×
W f,∞
0
| det a |
1
2 −s
Wf,∞
a
1
N∞ \H∞
a0
1
0
0
W F,∞ (a ) da da
(with h =
WF,∞ (a)
a
1
and m =
a0
1
)
[18] The L-functions L(s, f ⊗ F ) attached to cuspforms f and F by these zeta integrals are Euler products with local
factor equal to that of the L-functions L(s, πf × πF ) attached to the corresponding (irreducible cuspidal automorphic)
representations at all finite primes at which f and F are spherical. At other finite places the local factors of L(s, f ⊗F )
may be polynomial (in qv−s and qvs ) multiples of the corresponding local factor of L(s, πf × πF ). At archimedean
places, Lv (s, f ⊗ F )/Lv (s, πf × πF ) is entire, etc. Last, but not least, there are global constant factors sometimes
denoted ρf (1) which for newforms f for GL(2) are the ratios of the first Fourier coefficient of f to the Petersson norm
squared of f . See [Hoffstein-Lockhart 1994], [Bernstein-Reznikoff 1999], and [Sarnak 1985], [Sarnak 1992]. Here the
latter global normalizing constant is accommodated by a division by hF, F i1/2 , to make F/hF, F i1/2 run through an
orthonormal basis when F runs through an orthogonal basis.
9
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
0
since ϕ∞ (h) = | det a|s . Note that this depends only upon the archimedean data attached to f and F .
Thus, so far, the whole is
Z
Pé(g) |f (g)|2 dg
ZA Gk \GA
Z
X
=
F on GL(r−1)
Re (s)= 12
Γϕ∞ (s, s0 , f, F )
L(s0 + s, f ⊗ F ) L(1 − s, f ⊗ F )
dt
hF, F i
(with Re (s0 ) 0)
+ (continuous part)
Again, we want to meromorphically continue to s0 = 0.
[1.12.1] Remark: With or without detailed knowledge of the residual part of L2 (meaning that consisting
of square-integrable iterated residues of cuspidal-data Eisenstein series), automorphic forms in the residual
spectrum not admitting Whittaker models do not enter in this expansion.
2. Spectral expansion: reduction to GL2
The Poincaré series admits a spectral expansion in terms of Eisenstein series, cuspforms, and L-functions,
preparing for its meromorphic continuation. This section reduces the general spectral expansion to the case
r = 2.
[2.1] Poisson summation
Form the Poincaré series in two stages to allow application of Poisson
summation, namely
X
Pé(g) =
X
X
Zk Hk Uk \Gk
β∈Uk
ϕ(γg) =
Zk Hk \Gk
ϕ(βγg) =
X
X
Zk Hk Uk \Gk
ψ∈(Uk \UA )b
ϕ
bγg (ψ)
where
Z
ϕ
bg (ψ) =
UA
(for g ∈ GA )
ψ(u) ϕ(ug) du
[2.2] The leading term The inner summand for ψ = 1 gives a vector from which an extremely degenerate
Eisenstein series [19] for the (r − 1, 1) parabolic P r−1,1 = ZHU is formed by the outer sum. That is,
Z
g→
UA
ϕ(ug) du
r−1,1
is left equivariant by a character on PA
, and is left invariant by Pkr−1,1 , namely,
Z
Z
Z
−1
ϕ(upg) du =
ϕ(p · p up · g) du = δP r−1,1 (m) ·
ϕ(m · u · g) du
UA
UA
0 Z
det A s +1
= r−1 d
UA
UA
ϕ(ug) du
(where p =
A
0
∗
, A ∈ GLr−1 , d ∈ GL1 )
d
[19] These degenerate (spherical) Eisenstein series E r−1,1 are readily understood via Poisson summation, imitating
s
Riemann et alia. There is a simple pole at s = 1, with constant residue, and no other poles in Re (s) ≥ 1/r. That
first pole will be cancelled (when s0 goes to 0) by the continuous part of the spectral decomposition.
10
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
The normalization [20] is explicated by setting g = 1:
Z
Z
Z
Z
ϕ(u) du =
ϕ∞ ·
ϕfin =
UA
U∞
Ufin
Z
ϕ∞ · meas (Ufin ∩ Kfin ) =
U∞
ϕ∞
U∞
A natural normalization would have been that this value be 1, so the Eisenstein series here implicitly includes
the archimedean integral and finite-prime measure constant as factors:
Z
Z
X
ϕ(uγg) du
ϕ∞ · Esr−1,1
0 +1 (g) =
U∞
UA
γ∈Pkr−1,1 \Gk
As advance warning: the pole at s0 = 0 of this leading term will be cancelled by a contribution from the
continuous part of the spectral decomposition, below.
[2.3] Main terms: appearance of Q from GL2 The group Hk is transitive on non-trivial characters
on Uk \UA . As usual, for fixed non-trivial character ψ0 on k\A, let
ψ ξ (u) = ψ0 (ξ · ur−1,r )
(for ξ ∈ k × )
The spectral expansion of Pé with the obvious leading term removed, is


X
X
X

ϕ
bαγg (ψ ξ )
ξ∈k×
γ∈Pkr−1,1 \Gk α∈Pkr−2,1 \Hk
where P r−2,1 is the parabolic subgroup of H ≈ GLr−1 . Let



1r−2
1r−2
∗
}
U 00 = {
U 0 = {
1
1
Let

1
∗ }
1

1r−2
Θ = {

∗
∗
∗ }
∗
Then the expansion of the Poincaré series with leading term removed is


Z
X
X Z
ξ

ψ (u00 )
ϕ(u0 u00 γg) du0 du00 
γ∈Pkr−2,1,1 \Gk
0
UA
00
UA
ξ∈k×

=

X Z
X
X
γ∈Pkr−2,2 \Gk
α∈P 1,1 \Θk

ξ
00
UA
ξ∈k×
ψ (u00 )
Z
0
UA
ϕ(u0 u00 αγg) du0 du00 
Letting
Z
ϕ(g)
e
=
0
UA
ϕ(u0 g) du0
[20] Recall an integration-theory trick. Given a unimodular topological group G, use a topological group decomposition
G = P K with K compact open as follows. To integrate a right K-invariant function f on G, with the total mass of
R
R
K normalized to 1, we have G f = P f with a left Haar measure on P normalized so that the measure (in P ) of
K ∩ P is 1.
11
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
the expansion becomes
X
X Z
X
ξ
A
U 00
γ∈Pkr−2,2 \Gk α∈P 1,1 \Θk ξ∈k×
ψ (u00 ) ϕ(u
e 00 αγg) du00
We claim the equivariance

0
0
0
ϕ(pg)
e
= |det A|s +1 · |a|s · |d|−(r−1)s −(r−2) · ϕ(g)
e
A
(for p = 
∗
a
∗
d

 ∈ GA , with A ∈ GLr−2 )
A
This is verified by changing variables in the defining integral: let x ∈ r−1 and compute


 
 


1r−2
x
A b c
A b c + xd
A b c
1r−2
A−1 xd


=
=


1
a
a
a
1
d
d
d
1
1
0
0
0
Thus, | det A|s · |a|s · |d|−(r−1)s comes out of the definition of ϕ, and another | det A| · |d|2−r from the
change-of-measure in the change of variables replacing x by Ax/d in the integral defining ϕ
e from ϕ. Note
that
rs0 +(r−2)
(r−2)
0
0
0
a
2
|a|s · |d|−(r−1)s −(r−2) = | det
|− 2 ·(s +1) · |a/d|
d
Thus, letting
Φ(g) =
X Z
X
α∈Pk1,1 \Θk ξ∈k×
we can write
Pé(g) −
X
γ∈Pkr−1,1 \Gk
ξ
A
U 00
ψ (u00 ) ϕ(u
e 00 αg) du00
Z
UA
ϕ(uγg) du =
X
Φ(γg)
γ∈Pkr−2,2 \Gk
This is not an Eisenstein series for P r−2,2 in the strictest sense. An expression in terms of genuine Eisenstein
series is helpful in understanding meromorphic continuations.
Define a GL2 kernel ϕ(2) for a Poincaré series as follows. We require right invariance by the maximal compact
subgroups locally everywhere, and left equivariance
a
ϕ(2) (
· D) = |a/d|s · ϕ(2) (D)
d
(2)
Then the the archimedean data ϕ∞ is completely specified by


