MATH G9905 RESEARCH SEMINAR IN NUMBER THEORY
(FALL 2014)
LECTURE 4 (SEPTEMBER 30, 2014)
XIAOQING LI
UPPER BOUNDS FOR A RANKIN–SELBERG TYPE INTEGRAL
NOTES TAKEN BY PAK-HIN LEE
Abstract. In this talk, we will give an almost sharp upper bound for a Rankin–Selberg
type integral involving the Arthur truncated Eisenstein series on GL(n). Langlands–Shahidi
constant formula, Maass–Selberg relations and other tools on higher rank groups will be
used. This is a joint work with Goldfeld.
We will generalize Sarnak’s method to derive a logarithmic zero free region for Rankin–
Selberg L-functions on GL(n). His idea is to consider the integral
2
Z ∞Z 1 ×
1
2
ˆ
d z |ζ(1 + 2it)|.
+
it
|ζ(1 + 2it)|
E
z,
A
2
η
0
How can we generalize this integral to GL(2n)? A natural generalization is
2
Z 1 Z ∞Z ∞
Z ∞ Z 1
1
2
EˆA z, + it d× z
···
|L(1 + it, f × f )|
···
2
0
η1
η2
η2n−1 0
ˆ involves sums of the form
but
R1
R 1of EA
P this doesn’t work because the Fourier expansion
ˆ 2
γ∈U2n−1 \ SL2n−1 . A better solution is to replace 0 · · · 0 |EA | by a certain Rankin–Selberg
integral.
n×n
∗
Let G = GL(2n, R), Γ = SL(2n, Z) and K = O(2n, R). Let Pn,n (Z) =
⊂
0
n×n
In×n
∗
Γ be the maximal parabolic, N P =
be the unipotent radical, and M P =
0
In×n
n×n
0
be the standard Levi.
0
n×n
The cuspidal Eisenstein series is
X | det m1 | ns
E(z, s) :=
f (m1 )f (m2 )
| det m2 |
γ
γ∈Pn,n \Γ
where
m1and m2 come from the Iwasawa decomposition of z = nmk with n ∈ N P , m =
m1
∈ M P and k ∈ K, and f is a fixed Hecke–Maass form for GL(n, Z).
m2
Last updated: October 6, 2014. Please send corrections and comments to [email protected].
1
Arthur’s truncated Eisenstein series is
EˆA (z, s) := E(z, s) −
X
cP E(z, s)|γ .
γ∈Pn,n \Γ
h(z)|γ ≥A
m1 , and cP E is the constant term of E along Pn,n .
Here h(z) = det
det m2 There is no obvious connection between the Fourier expansions of EˆA and E. The idea is
to smooth out the sharp truncation, so we introduce a smooth version of Arthur’s truncated
Eisenstein series
n
1
Λ(2ns − 2n, f × f )
∗
EˆA (z, s) := E(z, s) − A 2 E z, s −
+
E(z, 2 − s)
2
Λ(1 + 2ns − 2n, f × f )
n
X
A2
ns
−
1−
f (m1 )f (m2 )
h(z)
n
h(z) 2
γ
γ∈P \Γ
h(γ·z)≥A
Λ(2ns − 2n, f × f )
−
Λ(1 + 2ns − 2n, f × f )
X
n(2−s)
h(z)
γ∈P \Γ
h(γ·z)≥A
n
A2
f (m1 )f (m2 ) .
1−
n
h(z) 2
γ
The truncated sum can be rewritten as
Z
n
n
X
A− 2 w
w
A2
1
ns
E z, s +
dw
h(z)
f (m1 )f (m2 ) =
1−
n
2πi (2) w(w + 1)
2
h(z) 2
γ
γ∈P \Γ
h(γ·z)≥A
which is smooth. Note that
Z
(2)
10
x−w
dw =
w(w + 1)
(
1
1−x
0
if x > 1,
if x ≤ 1.
2
Let β := tn , δ := β −1 (log log t)n , and g(x), ψ(x) ∈ Cc∞ ([1, 2]) be test functions. Define
Z ∞
2 Z
A
dA
det z
∗
ˆ
I := |L(1 + 2int, f × f )| ·
EA (z, 1 + it)g
ψ
d× z
β A
δ
P2n−1,1 \η 2n
0
(2n − 1) × (2n − 1) ∗
where P2n−1,1 is the maximal parabolic
and η 2n = GL(2n, R)/K·
0
∗
R× .
Our main theorem is
Theorem 1.
1
1
I f δ − 2 β 2 +n (log t)2
as t → ∞.
This bound is sharp modulo some log terms. Next time we will get a lower bound.
Recall that if φ is a Maass form, then the integral
Z
|φ|2 EP2n−1,1 (z, s)d× z
SL(2n,Z)\η 2n
is roughly equal to the Rankin–Selberg L-function Λ(s, φ × φ).
2
The maximal parabolic Eisenstein series is
X
EP2n−1,1 (z, w) :=
(det z)w |γ .
γ∈P2n−1,1 \ SL2n
Unfolding gives
I = |L(1 + 2int, f × f )|
Z ∞
2
Z
Z
dA
1
A
−w
ˆ∗ (z, 1 + it)g
EP2n−1,1 (z, w)d× zdw.
·
ψ̃(−w)δ ·
E
A
2πi (2)
β A
SL2n (Z)\η 2n
0
Shifting the w-line to 21 , we pick up the pole of EP2n−1,1 (z, w) at w = 1 and write
I = I1 + I2 ,
where
I1 = c|L(1 + 2int, f × f )|ψ̃(−1)δ
−1
Z
SL(2n,Z)\η
Z
2n
∞
0
2
A ˆ∗
dA ×
g
EA (z, 1 + it) d z
β
A
for some constant c, and
1
I2 = |L(1 + 2int, f × f )| ·
2πi
Z
ψ̃(−w)δ
−w
Z
( 21 )
SL2n (Z)\η
Z
2n
∞
0
2
· · · EP2n−1,1 (z, w)d× zdw.
We want to bound these two integrals.
Proposition 2. Suppose F is an automorphic function for SL(k, Z), k ≥ 4. Then F has a
Fourier expansion
 

