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OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
A Subconvexity Bound for AutomorphicL-functions for SL(3, Z)
Liangyi ZhaoJoint work with Stephan Baier
Nanyang Technological UniversitySingapore
andMax-Planck-Institut fur Mathematik
Bonn Germany
March 5, 2010
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
1 History of SubconvexityDegree One L-functionsHigher Degree L-functions
2 Statement of ResultsSubconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
3 Sketching the ProofsJutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
4 NotesDifferentiating Jutila’s Method and OursPotential Future Projects
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Degree One L-functionsHigher Degree L-functions
Subconvexity in General
The convexity bound for an L-function L(s) of degree d refersto the bound
L
(1
2+ it
)� |t|d/4, for |t| > 1.
This bound can be obtained from the functional equation ofL(s) and the Phragmen-Lindelof principle.
The generalized Lindelof hypothesis states that for any ε > 0
L
(1
2+ it
)�ε |t|ε, for |t| > 1.
In practice, even breaking the convexity bound is generallydifficult, but of great importance in many applications.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Degree One L-functionsHigher Degree L-functions
Riemann Zeta-function and Dirichlet L-functions
The convexity bound for ζ(s) is
ζ
(1
2+ it
)� |t|1/4, for |t| > 1.
The first subconvexity bound was obtained by H. Weyl.
ζ
(1
2+ it
)� |t|1/6 log3/2 |t|, for |t| > 2.
The best-known subconvexity bound for ζ(s) is due to M. N.Huxley.
Subconvexity bounds for Dirichlet L-functions in theconductor aspect were proved by D. A. Burgess.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Degree One L-functionsHigher Degree L-functions
L-functions of Higher Degree
For various types L-functions of degree 2, subconvexitybounds were obtained by A. Good, T. Meurman,Duke-Friedlander-Iwaniec, Blomer-Harcos-Michel.
Subconvexity for Rankin-Selberg L-functions onGL(2)× GL(2) were due to Sarnak,Kowalski-Michel-Vanderkam, Michel, Harcos-Michel,Michel-Venkatesh, Lau-Liu-Ye.
Recently, X. Li established a subconvexity bound for theGodement-Jacquent L-functions assoicated self-dual Maassforms (Gilbart-Jacquet lift from GL(2) to GL(3)) for SL(3,Z).
There are many subconvexity results in the literature. Weshall not attempt to make a complete list here.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
The Generalized Upper Half Plane
The generalized upper half plane H3 is defined as the cosetspace
H3 = GL(3,R)/〈O(3,R),R×〉.
SL(3,Z) acts on H3 by left multiplication. Every element z ofH3 can be represented by a matrix of the form z = xy , where
x =
1 x1,2 x1,3
0 1 x2,3
0 0 1
and y =
y1y2 0 00 y1 00 0 1
with y1, y2 > 0.
Fix (ν1, ν2) ∈ C2. We define the function Iν1,ν2 : H3 → C by
I(ν1,ν2)(z) = yν1+2ν21 y2ν1+ν2
2 , with z = xy as above.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
Jacquet’s Whittaker function
Moreover, for a ring R, let Un(R) be the group of n × n uppertriangular matrices with entries from R and 1’s on the diagonal.Jacquet’s Whittaker function is defined by
WJacquet (z ; (ν1, ν2), ψm1,m2)
=
∫U3(R)
I(ν1,ν2)
0 0 −10 1 01 0 0
uz
ψm1,m2d∗u,
where
ψm1,m2(u) = e(m1u1,2 + m2u2,3) and d∗u = du1,2du1,3du2,3.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
Maass forms for SL(3, Z)
Let S be the space consisting of smooth functions
f : GL(n,R)→ C.
For every n × n matrix α with real entries, define thedifferential operator acting on S by the rule
Dαf (g) :=∂
∂tf (g + t(g · α))
t=0
.
Let D3 be the associative algebra generated by all Dα’s,consisting of all linear combinations of Dα1 ◦ Dα2 ◦ · · · ◦ Dαk
with ◦ denoting composition. Let D3 be the center of D3.
I(ν1,ν2) is an eigenfunction of every differential operator D inD3 with DI(ν1,ν2) = λD I(ν1,ν2).