1r−2
1 x
ϕ(2)
= ϕ
e
1 x
∞
1
1
(with ϕ
e as above)
Then put
ϕ∗ (s, ϕ,
e D) =
X Z
ξ∈k×
UA
ξ
ψ (u) ϕ(2) (s, uD) du
(with U now the unipotent radical of P 1,1 in GL2 )
The corresponding GL2 Poincaré series with leading term removed is
X
Q(s, D) =
ϕ∗ (s, αD)
α∈Pk1,1 \GL2 (k)
12
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Thus, for
g =
A
∗
D
A
A
(with A ∈ GLr−2 ( ) and D ∈ GL2 ( ))
the inner integral
Z
g→
00
UA
ψ(u00 ) ϕ(u
e 00 g) du00
is expressible in terms of the kernel ϕ∗ for Q, namely,
0
X Z
(r−2)
rs + r − 2
ξ 00
00
00
s0 +1
− 2 ·(s0 +1)
∗
ψ (u ) ϕ(u
e g) du = | det A|
· | det D|
·ϕ
,D
00
2
UA
×
ξ∈k
Thus,
X
X Z
α∈Pk1,1 \Θk ξ∈k×
00
UA
ξ
00
00
00
ψ (u ) ϕ(u
e αg) du
s0 +1
= | det A|
−
· | det D|
(r−2)
·(s0 +1)
2
·Q
rs0 + r − 2
,D
2
Thus, to obtain a (not necessarily L2 ) spectral decomposition of the Poincaré series Pé (with main term
removed) we first determine the (L2 ) spectral decomposition of Q for r = 2, and then form P r−2,2 Eisenstein
series from the spectral fragments.
3. Spectral expansion for GL2
The spectral expansion of the Poincaré series for GL2 is easy, because the Poincaré series (with leading term
removed) is essentially a kernel for the linear map f → L(s0 + 21 , f ).
[3.1] Recall the set-up Take r = 2 and G = GL2 , and ϕ =
N
v
ϕv with ϕv constructed above. Namely,
at a finite prime v
a
ϕv (
∗
d
0
· θ) = |a/d|sv
(for θ ∈ Kv )
and at infinite v we have at least the left equivariance
0
a ∗
ϕv (
· g) = |a/d|sv · ϕv (g)
d
Let
∗
H={
1
}
Form a Pk -invariant function
Φ(g) =
U ={
X Z
ξ∈k×
Uk \UA
1
∗
}
1
∗
P ={
∗
}
∗
ψ 0 (ξ · u1,2 ) ϕ(ug) du
and the Poincaré series
Q(g) =
X
Φ(γg)
Pk \Gk
For GL2 , the L2 (Gk \G1 ) decomposition [21] has a cuspidal part, a continuous part, and a residual part.
Since Q is right KA -invariant and has trivial central character, all the spectral components will have these
attributes.
[21] We grant that suitable parameter choices put Pé (with a leading term, an Eisenstein series, removed) in L2 ,
allowing such a decomposition, which persists by analytic continuation even when Q is not in L2 . See estimates over
and (i) for GL2 , but with arbitrary level, in [Zhang 2005] and [Zhang 2006]. The general ideas of the spectral
decomposition for GL2 , at least over , have been understood for a long time. See [Selberg 1956], [Roelcke 1956],
[Godement 1966a], [Godement 1966b].
Q
Q
Q
13
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
[3.2] Cuspidal part Computation of hPé, F i for F a cuspform can be done directly. This pairing can also
be computed as for the pairing against Eisenstein series below, but the cuspform case allows an illuminating
direct computation. Let F be a spherical cuspform [22] with trivial central character, and Fourier-Whittaker
expansion
X
R
F (g) =
WF (γg)
(where WF (g) = Uk \UA ψ(u) F (ug) du)
γ∈Hk
Unwinding the Poincaré series gives [23]
Z
hPé, F i =
Z
ZA Gk \GA
Pé · F =
Z
ZA Gk \GA
Q·F =
ZA Hk \GA
Since F is a cuspform on GL2 , its Fourier-Whittaker expansion is F (g) =
Fourier expansion of F into the integral unwinds further to
Z
Y Z
YZ
ϕ(g) W F (g) dg =
ϕv (g) W F,v (g) dg ·
ZA \GA
v<∞
Zv \Gv
v|∞
P
ϕ·F
γ∈Hk
WF (γg). Substituting the
ϕv (g) W F,v (g) dg
Zv \Gv
At finite primes v, the right Kv -invariance implies (via an Iwasawa decomposition) that the v th integral is
Z
Z
ϕv (hu) W F,v (hu) dh du =
Hv ×Uv
Z
ϕv (h) W F,v (h) dh =
Hv
kv×
0
|a|sv
W F,v
a
1
da
since ϕv is supported on Hv Kv . We can adjust F by a scalar (without changing F/hF, F i1/2 so that the
non-archimedean Whittaker functions for the cuspform F give non-archimedean integrals over Hv which are
exactly the local L-factors Lv (s0 + 21 , F ). The archimedean integrals are not the usual gamma factors, but
still do depend only upon the local data of F . Thus,
Z
hPé, F i =
ϕ∞ · WF ,∞ · L(s0 + 21 , F )
(F a suitable cuspform)
Z∞ \G∞
[3.3] Poisson summation To isolate the leading term of the Poincaré series, as well as to determine the
continuous part of the spectral decomposition, recall the rewritten form of the Poincaré series via Poisson
summation:
X
X
X
X
X
Pé(g) =
ϕ(γg) =
ϕ(βγg) =
ϕ
bγg (ψ)
γ∈Zk Hk \Gk
γ∈Zk Hk Uk \Gk β∈Uk
where
Z
ϕ
bg (ψ) =
UA
ψ(u) ϕ(ug) du
γ∈Zk Hk Uk \Gk ψ∈(Uk \UA )b
(for g ∈ GA )
[3.4] The leading term The inner summand for trivial ψ gives a vector from which an Eisenstein series
is formed by the outer sum. That is,
Z
g→
UA
ϕ(ug) du
[22] As earlier, for us a cuspform generates an irreducible unitary representation of G .
A
[23] In this context, the pairing h, i can be taken to be an integral over Z G \G , since the integrand has trivial
k
A
central character.
14
A
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
is left equivariant by a character on PA , and is left invariant by Pk , namely,
Z
Z
Z
−1
ϕ(upg) du =
ϕ(p · p up · g) du = δP (m) ·
ϕ(m · u · g) du
UA
UA
s0 +1
= |a/d|
UA
Z
UA
ϕ(ug) du
(where p =
The normalization is understood by setting g = 1:
Z
Z
Z
ϕ(u) du =
ϕ∞ ·
UA
U∞
a
0
∗
)
d
Z
ϕfin =
Ufin
ϕ∞
U∞
The natural normalization would have been that this value be 1, so this Eisenstein series includes the
archimedean integral and finite-prime measure constant as factors:
Z
(leading term of Pé) =
ϕ∞ · Es0 +1
U∞
[3.5] Continuous part First, let
X
Q = Pé − (leading term) = Pé − (factor) · Es0 +1 =
X
ϕ
bγg (ψ)
γ∈Zk Hk Uk \Gk ψ6=1
be the Poincaré series with its leading term removed. Let
J1/k×)
κ = meas (
The residue of the zeta function of k at z = 1 is
Resz=1 ζk (z) =
κ
|Dk |1/2
(where Dk is the discriminant of k)
The continuous part of the Poincaré series is [24]
Z
1 X
hQ, Es,χ i · Es,χ ds
4πiκ χ Re (s)= 12
Computation of the pairing of Q against Eisenstein series is best approached indirectly, unlike the pairings
against cuspforms. [25] For GL2 , the simple transitivity of Hk on non-trivial characters allows Q to be
rewritten, with any fixed choice of non-trivial ψ, as
Q(g) =
X
X
ϕ
bβγg (ψ) =
γ∈Zk Hk Uk \Gk β∈Hk
X
ϕ
bγg (ψ)
γ∈Zk Uk \Gk
[24] This is the usual spectral inversion formula, as in many sources. For example, see [Godement 1966a] for ground
Q
Q
field . The volume constant κ necessary in the general case is inconspicuous when k = , because in that case its
value is 1. The 4πi is really 2 · 2πi, where the extra factor of 2 reflects the fact that we indicate an integral over the
entire vertical line, rather than half.
[25] To evaluate the integral of the Poinaré series (with leading term removed) against an Eisenstein series, the
computation here is more efficient than the approach which would unwind the Eisenstein series and then compute a
Mellin transform of the constant term of the Poincaré series.
15
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Integrate Q against an Eisenstein series [26] Es,χ with χ an unramified Hecke character χ.
Z
hQ, Es,χ i =
ZA Gk \GA
Q(g) E s,χ (g) dg
In the latter integral, unwind Q to obtain
Z
Z
ϕ
bg (ψ) E s,χ (g) dg =
ZA Uk \GA
Z
=
Z
=
ZA Uk \GA
Z
ZA UA \GA
ZA \GA
Z
ψ(u) ϕ(ug) E s,χ (g) du dg
Z
E
UA
UA
ψ(u) ϕ(ug) W s,χ (g) du dg =
E
Z
ZA UA \GA
E
(where Ws,χ
(g) =
ϕ(g) W s,χ (g) dg
E
UA
ϕ(ug) W s,χ (ug) du dg
R
Uk \UA
ψ(u) Es,χ (ug) du)
The Whittaker function of the Eisenstein series does factor over primes, into local factors depending only
upon the local data at v
O
E
E
Ws,χ
=
Ws,χ,v
v
Thus,
hQ, Es,χ i =
YZ
v
E
ϕv (g) W s,χ,v (g) dg
Zv \Gv
At finite v, using an Iwasawa decomposition and the vanishing of ϕv off Hv Uv , as in the integration against
cuspforms, the local factor is
Z
0
E
a
|a|sv W s,χ,v
da
1
kv×
However, for Eisenstein series, the natural normalization of the Whittaker functions differs from that used
for cuspforms, instead presenting the local Whittaker functions as images under intertwining operators.
Specifically, define the normalized spherical vector for data s, χv
ϕ◦v (pθ)
=
|a/d|sv
· χv (a/d)
(for p =
a
∗
d
in Pv and θ in Kv )
The corresponding (spherical) local Whittaker function for Eisenstein series is the integral [27]
E
Ws,χ,v
(g) =
Z
ψ(u) ϕ◦v (w◦ ug) du
Uv
where w◦ is the longest Weyl element. The Mellin transform of the Eisenstein-series normalization WvE
compares to the Mellin transform of the usual normalization Wv◦ of the Whittaker function (with the same
data s, χv ) as follows. Let o∗v be the local inverse different at v, and let dv ∈ kv× be such that
(o∗v )−1 = dv · ov
[26] As with the pairing against cuspforms, because the integrand has trivial central character, we can take the pairing
integral to be an integral over ZA Gk \GA .
[27] This integral only converges nicely for Re (s) 0, but admits a meromorphic continuation in s by various
means. For example, the relatively elementary algebraic form of Bernstein’s continuation principle applies, since the
dimension of the space of intertwining operators from the principal series to the Whittaker space is one-dimensional.
16
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Let d be the idele with v th component dv at finite places v and component 1 at archimedean places. Let
ν(x) = |x|. Then for finite v the v th local integral is [28]
Z
0
0
Lv (s0 + 21 , ν s χv ) · Lv (s0 + 12 , ν 1−s χv )
E
a 0
· |dv |−(s +1−s) χ(dv )
da = |d|1/2 ·
|a|sv W s,χ,v
2
0 1
Lv (2s, χv )
kv×
= |d|1/2 ·
0
Lv (s0 + s, χv ) · Lv (s0 + 1 − s, χv )
· |dv |−(s +1−s) χ(dv )
2
Lv (2s, χv )
Then
Z
hQ, Es,χ i =
ϕ∞ ·
E
W s,χ,∞
!
· |d|1/2 ·
Z∞ \G∞
0
L(s0 + s, χ) · L(s0 + 1 − s, χ)
· |d|−(s +1−s) χ(d)
2
L(2s, χ )
[3.6] Maintaining holomorphy The integral of hQ, Es,χ i · Es along the vertical line Re (s) = 1/2 is
anti-holomorphic in the first s and holomorphic in the second s, inconvenient when we want to move the line
of integration. To maintain holomorphy, as earlier, since s = 1 − s on Re (s) = 12 , we rewrite
Z
hQ, Es,χ i =
ZA Gk \GA
Pé · E1−s,χ
The latter expression is holomorphic in s and has the expected Euler product expansion for Re (1 − s) 0.
With this adjustment,
Z
0
L(s0 + 1 − s, χ) · L(s0 + s, χ)
E
· |d|1/2 ·
· |d|−(s +s) χ(d)
hQ, Es,χ i =
ϕ∞ · W1−s,χ,∞
2
L(2 − 2s, χ )
Z∞ \G∞
Thus, with the contour at Re (s) = Re (1 − s) = 1/2 and with Re (s0 ) 0,
Z
XZ
Pé =
ϕ∞ · Es0 +1 +
ϕ∞ · WF ,∞ · L(s0 + 21 , F ) ·
U∞
+
F
Z∞ \G∞
F
hF, F i
Z
L(s0 + 1 − s, χ) · L(s0 + s, χ)
X χ(d) Z
0
E
· |d|−(s +s−1/2) · Es,χ ds
ϕ∞ · W1−s,χ,∞
2
4πiκ Re (s)= 12 Z∞ \G∞