1 0 · · · 0 u1k
Z 1
Z 1  1 · · · 0 u2k  


.. 
. . ..
 z  d× u

F (z) =
···
F
.
.
.
 

0
0
 

1 u
k−1,k
1
+
X
X
1≤l≤k−2 mk−1 6=0
···
X
Z
X
Z
1
···
mk−l 6=0 γ∈P̃k−1,l \ SLk−1

1 0 ···
 1 · · ·


F


1
0 u1,k−l · · ·
0 u2,k−l · · ·
..
.
0
0
u1,k−1
u2,k−1
..
.
u1k
u2k
..
.
1
uk−1,k
1



 γk−1



z


1


· e(−mk−1 uk−1,k − · · · − mk−l uk−l,k−l+1 )d× u
Z 1
Z 1
X
X
X
+
···
···
mk−1 6=0
m1 6=0 γ∈Uk−1 \ SLk−1
0
0
3




1 u12 · · · u1,k−1 u1k


1 · · · u2,k−1 u2k  


γ
.
.
..



.
.
F
z
.
.
.



1


1
uk−1,k 
1
· e(−mk−1 uk−1,k − · · · − m1 u12 )d× u.
Here
P̃k−1,l

SLk−1−l ∗ ∗ · · ·




1 ∗ ···



1 ···
= 

..



.



∗ 


∗

∗ .