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
Maass forms for SL(3, Z)
A Maass form of type (ν1, ν2) for the group SL(3,Z) is afunction F satisfying the following properties:
F is a smooth function on H3 to C.
F (γz) = F (z) for all z ∈ H3 and γ ∈ SL(3,Z).
F is an eigenfunction of every differential operator D in D3
with corresponding eigenvalue λD .
F has a Fourier-Whittaker expansion of the form
F (z) =∑
γ∈U2(Z)\SL(2,Z)
∞∑m=1
∑n 6=0
am,n
|mn|
×WJacquet
(diag(|mn|,m, 1)
(γ
1
)z ; (ν1, ν2), ψ1,n/|n|
).
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
Maass forms for SL(3, Z)
It follows from the work of Luo-Rudnick-Sarnak that am,n
satisfy the bound
am,n � |mn|2/5+ε.
The generalized Ramanujan conjecture (GRC) would give that
am,n � |mn|ε.
For every Maass form F of type (ν1, ν2) for SL(3,Z), there isa dual Maass form F of type (ν2, ν1) whose Fouriercoefficients am,n satisfy the relation am,n = an,m.
A Maass form is said to be self-dual if F = F .
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
Godement-Jacquet L-functions
Let am,n be the (m, n)-th Fourier coefficient of a Maass formF for SL(3,Z). Also assume that these coefficients arenormalized so that a1,1 = 1.
The Godement-Jacquet L-function LF (s) is defined as
LF (s) =∞∑
n=1
a1,n
ns=∏p
(1− a1,p
ps+
ap,1
p2s− 1
p3s
)−1
, for <s > 1.
LF extends to an entire function on C.
This is a degree 3 L-function and hence has the convexitybound
LF
(1
2+ it
)� |t|3/4, for |t| > 1.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
Subconvexity bound for Godement-Jacquet L-functions
Recently, X. Li established a subconvexity bound forGodement-Jacquet L-functions associated with the specialclass of self-dual Maass forms for SL(3,Z).
LF
(1
2+ it
)�ε |t|11/16+ε, for |t| > 1.
We prove a subconvexity result for this L-function associatedwith a general Maass form for SL(3,Z).
LF
(1
2+ it
)�ε |t|18/25+ε, for |t| > 1. (1)
If we also assume the truth of GRC, then we have
LF
(1
2+ it
)�ε |t|99/140+ε, for |t| > 1. (2)
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Subconvexity bound for Godement-Jacquet L-functionsBounds for Dirichlet Polynomials
Bounds for Dirichlet Polynomials
The bounds (1) and (2) are consequences of following boundsfor Dirichlet polynomials with Fourier coefficients of theafore-mentioned Maass forms.
Let am,n be the (m, n)-Fourier coefficient of a Maass form Ffor SL(3,Z). Assume that ε, η > 0, 2 ≤ N ≤ n ≤ N ′ ≤ 2Nand N3/5+η ≤ |t| ≤ N5/1−η, we have∑
N<n≤N′
a1,nn−2πit �F ,ε,η N9/10+ε|t|3/25. (3)
Assuming the truth of GRC, we have, under the sameconditions as above,∑
N<n≤N′
a1,nn−2πit �F ,ε,η N6/7+ε|t|6/35. (4)
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
Proof of the Subconvexity Bounds
The subconvexity bounds (1) and (2) are obtained by applyingthe bounds for Dirichlet polynomials to the approximatefunctional equation for the Godement-Jacquet L-functions.
The most important range is
N � t3/2 or equivalently t � N2/3.
Note that 3/4 = 0.75, 18/25 = 0.72, 99/140 = 0.7071... and11/16 = 0.6875.
The bounds for Dirichlet polynomials follow from extending amethod of M. Jutila (with new ingredients) for the estimationof exponential sums with Fourier coefficients of a holomorphiccusp forms for SL(2,Z).
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
Jutila’s Method
M. Jutila developed a method to estimate sums of the form∑n∈I
a(n)e (f (n)) . (e(z) = exp(2πiz))
I is some interval, f is a smooth real-valued function satisfyingcertain conditions and a(n) is the divisor function or the n-thFourier coefficient of a holomorphic cusp form for SL(2,Z).