L(2 − 2s, χ )
χ
4. Continuation and holomorphy at s0 = 0 for GL2
The meromorphic continuation and non-obvious cancellation [29] in the spectral decomposition of the
Poincaré series for GL2 are reviewed in detail here, over an arbitrary number field, proving that the spectral
expression for the Poincaré series is holomorphic at s0 = 0.
To obtain the desired expression for integral moments, we must set s0 = 0, which on the spectral side
requires that the meromorphically continued Poincaré series be holomorphic at s0 = 0. The (obvious)
leading Eisenstein series Es0 +1 (without the archimedean factor) has a pole at s0 + 1 = 1, that is, at s0 = 0,
but we will see now that this is cancelled by part of the continuous spectrum of the Poincaré series. This
cancellation is independent of the choice of archimedean data ϕ∞ .
[28] This elementary comparison is reproduced in an appendix.
[29] This cancellation argument was given in detail over
Q in [Diaconu-Goldfeld 2006a].
17
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
[4.1] The leading term
From above, the leading term Eisenstein series, including the extra leading
archimedean factor (and measure constant from finite primes) is
Z
X Z
ϕ∞ · Es0 +1 (g) =
ϕ(uγg) du
U∞
γ∈Pk \Gk
UA
[4.2] Two residues of the continuous part
The only fragment of the continuous part of the spectral
decomposition of Q with poles to the right of the line Re (s) = 1/2 is that with trivial Hecke character
χ. Replacing (as above) s by 1 − s to maintain holomorphy in the integral [30] for hQ, Es i, the term with
relevant poles is [31]
Z
1
hQ, Es i · Es ds
4πiκ Re (s)= 12
Z
Z
ζ (s0 + 1 − s) · ζ (s0 + s)
0
1
k
k
E
=
ϕ∞ · W1−s,χ,∞
|d|−(s +s−1/2) Es ds
4πiκ Re (s)= 12 Z∞ \G∞
ζk (2 − 2s)
Aiming to analytically continue to s0 = 0, in the integral first take Re (s0 ) = 1/2 + ε, and move the contour
from Re (s) = 1/2 to Re (s) = 1/2−2ε. This picks up the residue of the integrand due to the pole of ζk (s0 +s)
at s0 + s = 1, that is, at s = 1 − s0 . This contributes
!
Z
ζk (2s0 ) · Resz=1 ζk (z) −1/2
1
E
· 2πi ·
ϕ∞ · Ws0 ,∞ ·
|d|
· E1−s0
4πiκ
ζk (2s0 )
Z∞ \G∞
!
!
Z
Z
1
1
= ·
ϕ∞ · WsE0 ,∞ · |d|1/2 · |d|−1/2 · E1−s0 = ·
ϕ∞ · WsE0 ,∞ · E1−s0
2
2
Z∞ \G∞
Z∞ \G∞
The vertical integral on Re (s) = 1/2 − 2ε is holomorphic in s0 in (at least) the strip
1
2
− ε ≤ Re (s0 ) ≤
1
2
+ε
(while Re (s) =
1
2
− 2ε)
Move s0 to Re (s0 ) = 1/2 − ε, and then move the vertical integral from the contour σ = 1/2 − 2ε back to the
contour σ = 1/2. This picks up (−1) times the residue at the pole of ζk (s0 + 1 − s) at 1, that is, at s = s0 ,
with another sign due to the sign on s inside this zeta function. Thus, we pick up the residue, namely
!
Z
0
1
1
Resz=1 ζk (z) · ζk (2s0 )
· 2πi ·
ϕ∞ · W1−s0 ,∞ ·
· |d|−2s + 2 · E1−(1−s0 )
0)
4πiκ
ζ
(2
−
2s
k
Z∞ \G∞
!
Z
0
1
ζk (2s0 )
ϕ∞ · W1−s0 ,∞ ·
· |d|−2s +1 · Es0
= ·
0
2
ζk (2 − 2s )
Z∞ \G∞
[4.3] Use of functional equation of Es The latter expression can be simplified by using the functional
equation of Es0 . Specifically, letting ζ∞ be the gamma factor for ζk ,
0
ζ∞ (2s0 ) · ζk (2s0 ) · Es0 = |d|2s −1 · ζ∞ (2 − 2s0 ) · ζk (2 − 2s0 ) · E1−s0
[30] The pairing h, i can be taken to be an integration over Z G \G , since the central character is trivial.
k
[31] Again, κ is the natural volume of
1/2
κ/Dk
J /k
1
A
A
×
, and is invisible in the case of k =
is the residue of ζk at 1, again invisible for groundfield .
Q
18
Q because in that case κ = 1. And
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
In the situation at hand, this gives
0
ζk (2s0 )
ζ∞ (2 − 2s0 )
· Es0 = |d|2s −1 ·
· E1−s0
0
ζk (2 − 2s )
ζ∞ (2s0 )
Thus, the powers of |d| cancel, and the second residue can be rewritten as
!
Z
1
·
2
ϕ∞ · W1−s0 ,∞
·
Z∞ \G∞
ζ∞ (2 − 2s0 )
· E1−s0
ζ∞ (2s0 )
These two contributions from the continuous part are
!
ζ∞ (2 − 2s0 )
ϕ∞ · Ws0 ,∞ +
· W1−s0 ,∞
· E1−s0
ζ∞ (2s0 )
Z∞ \G∞
Z
1
2
·
[4.4] Use of functional equation of archimedean Whittaker functions
The archimedean-place
(and finite-prime) Whittaker functions are normalized by presenting the Whittaker functions by the usual
intertwining integral (and analytically continuing), as follows. Let
ηs (umθ) = |a/d|sv
(for u ∈ U , m =
The normalization of the Whittaker function is
Z
E
ψ(u) ηs (w◦ ug) du
Ws,v (g) =
a
d
, θ ∈ Kv )
(for Re (s) 0, fixed non-trivial ψ)
Uv
Further, for v archimedean
E
Ws,v
a
1
Z
=
kv
s
Z
a
1−s
ψ 0 (x) ι
dx = |a|v
aa + xxι kv
v
by replacing x by ax, where ι is complex conjugation for v ≈
usual computation shows that
R
Ws,E
a
1
=
|a|1/2
π −s Γ(s)
∞
Z
1
1
e−π(t+ t )|a| ts− 2
0
ψ 0 (ax)
1
dx
|1 + xxι |sv
C and is the identity map for v ≈ R.
The
dt
1
=
× (invariant under s ↔ 1 − s)
t
ζR (2s)
and, similarly, [32]
Ws,EC
a
1
=
|a|
(2π)−2s Γ(2s)
Z
∞
1
e−2π(t+ t )|a| t2s−1
0
dt
1
=
× (invariant under s ↔ 1 − s)
t
ζC (2s)
Thus, in both cases,
ζv (2 − 2s)
E
· W1−s,v
ζv (2s)
[32] In the displayed identity for v ≈
|x| =
√
xxι .
a
1
E
= Ws,v
a
1
(for v archimedean)
C, an unadorned absolute value is not |x|C = xxι , but, rather, the standard
The appropriate measure is double the usual.
19
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
and
E
W1−s
=
ζ∞ (2s)
· WsE
ζ∞ (2 − 2s)
Thus, in fact, the second of the two residues from the continuous part is identical to the first. Thus, these
two residues combine to give
!
Z
ϕ∞ · Ws0 ,∞ · E1−s0
Z∞ \G∞
Thus, to prove that the leading term’s pole at s0 = 0 is cancelled by the poles of these residues at s0 = 0, we
must show that
!
Z
Z
· E1+s0 +
ϕ∞
· E1−s0 = (holomorphic at s0 = 0)
ϕ∞ · Ws0 ,∞
Z∞ \G∞
U∞
[4.5] Invocation of Fourier inversion Thus, to prove cancellation of poles at s0 = 0 it suffices to prove
a local fact, namely, that
Z
ϕv =
Z
ϕv · WsE0 ,v (for all archimedean v)
s0 =0
Zv \Gv
Uv
E
The Whittaker function Ws,v
can be presented in terms of a Fourier transform
E
Ws,v
a
1
b s,v (a)
= |a|1−s · Ψ
(where Ψ(x) = Ψs (x) = |1 + xxι |−s
v , v archimedean)
Let
Φ(x) = ϕv
1
x
1
Using the right Kv -invariance and an Iwasawa decomposition, and noting that Φ and Ψ are even,
Z
ϕv ·
WsE0 ,v
Z
Z
0
0
b
ψ(ax) dx da
|a|s Φ(x) · |a|1−s Ψ(a)
=
kv×
Zv \Gv
kv
Z
Z
=
kv×
Φ(a) Ψ(a) |a| da
b
b
Φ(a)
Ψ(a)
|a| da =
kv×
by Fourier inversion, since |a|da is an additive Haar measure. Writing the latter expression out, it is
Z
ϕv
1
a
1
kv
1
|a| da
|1 + aaι |sv0
·
The equality
Z
ϕv ·
WsE0 ,v
Zv \Gv
Z
=
kv×
ϕv
1
a
1
·
1
|a| da
|1 + aaι |sv0
holds at first only for Re (s0 ) 0, but then extends by analytic continuation for ϕv sufficiently integrable
on Nv . Thus, we can evaluate the latter integral at s0 = 0, obtaining
Z
ϕv ·
Zv \Gv
E
W0,v
Z
=
kv×
20
ϕv
1
a
1
|a| da
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
At archimedean places v, the normalization |a| da for an additive Haar measure, in terms of a multiplicative
Haar measure da, is correct. This proves that, indeed,
Z
Zv \Gv
Z
ϕv · WsE0 ,v s0 =0
=
ϕv
(for all archimedean v)
Uv
which yields the desired cancellation.
5. Spectral expansion for GLr
The spectral decomposition of the Poincaré series for r = 2 yields that for r > 2 by inducing. For r > 2,
nothing remains after all the non-L2 terms are removed. The non-L2 terms are induced from the genuinely
L2 spectral expansion of the Poincaré series on GL2 .
Before carrying out the spectral expansion for r = 2, we had found that
Pé(g) =
Z
ϕ∞ · Esr+1,1
0 +1 (g)
U∞
where
A
Φ
∗
D
0
= | det A|s +1 · | det D|− (r−2)·
X
+
Φ(γg)
γ∈Pkr−2,2 \Gk
s0 +1
2
· Q(
with Q from GL2 , and
rs0 + r − 2
, ϕ,
e D)
2
Z
ϕ(g)
e
=
A
(with A ∈ GLr−2 and D ∈ GL2 )
ϕ(u0 g) du0
Uk0 \U 0
Thus, formation of Pé (with its leading term removed) amounts to forming an Eisenstein series from Φ, with
analytical properties explicated by expressing Φ as a superposition of vectors generating irreducibles.
[5.1] Decomposition into irreducibles
From the decomposition of Q on GL2 for Re (s0 ) 0
s0 +1
s0 +1
rs0 + r − 2
A ∗
, ϕ,
e D)
Φ
= | det A|2· 2 · | det D|− (r−2)· 2 · Q(
0 D
2
!
X Z
0
0
rs0 + r − 2 1
F
2· s 2+1
− (r−2)· s 2+1
= | det A|
· | det D|
ϕ
e∞ · WF ,∞ · L(
+ 2, F) ·
2
hF,
Fi
P GL2 (k∞ )
F
+ | det A|2·
Z
Re (s)= 12
P GL2 (k∞ )
ϕ
e∞ ·
E
W1−s,χ,∞
· | det D|− (r−2)·
0
!
Z
s0 +1
2
·
s0 +1
2
X χ(d)
4πiκ
χ
0
L( rs +r−2
+ 1 − s, χ) · L( rs +r−2
+ s, χ)
rs0 +r−2
2
2
·|d|−( 2 +s−1/2) ·Es,χ (D) ds
2
L(2 − 2s, χ )
[5.2] Least continuous part of Poincaré series
Let
A
Φ s0 +1 ,F (
0
2
∗
D
· θ) = | det A|2·
s0 +1
2
· | det D|− (r−2)·
21
s0 +1
2
F (D)
(for θ ∈ KA )
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
and define a half-degenerate Eisenstein series [33]
X
(g) =
E r−2,2
s0 +1
2
,F
γ∈Pkr−2,2 \Gk
Φ s0 +1 ,F (γg)
2
Then the most-cuspidal (or least continuous) part of the Poincaré series is
!
0
X Z
L( rs +r−2
+ 12 , F )
2
ϕ
e · WF ,∞ ·
· E r−2,2
(cuspforms F on GL2 )
s0 +1
hF,
F
i
2 ,F
P GL2 (k∞ )
F
The summands of this expression have relatively well understood meromorphic continuations. As discussed
r−2,2
in an appendix, the half-degenerate Eisenstein series Es,F
has no poles in Re (s) ≥ 1/2. With s = (s0 +1)/2
0
this assures absence of poles in Re (s ) ≥ 0.
[5.3] Continuous part of the Poincare series
The Eisenstein series integral part of Q on GL2 gives
degenerate Eisenstein series attached to the (r − 2, 1, 1)-parabolic in GLr . This arises from a similar
consideration as for GL2 cuspforms, but for GL2 Eisenstein series, as follows.
Let Es,χ be the usual Eisenstein series for GL2 , and let
s0 +1
s0 +1
A ∗
0
· θ) = | det A|2· 2 · | det D|− (r−2)· 2 Es,χ (D)
Φ s +1 ,Es,χ (
0 D
2
(for θ ∈ KA )
and define an Eisenstein series
E r−2,2
s0 +1
2
For given s ∈
,Es,χ
X
(g) =
γ∈Pkr−2,2 \Gk
C, an easy variant of Godement’s criterion [34]
Φ s0 +1 ,Es,χ (γg)
2
proves convergence for sufficiently large Re (s0 ).
Then, ignoring the issue of interchange of sums and integrals, the Poincaré series has most continuous part
!
Z
X χ(d) Z
E
ϕ
e∞ · W1−s,χ,∞
4πiκ Re (s)= 12
P GL2 (k∞ )
χ
0
0
0
L( rs +r−2
+ 1 − s, χ) · L( rs +r−2
+ s, χ)
−( rs +r−2
+s−1/2)
2
2
2
ds
×
·
|d|
· E r−2,2
s0 +1
L(2 − 2s, χ2 )
2 ,Es,χ
That is, it is the analytically continued Es on the line Re (s) = 12 that enters. However, as usual, for
Re (s) 0 and Re (s0 ) 0 this iterated formation of Eisenstein series is equal to a single-step Eisenstein
series. The equality persists after analytic continuation.
Thus, let

