integral representations of...

Meromorphic continuationSelberg’s Analytic Continuation

Tate’s Thesis (1950): Adelic SettingAutomorphic L-functions

Automorphic forms and Langlands L−functions

Integral Representations of L−functions

Cetin Urtis

TOBB UNIVERSITY OF ECONOMICS AND TECHNOLOGY

October 21th, 2017

Cetin Urtis Integral Representations of L−functions




Outline

1 Meromorphic continuation

2 Selberg’s Analytic Continuation

3 Tate’s Thesis (1950): Adelic Setting

4 Automorphic L-functions

5 Automorphic forms and Langlands L−functions





Introduction

Integral representations of L-functions are not only crucial formeromorphic continuation of L-functions but also providesarithmetical and analytical results about L-functions.

In this series of lectures, some well-known integralrepresentations and some results obtained from these will bediscussed.

Note: I won’t promise I’ll cover all kinds of L-functions. Thisis not a talk on Langlangs program, it is only some part of it.





Some history of meromorphic continuation

Riemann, Dirichlet, Dedekind, Hecke zeta functions

Ranking-Selberg method: Try to get an L− function from aconvolution of an Eisenstein series and cuspforms or thetaseries. Shimura used theta functions; Garrett uses 3 cuspformsto get triple product L−function.

Jacquet-Langlands: Uses zeta integrals. Representationtheoretic treatment of the standard L−function.

Langlands-Shahidi method: Generalize Selberg’s method.Realize the L− function as a constant term of an Eisensteinseries on a larger group. You can only get a fixed and knownlist of L−functions.

PS-Rallis-Garrett: Doubling method (Pullback formula)





Riemann zeta function

The simplest L−function is the Riemann zeta function:

ζ(s) =∞∑n=1

1

ns

It is analytic for Re(s) > 1 and has meromorphic continuation(MC) to entire C,The completed zeta function ξ(s) = π−s/2Γ( s2 )ζ(s) has afunctional equation (FE): ξ(s) = ξ(1− s)Here gamma function is defined by the Mellin transform ofe−x :

Γ(s) =

∫ ∞0

e−xx s−1 dx

It has an Euler product (EP):

ζ(s) =∏p

(1− p−s)−1





Meromorphic continuation I

Riemann proved the meromorphic continuation of the completedzeta function ξ(s) by obtaining an integral representation.Mellin transform of theta series θ(z) =

∑n∈Z e

πin2z gives

ξ(s) =

∫ ∞0

y s/2 θ(iy)− 1

2

dy

y

Idea: Write the integral as∫ ∞0

=

∫ 1

0+

∫ ∞1

and use Jacobi’s transformation identity

θ(iy) =1√yθ

(− 1

iy

)Cetin Urtis Integral Representations of L−functions




Meromorphic continuation II

which is obtained by Poisson summation formula:∑n∈Z

f (n) =∑n∈Z

f (n)

Then the integral becomes:

ξ(s) =

∫ ∞1

(y s/2−1 + y (1−s)/2−1)θ(iy)− 1

2dy − 1

s− 1

1− s

Since the integral in this expression is convergent for any s, ξ(s)has MC with simple poles at s = 0 and s = 1.Moreover we have the FE: ξ(s) = ξ(1− s).





Dirichlet L-function and Hecke L-function I

Dirichlet L-function:

L(s, χ) =∞∑n=1

χ(n)

ns

χ is a Dirichlet character, i.e. hom. from (Z /N Z)∗ to Cwhich vanishes on integers sharing a common factor with N.

Dirichlet L-function also has MC and FE.





Dirichlet L-function and Hecke L-function II

A Hecke character is a generalization of a Dirichlet character tothe number field setting. Let F be a number field, v be itsnon-archimedean place. A Hecke char. is product of a family ofhom. χv : F ∗v → C∗:

χ(x) =∏v

χv (x)

There are two conditions:

1 χ must be trivial on F ∗

2 for almost all v , χv must be unramified, i.e. trivial on{x ∈ F ∗v : |x |v = 1}.





Dirichlet L-function and Hecke L-function III

Hecke L-function:

L(s, χ) =∑a

χ(a)

(N a)s=∑p

(1− χ(p)(N p)−s)−1

Hecke: MC, FE by using generalized θ-functions.