∗


1
The last term is called the nondegenerate term ND(F ). For details, see Goldfeld and
Ichino–Yamana.
Proposition 3 (Langlands–Shahidi). E(z, s) has constant term
!
det m1 n(1−s)
det m1 ns
f (m1 )f (m2 ) + cs f (m1 )f (m2 )
cP := det m2 det m2 with
Λ(2ns − n, f × f )
Λ(1 + 2ns − n, f × f )
along Pn,n . Along other parabolics, the constant terms are 0.
cs :=
Corollary 4.
E(z, s) =
!
det m1 ns det m1 n(1−s)
∗
+
c
f
(m
)f
(m
)
s
1
2 det m2 det m2 X
γ∈P̃2n−1,n−1 \ SL2n−1
+ ND(E).
(
γ
1)
Here
X B(m2n−1 , · · · , mn+1 )
WJ (M m2 ),
Qn−1 k(n−k)
2
m2n−1 6=0
mn+1 6=0
k=1 m2n−k
where B are the Fourier coefficients of f and WJ is the Jacquet–Whittaker function.
f ∗ (m2 ) =
X
···
Proposition 5. Let
X A(m1 , · · · , mk−1 )
F (z) :=
···
WJ (M z)
.
Qk−1
l(k−l)
γ
2
mk 6=0
γ∈Uk−1 \ SLk−1 m1 ≥1
l=1 |ml |
( 1)
X
k(k−1)
X
√
If maxi yi ≥ (t1+ ) 2 , mini yi ≥ 23 , and the Langlands parameters (α1 , . · · · , αk−1 ) satisfy
| Im αi | t, then F (z) (maxi yi )−N for N very big.
On a compact Riemannian surface, F (z) |λ| for the spectral eigenvalue λ. On a
non-compact surface, this is not true.
4
The Jacquet–Whittaker function has rapid decay
Wk,ν (z) (max yi )−N
i
under the above conditions.
Lemma 6 (Bump–Friedberg–Hoffstein).
Iν (g · z) =
k−2
Y
!
Pk−1
kek−i gz ∧ · · · ∧ ek−1 gzk−kνk−i−1
| det z|
i=0
iνk−i
i=0
for g ∈ SL(k, R). Here
Iν (z) =
k−1
Y k−1
Y
b νj
yi ij
i=1 j=1
(
ij
if i + j ≤ k,
with bij =
(k − i)(k − j) if i + j ≥ k,
and ei = (0, · · · , 0, 1, 0, · · · , 0).
Note that kek−i M ∧ · · · ∧ ek M k2 is the sum of squares of all the (i + 1) × (i + 1) minors
of the last i + 1 rows of M .
This bound I1 . To bound I2 we need to use the maximal parabolic Eisenstein series
EP2n−1,1 (z, w)
which is essentially the Epstein zeta function. The completed Eisenstein series
EP∗ 2n−1 (z, w) = π −nw Γ(nw)ζ(2nw)EP2n−1,1 (z, w)
has integral representation
Z ∞
Z ∞
1
1
1
n(1−w) du
nw du
(θw0 t z−1 w0 (u) − 1)u
+
−
+
.
(θz (u) − 1)u
u
u
n 1−w w
1
1
Here w0 is the long element in the Weyl group, and the theta function is
X
2
2
2
e−π(b1 +b2 +···+b2n )u
θz (u) =
(a1 ,··· ,a2n )∈Z2n
where
b 1 = a1 Y 1 ,
b2 = (a1 x12 + a2 )Y2 ,
..
.
b2n = (a1 x1,2n + a2 x2,2n + · · · + a2n )Y2n ,
and
1
Yk = y1 · · · y2n−k (y12n−1 · · · y2n−1 )− 2n .
√
Proposition 7. For yi ≥ 23 , 1 ≤ i ≤ 2n − 1, Re w = 12 , we have
X 1
2n−k 12
k−1
EP2n−1,1 (z, w) (y1 y22 · · · y2n−k
) (y2n−k+1
· · · y2n−1 ) 2
2≤k≤2n
2n−k
+(y1 y22 · · · y2n−k
)
k−1
2n
5
k−1
(y2n−k+1
· · · y2n−1 )
2n−k+1
2n
ln y1 · · · y2n−1 .
Recall the Maass–Selberg relation
r+s−1
r−s
s−r
1−r−s
A
A
A
A
2
+ cs
+ cr
+ cr cs
.
hEˆA (·, r), EˆA (·, s)i = |hf, f i|
r+s−1
r−s
s−r
1−r−s
We can apply this to the smooth truncation above, which is just a linear combination of
Arthur’s truncation.
We need an upper bound for the Rankin–Selberg L-function.
Lemma 8.
L(1 + 2int, f × f ) n,f log(|t| + 1)
and
L0 (1 + 2int, f × f ) n,f log2 (|t| + 1).
These are all the ingredients of the proof.
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