A ∗
Φs1 ,s2 ,s3 ,χ ( 0 m2
0
0

∗
∗  · θ)
m3
= | det A|s1 · |m2 |s2 χ(m2 ) · |m3 |s3 χ(m3 )
(for θ ∈ KA and A ∈ GLr−2 )
[33] Visibly, this Eisenstein series is a P r−2,2 Eisenstein series degenerate on the (upper-left) Levi component GL
r−2
for r > 3, while cuspidal on the (lower-right) Levi component GL2 . Basic analytic properties of such Eisenstein series
are accessible by relatively elementary methods, as noted in the appendix on half-degenerate Eisenstein series.
[34] A neo-classical version of Godement’s criterion is in [Borel 1966].
22
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
and
X
Esr−2,1,1
=
1 ,s2 ,s3 ,χ
Φs1 ,s2 ,s3 ,χ (γg)
γ∈Pkr−2,1,1 \Gk
Then, ignoring the issue of interchange of sums and integrals, the Poincaré series has most continuous part
!
Z
X χ(d) Z
E
ϕ
e∞ · W1−s,χ,∞
4πiκ Re (s)= 12
P GL2 (k∞ )
χ
0
0
L( rs +r−2
+ 1 − s, χ) · L( rs +r−2
+ s, χ)
rs0 +r−2
2
2
×
· |d|−( 2 +s−1/2) · E r−2,1,1
ds
0
0
0
2
2· s 2+1 , s−(r−2)· s 2+1 , −s−(r−2)· s 2+1 , χ
L(2 − 2s, χ )
[5.4] Comment It is remarkable that there are no further terms in the spectral expansion of Pé, beyond
the main term, the cuspidal GL2 part induced up to GLr , and the continuous GL2 part induced up to GLr .
6. Continuation and cancellation for GLr
Take r > 2. In meromorphically continuing the Poincaré series to s0 = 0 the leading term
Z
ϕ∞ · Esr+1,1
0 +1 (g)
U∞
seems to create an obstruction, having a pole at s0 = 0 (coming from the pole of the Eisenstein series). As
with GL2 , it is not obvious, but this pole is cancelled by poles coming from the trivial-χ integral of Eisenstein
series
!
Z
Z
1
rs0 + r − 2
E
ϕ
e∞ (
) · W1−s,∞
4πiκ Re (s)= 12
2
P GL2 (k∞ )
0
0
ζ( rs +r−2
+ 1 − s) · ζ( rs +r−2
+ s)
rs0 +r−2
2
2
×
· |d|−( 2 +s−1/2) · E r−2,1,1
ds
0
0
0
2· s 2+1 , s−(r−2)· s 2+1 , −s−(r−2)· s 2+1
ζ(2 − 2s)
Unlike the GL2 case, for r > 2 the relevant poles of this integral are due to the Eisenstein series, not to the
zeta functions.
[6.1] Residue of leading term First, recall from the appendix that the leading term has residue at s0 = 0
given explicitly by the residue of the Eisenstein series, namely [35]
Z
κ · |d|r/2
Z
Z
r+1,1
r+1,1
0
0
ϕ∞ ·
Ress =0
ϕ∞ · Es0 +1 (g) =
ϕ∞ · Ress =0 Es0 +1 (g) =
r · ξ(r)
U∞
U∞
U∞
[6.2] The initial situation
The spectral decomposition initially holds for Re (s0 ) 0 while the integral
[35] As usual, the function ξ(s) is the zeta function of the underlying number field with gamma factors added, but
not attempting to compensate for the conductor or epsilon factor (the latter being absent here). Thus, the functional
equation is
1
ξ(s) = |d|1/2 · |d|−(1−s) · ξ(1 − s) = |d|s− 2 · ξ(1 − s)
where d is a finite idele generating the local different everywhere. The first |d|1/2 is the product of standard measures
of local integers at ramified primes, and |d|−(1−s) arises because the Fourier transform of the characteristic function
of the local integers is the characteristic function of the local inverse different.
23
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
is along Re (s) = 1/2. For brevity, let the three complex arguments to the Eisenstein series be denoted
s1 = 2 ·
s0 + 1
2
s2 = s − (r − 2)
s0 + 1
2
s3 = −s − (r − 2)
s0 + 1
2
For Re (s0 ) 0, certainly Re (s1 − s2 ) > r − 1 and Re (s1 − s3 ) > r, although Re (s2 − s3 ) = 1, since
Re (s) = 1/2. Thus, for convergence, the Eisenstein series Esr−2,1,1
must be understood as formed by inducing
1 ,s2 ,s3
r−2,2
1,1
up along P
the meromorphic continuation of Es2 ,s3 . Godement’s criterion [36] yields convergence.
[6.3] Move s0 to the left edge
Aiming to move s0 toward 0, first move s0 as far to the left as possible
without violating either of the two conditions