If χ is the trivial character, then L(s, χ) specialized toDedekind zeta-function.

For F = Q this reduces to ζ(s) and if χ is of finite order itbecomes the Dirichlet L-function.





Standard L−functions attached to cusp forms I

To a cuspform of weight 2k

f (z) =∑n>0

ane2πinz

we attach an L−function

L(f , s) =∑n>0

anns

Similary, there is an integral representation of the completedL−function.

Λ(f , s) = (2π)−sΓ(s)L(f , s) =

∫ ∞0

y s f (iy)dy

y= (−1)kΛ(f , 2k− s)





Standard L−functions attached to cusp forms II

Therefore, we have MC, FE. Moreover if f is an eigenfunction ofthe Hecke operators then there is EP:

L(f , s) =∏p

(1−app−s+p2k−1−2s)−1 =∏p

(1−αpp−s)−1(1−βpp−s)−1

Since there are two factors in each p, this is an degree 2L−function.





Rankin-Selberg L−functions ILet f , g be two cuspforms of weight 2k with Fourier expansions:

f (z) =∑n>0

ane2πinz

andg(z) =

∑n>0

bne2πinz

Let Γ = SL2(Z ) and P =

{(a b0 d

)∈ Γ

}.

Consider the following Eisenstein series

Es(z) =∑γ∈P\Γ

Im(γz)s =∑

(m,n)=1

y s

|mz + n|2s





Rankin-Selberg L−functions II

Some properties of Eisenstein series:

abs. convergent for Re(s) > 1 and has MC,

It is Γ-invariant,

it has FE: ξ(2s)Es = ξ(2− 2s)E1−s ,

it is moderate growth: |Es(x + iy)| ≤ Cyn for some n and C .

Space of cuspforms of weight 2k is a Hilbert space with thePeterson inner product:

〈f , g〉 =

∫Γ\h

f (z)g(z)y2k dxdy

y2.





Rankin-Selberg L−functions III

By considering the Rankin-Selberg convolution we have

Theorem

〈f · Es , g〉 = (4π)−(s+2k−1)Γ(s + 2k − 1)∑n≥1

anbnns+2k−1

ξ(2s)〈f · Es , g〉 has an MC with poles at most at s = 0, 1.

The sum in the expression is called the Rankin-Selberg L−function:

L(f ⊗ g) =∑n≥1

anbnns+2k−1





Rankin-Selberg L−functions IV

If f , g are both Hecke eigenfunctions then Rankin L−function hasEP of degree 4.

L(f ⊗ g , s) =∏p

(1− αpγpp−s)−1(1− αpδpp

−s)−1

(1− βpγpp−s)−1(1− βpδpp−s)−1

In 2000 Ramakrishnan proved that the Rankin-Selberg L-functionsare also attached to some automorphic object, namely anautomorphic form on GL(4) as predicted by the Langlandsfunctoriality conjectures.





Rankin-Selberg L−functions V

Proof.

If ϕ on h is an integrable P−inv. function then by Fubini’stheorem we have∫

P\hϕ(z)

dxdy

y2=

∫Γ\h

∑γ∈P\Γ

ϕ(γz)dxdy

y2

For φ(z) = y s f (z)g(z)y2k we have∫P\h

y s f (z)g(z)y2k dxdy

y2= 〈f · Es , g〉





Rankin-Selberg L−functions VI

Proof.

The integral is absolutely convergent for all s ∈ C away from thepoles of the Eisenstein series, since the Eisenstein series is ofmoderate growth and the cuspforms are of rapid decay.





Selberg’s method I

Selberg’s method can be used to obtain the analytic continuationand functional equations of the L-functions that arise in theconstant terms of Eisenstein series.Let’s rewrite the Eisenstein series:

Es(z) =1

2ζ(2s)

∑m,n)6=(0,0)

y s

|mz + n|2s

It’s Fourier expansion is given by

Es(z) =∑m∈Z

am(y , s)e2πimx





Selberg’s method II

where

am(y , s) =

∫ 1

0Es(x + iy)e−2πimx dx

By using Bruhat decomposition we find

a0(y , s) = y s + φ(s)y1−s

where the “constant term” is

φ(s) =ξ(2s − 1)

ξ(2s)

and

am(y , s) = 2

√yKs− 1

2(2π|m|y)

π−sΓ(s)ζ(2s)|m|s−1σ1−2s(m)





Selberg’s method III

where Ks(y) is the K-Bessel function:

Ks(y) =1

2

∫ ∞0

e−y(t+t−1)/2tsdt

t

Note that as y →∞, Ks(y) decays exponentially andKs(y) = K−s(y).