As application, Jutila re-proved the bound for the 11-thmoment of ζ(s) due to Heath-Brown and the subconvexity forL-functions associated with holomorphic cusp form forSL(2,Z) due to Good.
T. Meurman extended this method to exponential sums withFourier coefficients of Maass cusp forms for SL(2,Z).
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
An Averaging Process
We start by considering a general exponential sum of the form∑N<n≤N′
aq,ne(f (n)), N < N ′ ≤ 2N.
f satisfies the following conditions.1 f is real-valued on [N/2, 3N], f ′(x) < 0 and f ′′(x) > 0 for
x ∈ [N/2, 3N].2 f extends to a holomorphic function in
D = {z : N/2 ≤ <z ≤ 3N, |=z | ≤ N}.3 f ′(z) � NΛ for z ∈ D.4 f ′′(x) � Λ for x ∈ [N/2, 3N].5 f (j)(x)� N1−jΛ for all j ≥ 3.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
An Averaging Process
We introduce a smooth weight in this averaging process.
We have∑N<n≤N′
aq,ne(f (n))
=1
W
∑n
∑n−M≤m≤n+M
w(m − n)aq,me(f (m)) + error terms.
(5)
w(y) is a smooth weight; in fact, it is the product of twosmooth weights W (y), a Gaussian-like function, and Υ(y/M),a smooth function with compact support in [−M,M] andΥ(y) = 1 for −1/2 ≤ y ≤ 1/2. W =
∑−M≤m≤M w(m) is the
total weight.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
An Averaging Process
The introduction of the weight function here will beadvantageous later.
The error terms in (5) can be estimated using either GRC orestimate for the mean-square of aq,n. GRC gives a betterbound.
Next, given y ∈ −f ′([N/2, 3N]), let x0(y) be the uniquesolution to
−f ′(x0(y)) = y .
By Dirichlet approximation, we have, for each N < n ≤ N ′,∣∣∣∣f ′(n) +l
k
∣∣∣∣ ≤ 1
kK
for some K ≥ 1(to be chosen later), k , l ∈ Z, 1 ≤ k ≤ K and(l , k) = 1.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
An Averaging Process
The above approximation and the conditions on f give∣∣∣∣n − x0
(l
k
)∣∣∣∣� 1
kKΛ.
So each n in question can be written as
n =
[x0
(l
k
)]+ r with r � 1
kKΛ.
Therefore, the double sum over m and n in (5) is replaced by
1
W
∑(k,l ,r)∈Z
∑m∈I (l/k,r)
w
(m − x0
(l
k
))aq,me(f (m)) + errors,
(6)where Z is an appropriate set of triples (k, l , r) andI (l/k , r) = [n −M, n + M].
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
An Averaging Process
This averaging process is reminiscent to a similar process doneby Bombieri and Iwaniec in their proof of a subconvexitybound for ζ(s).
We still would like to make the summation range of m in (6)independent of r for convenience. This replacement ofI (l/k , r) by I (l/k , 0) = I (l/k) will introduce further errorterms.
To control all the errors incurred above, we need to imposeconditions on the sizes of the parameters M and K .
Moreover, to estimate these errors, we can either assume GRCor use the mean-square estimate aq,n. GRC gives betterestimates.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
Application of the Voronoi Summation Formula
After the averaging process, it suffices to consider the sum
∑K0<k≤K
1
k
∑l�kΛN(l ,k)=1
∣∣∣∣∣∣∑
m∈I (l/k)
w
(m − x0
(l
k
))aq,me(f (m))
∣∣∣∣∣∣ ,(7)
where K0 is a parameter to be chosen later.
We shall apply the twisted (with an additive character)Voronoi summation formula to the inner-most sum of (7).
The Voronoi summation formula for GL(3) automorphic formhas been established by S. D. Miller and W. Schmid andre-proved by D. Goldfeld and X. Li by a different method.
This summation formula is a key ingredient of ourproceedings.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
Application of the Voronoi Summation Formula
Now the inner-most sum of (7) becomes the sum of foursums, each of which is of a form similar to
k∑d |kq
∞∑n=1
an,d
ndS
(ql , n;
qk
d
)Φ0
(nd2
k3q
).