 Re (s1 − s2 ) > r − 1 (first condition)

Re (s1 − s3 ) > r
(second condition)
Let σ = Re (s) and σ 0 = Re (s0 ). Expanded, the first condition is
σ0 + 1
σ0 + 1
2·
− σ − (r − 2) ·
>r−1
2
2
which simplifies to
r·
and then
σ0 + 1
>σ+r−1
2
σ0 + 1
σ
1
> +1−
2
r
r
and finally
2(1 − σ)
r
Similarly, the second condition becomes
σ0 > 1 −
(first condition, simplified)
2σ
r
σ0 > 1 −
(second condition, simplified)
For σ = Re (s) = 1/2, which is where we start, these two conditions are equivalent, namely
σ0 > 1 −
1
r
(either condition, at σ = 21 )
Thus, move s0 close to the left edge of the region defined by these conditions: for small ε > 0 move σ 0 = Re (s0 )
to σ 0 = 1 − 1r + ε.
[6.4] First contour shift and residue Now solve the two inequalities for σ in terms of σ0 . These are

σ
r
2
<
· σ0 −
r−2
2
> − 2r · σ 0 +
r
2
With σ 0 = 1 − 1r + ε, move σ to the left, from
first) to be violated:
 1 r+1
 2 − 2 · ε < 12 + 2r · ε =
1
2

