A Subconvexity Bound for Automorphic L-functions for …lzhao/Presentations/subconv.pdf ·...

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)



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




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.




Notes


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.




Notes


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.




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.




Notes


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.




Notes


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).




Notes



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|

).




Notes



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 .




Notes


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.




Notes


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)




Notes


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)




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).




Notes


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).




Notes


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.




Notes



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.




Notes



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.




Notes



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].




Notes



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.




Notes


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.




Notes


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.




Notes


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.




Notes


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.




Notes

Differentiating Jutila’s Method and OursPotential Future Projects


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.




Notes



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.




Notes


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.




Notes


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.


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