In the above expression, S(ql , n; qk/d
)is the Kloosterman
sum and Φ0
(nd2/(k3q)
)is an integral involving Gamma
factors and a certain Mellin transform.
From the works of X. Li, Φ0 can be estimated if n is small andapproximated by a trigonometric (or exponential) integralwhen n is large.
Futher conditions on M need to be imposed to ensure thatabove estimates are sufficient.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
Treatments of the Exponential Integrals
Now to treat the exponential integral, we use the method ofstationary phase.
The situation in which we do have a stationary point will givethe biggest contribution.
The cases in which a stationary point does not exist will give,using a result of Heath-Brown, a small contribution, becauseof one of the weight functions introduced earlier.
The Gaussian weight functions introduced in the beginning isuseful here to control the boundary terms in the stationaryphase approximation. (A result quoted from Ivic.)
More conditions on M and Λ need to be imposed.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Jutila’s MethodAn Averaging ProcessTransforming Short Exponential Sums by Voronoi SummationTreatments of the Exponential IntegralsEstimating the Main Term
Estimating the Main Term
Now it still remains to consider the main term, thecontribution of the terms with stationary points. It suffices toestimate sums of the form
∑k�Q
∑l�L
(l ,kd)=1
∣∣∣∣∣∣∑
n∈Jk,l,d
an,d
n1/3S(l , n; k)e (pkd ,l ,d ,n(ykd ,l ,d ,n))
∣∣∣∣∣∣ .The summation condition n ∈ Jk,l ,d above is translated fromthe condition under which a stationary point exists.
The coefficients an,d ’s are now removed using Cauchy’sinequality.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Differentiating Jutila’s Method and OursPotential Future Projects
Estimating the Main Term
Opening up the modulus square, we are led to consider thesum ∑
k1�Q
∑l1�L
(l1,k1)=1
∑k2�Q
∑l2�L
(l2,k2)=1
vk1,l1vk2,l2Ud(k1, k2, l1, l2),
where vk,l ’s are complex numbers of modulus 1 and
Ud(k1, k2, l1, l2) =∑n
S(l1, n; k1)S(l2, n; k2)e(· · · ),
with the appropriate conditions on n in Ud .
To estimate Ud , we break n into residue classes modulo k1k2
and then apply Poisson summation.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Differentiating Jutila’s Method and OursPotential Future Projects
Estimating the Main Term
This leads to a problem of estimating an exponential integraland counting solutions to certain congruence equations, bothof which can be resolved using classical means.
We still need to sum over k1, k2, l1 and l2. To this end, oneneeds to resolve a spacing problem. For this, we use a resultof Fouvry and Iwaniec.
Finally, we collect everything and check that it’s possible tochoose K , Λ, M to simultaneously satisfy all conditions wehave imposed on them.
Optimizing the parameters, we get the bound for the Dirichletpolynomial.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Differentiating Jutila’s Method and OursPotential Future Projects
Differentiating Jutila’s Method and Ours
The effect of our averaging process is to reduce the problemto considering weighted exponential sums with coefficents a1,n
over short intervals. In place of the averaging process, Jutilasimply split his sums into short ones. This is sufficient in hissituation, but problematic in ours.
Jutila also introduced a weight, but his weight is not smoothas it has only finitely many continuous derivatives. To useVoronoi summation formula available to us, we must havesmooth weights.
The presence of Kloosterman sums after applying Voronoisummation is not in Jutila’s method. We resolve this by anapplication of Poisson summation.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)
OutlineHistory of Subconvexity
Statement of ResultsSketching the Proofs
Notes
Differentiating Jutila’s Method and OursPotential Future Projects
Potential Future Projects
A potential future project would be the generalize our methodto GL(n) automorphic forms with n > 3, i.e. to prove asubconvexity bound of the form
L
(1
2+ it
)� |t|n/1−δ(n), for |t| > 1.
Voronoi-type summation formulas, a key ingredient in ourmodus operandi, were proved by D. Goldfeld and X. Li.
Another application of our method could be to obtain newmoment estimates for automorphic L-functions.
Conceivably, our method (at the bottom a method toestimate exponential sums with GL(3) coefficients) can beuseful in many other problems.
Liangyi Zhao Joint work with Stephan Baier A Subconvexity Bound for Automorphic L-functions for SL(3, Z)