1
2
−
r+1
2
σ
· ε 6>
1
2
−
r
2
·ε
(first condition, rearranged)
(second condition, rearranged)
to
r
2
1
2
−
· (1 −
r+1
2 ε.
1
r
= − 2r · (1 −
This causes the second condition (but not the
+ ε) −
1
r
r−2
2
+ ε) +
r
2
[36] A neo-classical version of Godement’s criterion appears in [Borel 1966].
24
(first condition holds)
(second condition fails)
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Thus, we pick up 2πi times the residue of the pole of the Eisenstein series at
s=−
r
r 0 r
· s + = · (1 − s0 )
2
2
2
The corresponding residue of the P r−2,1,1 Eisenstein series is (from the appendix)
r−1
ξ(s2 − s3 − 1)
κ · |d| 2
·
· Esr−1,1
1 −1,s2 −1
ξ(r − 1)
ξ(s2 − s3 )
r−1
=
κ · |d| 2
ξ(r(1 − s0 ) − 1)
·
· Esr−1,1
0 ,−(r−1)s0
ξ(r − 1)
ξ(r(1 − s0 ))
With s = r(1 − s0 )/2, the zeta functions in the integral of Eisenstein series become
0
0
ζ( rs +r−2
+ 1 − 2r (1 − s0 )) · ζ( rs +r−2
+ 2r (1 − s0 ))
ζ(rs0 ) · ζ(r − 1)
2
2
=
ζ(2 − r(1 − s0 ))
ζ(2 − r + rs0 )
That is, apart from the 2πi and archimedean factor, we have picked up
r−1
(2πi) ·
ζ(rs0 ) · ζ(r − 1) κ · |d| 2
ξ(r(1 − s0 ) − 1)
·
·
· Esr−1,1
0
0
ζ(2 − r + rs )
ξ(r − 1)
ξ(r(1 − s0 ))
From the appendix,
κ
r
0
The only other finite-prime zeta which misbehaves at s = 0 is the ζ(2 − r + rs0 ) in the denominator. From
the functional equation
1
ξ(s)
|d|s− 2 · ξ(1 − s)
ζ(s) =
=
ζ∞ (s)
ζ∞ (s)
=−
Re s0 =0 ξ(rs0 ) · Esr−1,1
0
we have
0
0
1
1
1
ζ∞ (2 − r + rs0 )
ζ∞ (2 − r + rs0 )
2 −(2−r+rs ) ·
2 −2+r−rs ·
=
|d|
=
|d|
ζ(2 − r + rs0 )
ξ(1 − (2 − r + rs0 ))
ξ(r − 1 − rs0 )
Thus, the iterated residue at s0 = 0 of this residue (without 2πi and without the archimedean factor) is
Ress0 =0
ζ∞ (2 − r + rs0 )
ζ∞ (rs0 )
= − Ress0 =0
·
3r
ζ(r − 1) · κ · |d| 2 −2 · ξ(r − 1) κ · −
ξ(r − 1) · ξ(r − 1) · ξ(r)
r
ζ∞ (2 − r + rs0 )
ζ∞ (rs0 )
·
3r
κ
κ · |d| 2 −2
·
ζ∞ (r − 1) · ξ(r)
r
[6.5] The archimedean factor Take
r
· (1 − s0 )
2
and use the functional equation for the GL2 archimedean Whittaker function, namely
s=
E
W1−s
=
Then the archimedean integral is
Z
P GL2 (k∞ )
ϕ
e∞ (
ζ∞ (2s)
· WsE
ζ∞ (2 − 2s)
rs0 + r − 2
ζ∞ (r · (1 − s0 ))
)·
· W rE(1−s0 ),∞
2
2
ζ∞ (2 − r · (1 − s0 ))
25
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Continuing this discussion shows that the continuous-spectrum part of the spectral decomposition cancels
the pole of the singular term, giving the meromorphic continuation to s0 = 0.
7. Appendix: half-degenerate Eisenstein series
A
A
Take q > 1, and let f be a cuspform on GLq ( ), in the strong sense that f is in L2 (GLq (k)\GLq ( )1 ), and
f meets the Gelfand-Fomin-Graev conditions
Z
f (ng) dn = 0
(for almost all g)
Nk \NA
and f generates an irreducible representation of GLq (kν ) locally at all places ν of k. For a Schwartz function
Φ on q×r and Hecke character χ, let
Z
q
ϕ(g) = ϕχ,f,Φ (g) = χ(det g)
f (h−1 ) χ(det h)r Φ(h · [0q×(r−q) 1q ] · g) dh
A
A
GLq ( )
This function ϕ has the same central character as f . It is left invariant by the adele points of the unipotent
radical
1r−q ∗
N ={
}
(unipotent radical of P = P r−q,q )
1r
The function ϕ is left invariant under the k-rational points Mk of the standard Levi component of P ,
a
M ={
: a ∈ GLr−q , d ∈ GLr }
d
To understand the normalization, observe that
Z
r
ξ(χ , f, Φ(0, ∗)) = ϕ(1) =
A
f (h−1 ) χ(det h)r Φ(h · [0q×(r−q) 1q ]) dh
GLq ( )
is a zeta integral as in [Godement-Jacquet 1972] for the standard L-function attached to the cuspform f (or
perhaps a contragredient). Thus, the Eisenstein series formed from ϕ includes this zeta integral as a factor,
so write
X
P
ξ(χr , f, Φ(0, ∗)) · Eχ,f,Φ
(g) =
ϕ(γ g)
(convergent for Re (χ) >> 0)
γ∈Pk \GLr (k)
Now prove the meromorphic continuation via Poisson summation:
P
ξ(χr , f, Φ(0, ∗)) · Eχ,f,Φ
(g)
q
= χ(det g)
X
γ∈Pk \GLr (k)
= χ(det g)q
Z
A
GLq (k)\GLq ( )
Z
A
X
f (h) χ(det h)−r
Φ(h−1 · [0 α] · g) dh
α∈GLq (k)
f (h) χ(det h)−r
GLq (k)\GLq ( )
X
y∈kq×r ,
Φ(h−1 · y · g) dh
full rank
The Gelfand-Fomin-Graev condition on f will compensate for the otherwise-irksome full-rank constraint.
Anticipating that we can drop the rank condition suggests that we define
X
ΘΦ (h, g) =
Φ(h−1 · y · g)
y∈kq×r
26
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
As in [Godement-Jacquet 1972], the non-full-rank terms integrate to 0: [37]
[7.0.1] Proposition: For f a cuspform, less-than-full-rank terms integrate to 0, that is,
Z
A
X
f (h) χ(det h)−r
GLq (k)\GLq ( )
y∈kq×r ,
Φ(h−1 · y · g) dh = 0
rank <q
Proof: Since this is asserted for arbitrary Schwartz functions Φ, we can take g = 1. By linear algebra, given
y0 ∈ k q×r of rank `, there is α ∈ GLq (k) such that
y`×r
α · y0 =
(with `-by-r block y`×r of rank `)
0(q−`)×r
Thus, without loss of generality fix y0 of the latter shape. Let Y be the orbit of y0 under left multiplication
by the rational points of the parabolic
`-by-`
∗
`,q−`
P
={
} ⊂ GLq
0
(q − `)-by-(q − `)
This is some set of matrices of the same shape as y0 . Then the subsum over GLq (k) · y0 is
Z
Z
X
X
−r
−1
f (h) χ(det h)
Φ(h · y) dh =
f (h) χ(det h)−r
Φ(h−1 · y) dh
A
GLq (k)\GLq ( )
A
Pk`,q−` \GLq ( )
y∈GLq (k)·y0
y∈Y
Let N and M be the unipotent radical and standard Levi component of P `,q−` ,
1`
∗
`-by-`
0
N=
M=
0 1q−`
0
(q − `)-by-(q − `)
Then the integral can be rewritten
as an iterated integral
Z
X
f (h) χ(det h)−r
Φ(h−1 · y) dh
A
Nk Mk \GLq ( )
Z
=
A
NA Mk \GLq ( ) y∈Y
Z
=
y∈Y
XZ
X
A
NA Mk \GLq ( ) y∈Y
Nk \NA
f (nh) χ(det nh)−r Φ((nh)−1 · y) dn dh
−r
χ(det h)
−1
Φ(h
!
Z
· y)
Nk \NA
f (nh) dn
dh
[37] There are issues of convergence. First, for Re (χ) sufficiently large, the integral
χ(det g)q
X
Z
A
f (h) χ(det h)−r Θ(h, g) dh
γ∈Pk \GLr (k) GLq (k)\GLq ( )
J
is absolutely convergent.
Also, we have the integrals analogous to integrals over k× \ + .
That is, let
GL+
=
{h
∈
GL
(
)
:
|
det
h|
≥ 1}. Then, for arbitrary χ, using the fact that cuspforms f are of rapid decay
q
q
in Siegel sets,
Z
X
X
χ(det g)q
f (h) χ(det h)−r
Φ(h−1 · y · g) dh
A
+
γ∈Pk \GLr (k) GLq (k)\GLq
y∈kq×r , full rank
is absolutely convergent.
27
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
since all fragments but f (nh) in the integrand are left invariant by NA . But the inner integral of f (nh) is
0, by the Gelfand-Fomin-Graev condition, so the whole is 0.
///
Let ι denote the transpose-inverse involution(s). Poisson summation gives
X
ΘΦ (h, g) =
Φ(h−1 · y · g)
y∈kq×r
= | det(h−1 )ι |r | det g ι |q
X
b ι )−1 · y · g ι ) = | det(h−1 )ι |r | det g ι |q Θ b (hι , g ι )
Φ((h
Φ
y∈kq×r
As with ΘΦ , the not-full-rank summands in ΘΦ
b integrate to 0 against cuspforms. Thus, letting
A
A
GL−
q = {h ∈ GLq ( ) : | det h| ≤ 1}
GL+
q = {h ∈ GLq ( ) : | det h| ≥ 1}
P
ξ(χr , f, Φ(0, ∗)) · Eχ,f,Φ
(g) = χ(det g)q
Z
f (h) χ(det h)−r ΘΦ (h, g) dh
A
GLq (k)\GLq ( )
q
Z
−r
= χ(det g)
f (h) χ(det h)
GLq (k)\GL+
q
χ(det g)q
=
q
ΘΦ (h, g) dh + χ(det g)
Z
GLq (k)\GL+
q
+ χ(det g)q
Z
GLq (k)\GL−
q
Z
GLq (k)\GL−
q
f (h) χ(det h)−r ΘΦ (h, g) dh
f (h) χ(det h)−r ΘΦ (h, g) dh
ι ι
| det(h−1 )ι |r | det g ι |q f (h) χ(det h)−r ΘΦ
b (h , g ) dh
By replacing h by hι in the second integral, convert it to an integral over GLq (k)\GL+
q , and the whole is
Z
P
f (h) χ(det h)−r ΘΦ (h, g) dh
ξ(χr , f, Φ(0, ∗)) · Eχ,f,Φ
(g) = χ(det g)q
GLq (k)\GL+
q
+ νχ−1 (det g ι )q
Z
GLq (k)\GL+
q
ι
f (hι ) νχ−1 (det hι )−r ΘΦ
b (h, g ) dh
Since f ◦ ι is a cuspform, the second integral is entire in χ. Thus, we have proven
P
ξ(χr , f, Φ(0, ∗)) · Eχ,f,Φ
is entire
[7.1] Remark Except for the extreme case q = r − 1, these Eisenstein series are degenerate, so occur only
as (iterated) residues of cuspidal-data Eisenstein series. Assessing poles of residues is less effective in the
present special circumstances than the above argument.
8. Appendix: Eisenstein series for GL2
We compute Mellin transforms of the Whittaker functions attached to Eisenstein series for GL2 (with trivial
central character, spherical, over a number field k). The natural normalization of Eisenstein series adds a
further local L-factor to the Mellin transform (as well as a measure constant at ramified primes), by contrast
to the usual Mellin transform of the usual normalization Wv◦ of the spherical Whittaker function. And at
the end we give a precise function equation.
[8.1] The standard Eisenstein series Let G = GL2 and
P ={
∗
0
∗
}
∗
N ={
1
0
28
∗
}
1
∗
M ={
0
0
}
1
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
and let Z be the center. An absolutely unramified Hecke character can be decomposed as χ(α) · |α|s
corresponding to the usual decomposition
J/k× ≈ J1/k× × (0, +∞)
The standard normalization of the spherical function (in the tensor product of principal series) is ϕ =
with
a ∗
ϕv (
· θ) = |a/d|sv · χv (a/d)
(θ in the standard maximal compact of GL2 (kv ))
0 1
N
v
ϕv
Thus, ϕv (1) = 1 for all v. The standard Eisenstein series attached to data s, χ is
X
E(g) = Es,χ (g) =
ϕ(γg)
Pk \Gk
A
[8.2] The Eisenstein Whittaker functions Let ψ be the standard non-trivial character on /k. Let ov
be the local integers at a finite place v, let p be the rational prime lying under v. The dual lattice to ov with
respect to trace is
o∗v = {α ∈ kv : tr kQvp (αβ) ∈ p : for all β ∈ ov }
Z
This dual lattice o∗v is exactly the kernel of the standard ψ. Let
W E (g) =
Z
Nk \NA
ψ(n) E(ng) dn
where the measure is normalized so that the total measure of Nk \NA is 1. As usual, this unwinds to
Z
Z
Nk \NA
ψ(n) ϕ(ng) dn +
NA
ψ(n) ϕ(w◦ ng) dn
As ψ is non-trivial, the first of the two integrals is 0. The second integral has an Euler product, with v th
factor
Z
WvE (g) =
ψ(n) ϕv (w◦ ng) dn
Nv
[8.3] Archimedean local computations
At archimedean places, for present application we do not need the Mellin transform of the Whittaker
functions. Instead, we need to present the archimedean-place Eisenstein-Whittaker functions in a form
that emphasizes local functional equations. As above, let
a ∗
ϕv (
0 1
· θ) = |a/d|sv · χv (a/d)
(θ in the standard maximal compact of GL2 (kv ))
and
WvE (g)
Z
=
ψ(n) ϕv (w◦ ng) dn
Nv
By the right Kv invariance, Zv -invariance, and left Nv equivariance, to understand WvE it suffices to take g
of the form
y
g=
1
29
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
For v complex and x ∈ kv , let xσ = x, and for v real let σ be the identity map. For archimedean v, the
standard Iwasawa decomposition is
σ
√
0 1
1 x
y
−y/r xσ /r
x /r −y σ /r
·
·
=
·
(where r = xxσ + yy σ )
1 0
1
1
0
r
y/r
x/r
Thus,
ϕv
0
1
1
0
1
x
1
y
1
= ϕv
−y/r
0
∗
r
s
s y
y
= 2 = σ
σ
r v
xx + yy v
The Fourier-Whittaker transform is
s
s