Theorem (Selberg,1962)

Es(z) has a meromorphic continuation to the whole complexs-plane and satisfies the functional equation

Es(z) = φ(s)E1−s(z)





Selberg’s method IV

Proof: Each term is mero and satisfies the FE. The sum convergesrapidly.Note: By using spectral theory, it is possible to prove it withoutknowing ζ is mero and having the FE.Idea: Use Eisenstein series to prove other way. a0 and a1 providemero cont. and FE of L-function.





Selberg’s method V

An application:Eisenstein series can be used to show that non-vanishing ofζ(1 + it) which is equivalent to the prime number theorem.Consider the general Fourier coefficient am:

am(y , s) = 2

√yKs− 1

2(2π|m|y)

π−sΓ(s)ζ(2s)|m|s−1σ1−2s(m)

Numerator is entire and Es(z) is holomorphic on Re(s) = 1/2therefore ζ(1 + it) does not vanish.





Ring of adeles I

J. Tate reinterpreted the methods of Riemann and Hecke in termsof harmonic analysis on the adeles A of a number field F .Simplest example for adeles is AQ: adele ring of Q. Let x ∈ Q andp be a prime.

p−adic valuation: |x |p := p−ordp(x) (and metric defined)

The field of p−adic numbers Qp is the completion of Q underthis metric. Similarly its ring of integers Zp is defined.

Completion at infinity (archimedean place) Q∞ = R





Ring of adeles II

Adele ring of Q is the restricted direct product of Qp withrespect to Zp:

AQ = Π′p Qp

This means that the adeles are infinite-tuples of the form

a = (a∞; a2, a3, a5, a7, ...), ap ∈ Qp

such that all but finitely many ap lie in Zp.





Ring of adeles III

The completed zeta function is now adelic zeta function:

ξ(s) = π−s/2Γ(s

2)ζ(s)

non-archimedean part: ζ(s) =∏

p<∞ ζp(s) =∏

p(1− p−s)−1

archimedean part: ζ∞(s) = π−s/2Γ( s2 )

We can replace Q with any number field F . Also we can considergroups defined over ring of adeles. Eg. GLn(AF ).





Adelic integrals I

Tate considers generalized ζ-integral:

ζ(f , c) =

∫Af (a)c(a) d∗a

where

c(a) is any quasi-character of A, that is it a continous hom.from A∗ to C∗ which is trivial on Q∗. (Eg. |a|A.)

d∗a is Haar measure which is product of local Haar measuresd∗x∞ = dx

|x | and d∗xp (normalized so that Z∗p has measure 1.

Let c(a) = |a|sA and f (a) =∏

fp(ap) where f∞(x) = e−πx2

andfp(x) = χZp(x) Then, ζ(f , | · |sA recovers Riemann’s integral asfollows:





Adelic integrals II





Adelic integrals III

Euler product:





Automorphic forms as group representations I

SL2(R) acts on the upper half plane h by linear fractionaltransformations. Since the stabilizer in SL2(R) of i is the rotationgroup SO(R), we have the identification:

h ' SL2(R)\SO2(R)

If f (z) is a modular form of weight k for Γ = SL2(Z), it defines afunction φ = φf on SL2(R):

φf (g) = (ci + d)−k f (gz)





Automorphic forms as group representations II

Properties:

1 φ(gr(θ)) = e−ikθφ(g) for r(θ) =

(cos θ sin θ− sin θ cos θ

)2 φ(γg) = φ(g) for all γ ∈ Γ

3 there is a second-order differential (Laplace or Casimir)operator ∆ s.t.