s
Z
Z
Z
y
y
1
dx = |y|v
dx = |y|1−s
dx
ψ(x) σ
ψ(xy)
ψ(xy)
v
σ
σ
σ
σ
xx + yy v
yy (xx + 1) v
xx + 1 v
kv
kv
kv
For v real, the following computation is familiar, but the normalizations are important, so we recall it. Using
the identity
Z ∞
dt
1
−s
e−tu ts
(for u > 0 and Re (s) > 0)
u =
Γ(s) 0
t
rewrite the Fourier-Whittaker transform as
√
√ Z Z
Z Z
2
2
1
|y|1−s ∞
dt
π
|y|1−s π ∞
dt
ψ(xy) e−t(1+x ) ts dx
ψ(xy √ ) e−t e−πx ts− 2 dx
=
Γ(s) 0
t
Γ(s)
t
t
R
0
R
=
√ Z
|y|1−s π ∞ −t −πy2 · π s− 1 dt
t t
2
e e
Γ(s)
t
0
(Fourier transform after replacing x by x ·
Replacing t by t · πy turns the whole into
√ Z
Z ∞
√
y
1
1 dt
1
|y|1−s π ∞ −π(t+ 1 )|y| s− 1 dt
t
e
t 2
= −s
e−π(t+ t )|y| ts− 2
(πy)s− 2 ·
Γ(s)
t
π Γ(s) 0
t
0
√
√π )
t
(v real)
For v complex, the computation is less familiar, but follows the same course. Keep in mind that the complex
norm fitting into the product formula is the square of the usual:
|y|C = |yy|R = |y|2
The character is
C
ψC (xy) = ψR (tr C/R (xy))
R
C R
In particular, for x, y ∈ , the trace tr C/R (xy) is double the standard -valued pairing on
≈ 2 . This
2 appears explicitly in the Gaussian after a Fourier transform. And the measure on
is twice the usual
Lebesgue measure on 2 . We write this factor explicitly from the outset. The same trick with the gamma
function rewrites the Fourier-Whittaker transform as
√
Z ∞Z
Z Z
σ
2|y|1−s
dt
2|y|2−2s π ∞
π
dt
−t(1+xxσ ) s
C
ψ(xy) e
t dx
=
ψ(xy √ ) e−t e−πxx t2s−1 dx
Γ(2s) 0
t
Γ(2s)
t
t
C
0
R
Z
√
2|y|2−2s π ∞ −t −π4yyσ · π 2s−1 dt
t t
=
e e
(Fourier transform after replacing x by x · √πt )
Γ(2s)
t
0
R
C
Again, the 4 = 22 in 4yy σ appears because of the normalization of the pairing. Replacing t by t · 2πy turns
the whole into
Z
Z ∞
1
2|y|2−2s π ∞ −2π(t+ 1 )|y| 2s−1 dt
|y|
dt
t
(2πy)2s−1
e
t
=
e−2π(t+ t )|y| t2s−1
(v complex)
−2s Γ(2s)
Γ(2s)
t
(2π)
t
0
0
30
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
In both cases, we have
E
ζv (2s) · Ws,v
= (invariant under s → 1 − s)
[8.4] The Mellin transform The global Mellin transform of W E factors
Z
J
s0
|a|
E
Ws,χ
a 0
0 1
da =
YZ
v
s0
|a|
kv×
E
Ws,χ,v
a 0
0 1
da
To compute this, we cannot simply change the order of integration, since this would produce a divergent
integral along the way. Instead, we present [38] the vectors ϕv in a different form. Let Φv be any Schwartz
function on kv2 invariant under Kv (under the obvious right action of GL2 ), and put [39]
ϕ0v (g) = χ(det g)| det g|s ·
Z
kv×
χ(t)2 |t|2s · Φ(t · e2 · g) dt
where e2 is the second basis element in k 2 . This ϕ0v has the same left Pv -equivariance as ϕv , namely
ϕ0v (
a ∗
0 d
· g) = ϕ0v (g) · |a/d|s χ(a/d)
For Φ invariant under the standard maximal compact Kv of GL2 (kv ), this function ϕ0v is right Kv -invariant.
By the Iwasawa decomposition, up to constant multiples there is only one such function, so
ϕ0v (g) = ϕ0v (1) · ϕv (g)
(since ϕv (1) = 1)
and
ϕ0v (1)
Z
=
kv×
χ(t)2 |t|2s · Φ(t · e2 ) dt = ζv (2s, χ2 , Φ(0, ∗))
(a Tate-Iwasawa zeta integral)
Thus, it suffices to compute the local Mellin transform of
E
ϕ0v (1) · Ws,χ,v
(m) =
Z
ψ(n) ϕ0v (w◦ nm) dn = χ(a)|a|s
Z
Z
ψ(x)
kv
Z
ψ(n)
Nv
Nv
= χ(a)|a|s
Z
χ2 (t)|t|2s Φ(ta tx) dt dx
kv×
χ2 (t)|t|2s Φ(t · e2 · w◦ nm) dt dn
(with m =
kv×
a 0
)
0 1
At finite primes v, we may as well take Φ to be
Φ(t, x) = chov (t) · chov (x)
(chX = characteristic function of set X)
Then ϕ0v (1) is exactly an L-factor
ϕ0v (1) = ζv (2s, χ2 , chov ) = Lv (2s, χ2 )
[38] This variant presentation of the vector used to form the Eisenstein series is essentially a globally split theta
correspondence. At a more elementary level, this trick is a non-archimedean analogue of classical computations
involving Bessel functions.
[39] The leading χ(det g)| det g|s in this integral maintains the triviality of the central character.
31
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
and [40]
E
ϕ0v (1) · Ws,χ,v
a 0
0 1
= χ(a)|a|s
Z
Z
ψ(x) chov (tx)
kv
= χ(a)|a|s meas (ov )
= |dv |1/2
· χ(a)|a|s
v
Z
kv×
Z
kv×
kv×
χ2 (t)|t|2s chov (ta) dt dx
cho∗v (1/t)χ2 (t)|t|2s−1 chov (ta) dt
cho∗v (1/t)χ2 (t)|t|2s−1 chov (ta) dt
where dv ∈ kv× is such that (o∗v )−1 = dv · ov . We can compute the Mellin transform
Z
0
|a|s · χ(a)|a|s
Z
kv×
kv×
cho∗v (1/t)χ2 (t)|t|2s−1 chov (ta) dt da
Replace a by a/t, and then t by 1/t to obtain a product of two zeta integrals
Z
Z
s0
s
s−1−s0
|a| · χ(a)|a| chov (a) da ·
cho∗v (1/t)χ(t)|t|
dt
kv×
kv×
)
= ζv (s0 + s, χ, chov ) · ζv (s0 + 1 − s, χ, cho∗−1
v
0
= Lv (s0 + s, χ) · Lv (s0 + 1 − s, χ) · |dv |−(s +1−s) χ(dv )
E
is
Thus, dividing through by ϕ0v (1) and putting back the measure constant, the Mellin transform of Ws,χ,v
Z
0
0
Lv (s0 + s, χ) · Lv (s0 + 1 − s, χ)
a 0
E
|a|s Ws,χ,v
da = |dv |1/2
·
· |dv |−(s +1−s) χ(dv )
v
2)
×
0
1
L
(2s,
χ
v
kv
Let d be the idele whose v th component is dv for finite v and whose archimedean components are all 1. The
product over all finite primes v of these local factors is
Z
J
s0
|a|
fin
E
Ws,χ
a 0
0 1
da = |d|1/2 ·
0
L(s0 + s, χ) · L(s0 + 1 − s, χ)
· |d|−(s +1−s) χ(d)
L(2s, χ2 )
In our application, we will replace s by 1 − s and χ by χ, giving
Z
J
s0
|a|
fin
E
W1−s,χ
a 0
0 1
da = |d|1/2 ·
0
L(s0 + 1 − s, χ) · L(s0 + s, χ)
· |d|−(s +s) χ(d)
2
L(2 − 2s, χ )
In particular, with χ trivial,
Z
J
0
fin
E
|a|s W1−s
[8.5] Functional equation
a 0
0 1
da = |d|1/2 ·
0
ζk (s0 + 1 − s) · ζk (s0 + s)
· |d|−(s +s)
ζk (2 − 2s)
With χ trivial, Poisson summation gives the functional equation for the
spherical Eisenstein series:
b ∗)) · E1−s (g ι )
ξ(2s, Φ(0, ∗)) · Es (g) = ξ(2 − 2s, Φ(0,
1/2
[40] The Fourier transform of the characteristic function ch
times the characteristic function of o∗v .
ov of ov is |dv |
32
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
where g ι is g-transpose-inverse. Peculiar to GL2 is that
wg ι w−1 = (det g)−1 · g
(where w =
0
1
1
)
0
Since w lies in the standard maximal compact locally everywhere, and is in GL2 (k), and since the Eisenstein
series is spherical everywhere and has trivial central character,
b ∗)) · E1−s
ξ(2s, Φ(0, ∗)) · Es = ξ(2 − 2s, Φ(0,
Take Φ to be Kv -invariant locally everywhere, Taking Φ to be the standard Gaussian at archimedean places,
and the characteristic function of o2v at finite places, this gives
ξ(2s) · Es = |d| · |d|2s−2 · ξ(2 − 2s) · E1−s = |d|2s−1 · ξ(2 − 2s) · E1−s
where the first factor |d| is the product of the measures of the o2v . Recall the functional equation of ξ
1
ξ(s) = |d|s− 2 · ξ(1 − s)
Then
ξ(2s) · Es = ξ(2s − 1) · E1−s
9. Appendix: residues of degenerate Eisenstein series for P n−1,1
We prove meromorphic continuation and determine some residues of some very degenerate Eisenstein series
(sometimes called Epstein zeta functions). We need to recall some specifics about these well-known examples.
Let
(n − 1)-by-(n − 1)
∗
P = P n−1,1 = {
}
0
1-by-1
A
View n and k n as row vectors. Let e1 , . . . , en the standard basis for k n . The parabolic P is the stabilizer
in GLn of the line ken . Given a Hecke character of the form χ(α) = |α|s and a Schwartz function Φ on n ,
let
Z
s
ϕ(g) = | det g|
|t|ns Φ(t · en · g) dt
A
J
ns
The factor |t| in the integrand and the leading factor | det g|s combine to give the invariance ϕ(zg) = ϕ(g)
for z in the center ZA of G = GLn . By changing variables in the integral observe the left equivariance
ϕ(pg) = | det pg|s
Z
J
|t|ns Φ(t · en · pg) dt = | det A|s |d|−(n−1)s · ϕ(g)
(for p =
A
∗
d
∈ PA )
The normalization is not ϕ(1) = 1 but
Z
ϕ(1) =
J
|t|ns Φ(t · en ) dt
(Tate-Iwasawa zeta integral at ns)
Denote this zeta integral by ξ = ξ(ns, Φ(0, ∗)), indicating that it only depends upon the values of Φ along the
last coordinate axis. Thus, by comparison to the standard spherical Eisenstein series Es (g) corresponding
to this sth degenerate principal series, the Eisenstein series associated to ϕ has a factor of ξ(ns, Φ(0, ∗))
included, namely
X
ξ(ns, Φ(0, ∗)) · Es (g) =
ϕ(γg)
γ∈Pk \Gk
33
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Poisson summation proves the meromorphic continuation of this Eisenstein series, as follows. Let
J+ = {t ∈ J
J− = {t ∈ J
: |t| ≥ 1}
: |t| ≤ 1}
and g ι = (g > )−1 (transpose inverse)
X
ξ(ns, Φ(0, ∗)) · Es (g) =
ϕ(γg) = | det g|
γ∈Pk \Gk
X
s
= | det g|
γ∈Pk \Gk
Z
k× \
J
|t|
X
ns
Z
X
s
J
γ∈Pk \Gk
Φ(t · λen · γg) dt = | det g|
s
|t|ns Φ(t · x · γg) dt
Z
k× \
λ∈k×
J
X
|t|ns
Φ(t · x · g) dt
x∈kn −0
Let
Θ(g) =
X
Φ(t · x · g)
x∈kn
Then
ξ(ns, Φ(0, ∗)) · Es (g) = | det g|s
Z
k× \
J
|t|ns [Θ(g) − Φ(0)] dt + | det g|s
Z
k× \
+
J
J
|t|ns [Θ(g) − Φ(0)] dt
−
C
The usual estimate shows that the integral over k × \ + converges absolutely for all s ∈ . Rewrite the
second part of the integral as an analogous integral over k × \ + . Poisson summation gives
X
X
b −1 · x · g ι ) + |t|−n | det g|−1 Φ(0)
b
Φ(t · x · g) + Φ(0) = |t|−n | det g|−1
Φ(t
x∈kn −0
J
x∈kn −0
Let
Θ0 (g ι ) =
X
b · x · g ι )·
Φ(t
x∈kn
J− turns this integral into
b
Removing the Φ(0) and Φ(0)
terms and replacing t by t−1 in the integral over k × \
Z
s−1
b
| det g|
|t|n(1−s) [Θ0 (g ι ) − Φ(0)]
dt
k× \
− | det g|s Φ(0)
×
J
Z
J
k× \ −
J
+
b
|t|ns dt + | det g|s−1 Φ(0)
Z
J
|t|n(s−1) dt
k× \ −
+
The integral over k \
is entire. Thus, the non-elementary part of the integral is converted into two entire
integrals over k × \ + together with two elementary integrals that give the only possible poles:
Z
Z
s
n
s−1 b
ξ · Es (g) = (entire) − | det g| Φ(0)
|t| dt + | det g| Φ(0)
|t|n(s−1) dt
J
k× \
With
J
−
k× \
Z
κ=
k× \
J
1 dt
1
the relatively elementary integrals can be evaluated
Z
k× \
J
|t|
ns
! Z
1 dt ·
Z
dt =
−
k× \
J
1
κ
tns dt =
ns
1
0
|t|n(s−1) dt =
κ
n(s − 1)
Similarly,
Z
k× \
J
−
34
J
−
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
That is,
ξ(ns, Φ(0, ∗)) · Es = (entire) −
b
κ Φ(0)
κ Φ(0)
+
ns
n(s − 1)
Thus, the residue at s = 1 of Es is
Ress=1 Es =
b
κ Φ(0)
n · ξ(n, Φ(0, ∗))
Let d be an idele such that dv generates the local different at a finite place v, and is trivial at archimedean
places. Let Φ be the standard Gaussian at archimedean places (so its integral is 1), and the characteristic
function of onv at finite places v. With the standard measure on
we have
A
b
Φ(0)
= |d|n/2
ξ(n, Φ(0, ∗)) = ξ(n)
where ξ is the usual zeta function with standard gamma factors, but without any epsilon factor or accounting
for conductors. The residue at s = 1 is
Ress=1 E =
κ · |d|n/2
n · ξ(n)
At s = 0, the relevant residue is
Ress=0 ξ(ns, Φ(0, ∗)) · Es = −
κ Φ(0)
n
10. Appendix: degenerate Eisenstein series for P r−2,1,1
Let P = P r−2,1,1 , and

A ∗
a
ϕs1 ,s2 ,s3 

∗
∗  = | det A|s1 · |a|s2 · |d|s3
d
A
(with A ∈ GLr−2 , a, d ∈ GL1 )
A
and extend ϕ to a function on G( ) = GLr ( ) by requiring right KA -equivariance. Define an Eisenstein
series on G = GLr by
X
Es1 ,s2 ,s3 (g) =
ϕs1 ,s2 ,s3 (γg)
γ∈Pkr−2,1,1 \Gk
[10.1] Symmetry in s2 and s3
This can be rewritten as an interated sum
X
Es1 ,s2 ,s3 (g) =
ϕs1 ⊗Es1,1,s (γg)
2
3
γ∈Pkr−2,2 \Gk
where
ϕ
A
s1 ⊗Es1,1
2 ,s3
∗
D
= | det A|s1 · Es1,1
(D)
2 ,s3
(with A ∈ GLr−2 and D ∈ GL2 )
and Es1,1
is the GL2 Eisenstein series
2 ,s3
Es1,1
(g) =
2 ,s3
X
ϕs2 ,s3 (γg)
(with ϕs2 ,s3
γ∈Pk1,1 \GL2 (k)
35
a
∗
d
= |a|s2 |d|s3 )
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
Since
|a|s2 |d|s3 = |ad|
s2 +s3
2
· |a/d|
s2 −s3
2
this GL2 Eisenstein series can be expressed in terms of an Eisenstein series with trivial central character,
namely
s2 +s3
Es1,1
(g) = | det g| 2 · E s2 −s3 (g)
2 ,s3
2
where
X
Es (g) =
ϕs (γg)
with
ϕs
a
γ∈Pk1,1 \GL2 (k)
∗
d
= |a/d|s
From the functional equation
ξ(2s) · Es = ξ(2s − 1) · E1−s
the P 1,1 Eisenstein series Es1,1
has a functional equation under
2 ,s3
(s2 , s3 ) = ( 21 , − 12 ) + (s2 − 21 , s3 + 12 ) → ( 12 , − 12 ) + (s3 + 12 , s2 − 12 ) = (s3 + 1, s2 − 1)
given by
Es1,1
=
2 ,s3
ξ(s2 − s3 − 1)
· Es1,1
3 +1,s2 −1
ξ(s2 − s3 )
Thus,
Esr−2,1,1
=
1 ,s2 ,s3
ξ(s2 − s3 − 1)
· Esr−2,1,1
1 ,s3 +1,s2 −1
ξ(s2 − s3 )
[10.2] Pole at s1 − s2 = r − 1 There is another iterated sum expression
X
Esr−2,1,1
(g) =
1 ,s2 ,s3
ϕEsr−2,1
(γg)
,s ⊗s3
1
2
γ∈Pkr−1,1 \Gk
where
ϕEsr−2,1
,s ⊗s3
1
A
2
∗
d
= Esr−2,1
(A)
1 ,s2
(with A ∈ GLr−1 and d ∈ GL1 )
and Esr−2,1
is the GLr−1 Eisenstein series
1 ,s2
Esr−2,1
1 ,s2
A
∗
d
X
=
ϕs1 ,s2 (γg)
(with ϕs1 ,s2
γ∈Pkr−2,1 \GLr−1 (k)
Since
s1
s2
|A| |d|
A
∗
d
= |A|s1 |d|s2 )
s1 −s2
det A r−1
(r−2)s1 +s2
· | det A · d| r−1
= r−2 d
we can express Esr−2,1
in terms of an Eisenstein series with trivial central character, as
1 ,s2
Esr−2,1
(h) = | det h|
1 ,s2
(r−2)s1 +s2
r−1
· E s1 −s2 (g)
r−1
where
X
Es (h) =
ϕs (γh)
Pkr−2,1 \GLr−1 (k)
with
ϕs
A
∗
d
det A s
(g) = r−2 d
(for A ∈ GLr−2 and d ∈ GL1 )
36
Diaconu-Garrett-Goldfeld: Moments for L-functions for GLr × GLr−1
From the previous appendix, the Eisenstein series Es for P r−2,1 has a pole at s = 1 with constant residue
r−1
κ · |d| 2
Ress=1 Es =
(r − 1) · ξ(r − 1)
Thus, at
s1 −s2
r−1
(h) has residue
= 1, that is, at s1 − s2 = r − 1, Esr−2,1
1 ,s2
r−1
Ress1 −s2 =r−1 Esr−2,1
(h)
1 ,s2
= | det h|
(r−2)s1 +(s1 −(r−1))
r−1
r−1
κ · |d| 2
κ · |d| 2
·
= | det h|s1 −1 ·
(r − 1) · ξ(r − 1)
(r − 1) · ξ(r − 1)
Thus, since
s1
| det A|
s1 −(r−1)
· |a|
= | det
A
a
|
s1 −1
det A 1
· r−2 a
the residue is
(for A ∈ GLr−2 and a ∈ GL1 )
r−1
Ress1 −s2 =r−1 Esr−2,1,1
=
1 ,s2 ,s3
κ · |d| 2
· Esr−1,1
1 −1,s3
(r − 1) · ξ(r − 1)
[10.3] Residue at s1 − s3 = r The functional equation in s2 and s3 (from GL2 ) is
Esr−2,1,1
=
1 ,s2 ,s3
ξ(s2 − s3 − 1)
· Esr−2,1,1
1 ,s3 +1,s2 −1
ξ(s2 − s3 )
at s1 − s3 = r
at s − 1 − s2 = r − 1 give a pole of Esr−2,1,1
This functional equation and the pole of Esr−2,1,1
1 ,s2 ,s3
1 ,s2 ,s3
with residue
ξ(s2 − s3 − 1)
Ress1 −s3 =r Esr−2,1,1
= Ress1 −s3 =r
· Esr−2,1,1
1 ,s2 ,s3
1 ,s3 +1,s2 −1
ξ(s2 − s3 )
r−1
=
ξ(s2 − s3 − 1)
κ · |d| 2
·
· Esr−1,1
1 −1,s2 −1
(r − 1) · ξ(r − 1)
ξ(s2 − s3 )
r−1
=
κ · |d| 2
ξ(s2 − (s1 − r) − 1)
·
· Esr−1,1
1 −1,s2 −1
(r − 1) · ξ(r − 1)
ξ(s2 − (s1 − r))
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