∆φ = −1

4k(k − 2)φ

4 cuspidal condition: φ ∈ L2(Γ\SL2(R)) and the zeroth Fouriercoefficient vanishes:∫

Z \Rφ

((1 x0 1

)g

)dx = 0





Automorphic forms as group representations III

A function f on G (A) is automorphic if it satisfies the followingconditions:

1 f (g∞, g0) is smooth: infinitely-differentiable function in g∞and uniformly locally constant in g0.

2 f (γg) = f (g) for all γ ∈ G (k) (left G (k)-invariance)

3 f is right K -finite, that is, the right translates of f byelements of K span a finite dimensional space of functions,

4 f is z-finite, where z is the center of universal envelopingalgebra U(g) of the Lie algebra g = Lie(G∞).

5 f is of moderate growth (slowly increasing): there is an integern and a constant C such that for all g , |f (g)| ≤ C‖g‖n





Automorphic forms as group representations IV

If an automorphic form f also satisfies the condition

6 For every parabolic subgroup P of G with unipotent radical Nfor almost all g ∈ G (A)∫

N(k)\N(A)f (ng) dn = 0

then f is said to be a cuspform.

An irreducible unitary representation of G (A) is automorphic(respectively cuspidal) if it is isomorphic to a subquotient of arepresentation of G (A) in the space of automorphic forms(respectively cusp forms) on G (A).





Automorphic L−functions Iπ: irreducible automorphic cuspidal repr. of G (A).

π =⊗v

πv

where πv is an irreducible representation of Gv , almost everywherelocally spherical (unramified).S : the archimedean places together with all finite places v atwhich πv is not spherical.LG : L−group of G . Some examples:

G LG

GLn GLn(C)Spn SO2n+1(C)

SO2n+1 Sp2n(C)





Automorphic L−functions II

r : an algebraic representation of LG .For each v 6∈ S , let λv (πv ) be the conjugacy class in LGv

associated to the restriction of π to Gv . Define the local factors by

L(s, πv , rv ) =1

det(1− r(λ(πv )) q−sv )

where qv is the order of the residue field of kv . The (restricted)global L-function is the infinite product

LS(s, π, r) =∏v 6∈S

L(s, πv , rv )





Automorphic L−functions III

Theorem (Langlands)

Let π be an irreducible admissible representation unitarizablerepresentation of G (A) and r be a representation of LG . ThenLS(s, π, r) converges absolutely for Re(s) sufficently large.





A Sample of References I

Classical Theory

Topics in Classical Automorphic Forms, Henryk Iwaniec

Modular Forms Miyake

Diamond-Shurman A First Course in Modular Forms

Adelic Theory

Automorphic Forms onx Adele Groups, Gelbart

Tate’s thesis

An Introduction to the LANGLANDS PROGRAM, Bernstein,Gelbart (It starts from the classical theory to automorphictheory)

Some other useful things





A Sample of References II

Garrett’s web page:http://www-users.math.umn.edu/ garrett/m/v/ (many usefulnotes, papers)

Riemann’s Zeta Function and Beyond, Gelbart and Miller

An elementary introduction to the Langlands program,Gelbart.

Some more theoretical papers

R.P. Langlands, Euler Products, Yale University Press, JamesK. Whitmore Lectures, 1967.

R.P. Langlands, On the Functional Equations Satisfied byEisenstein Series, Lecture Notes in Mathematics no. 544,Springer-Verlag, New York, 1976.


http://www-users.math.umn.edu/~garrett/m/v/




A Sample of References III

A. Borel and W. Casselman, Automorphic Forms,Representations, and L-Functions: Symposium in PureMathematics. Volume 33. (1979)

P.B. Garrett, Pullback of Eisenstein series; applications, inAutomorphic Forms of Several Variables, ed. I Satake and Y.Morita, Birkhauser, Boston, 1984.

P.B. Garrett, Decomposition of Eisenstein series: Rankin tripleproducts. Annals of Mat. 125 (1987), pp. 209-237.

S. Gelbart, I. Piatetski-Shapiro, S. Rallis, ExplicitConstructions of Automorphic L-functions, Lecture Notes inMathematics no. 1254, Springer, New York, 1987.

S. Gelbart and F. Shahidi, Analytic Properties of AutomorphicL-functions, Academic Press, New York, 1988.





A Sample of References IV

D. Lanphier and C. Urtis, Arithmeticity of holomorphiccuspforms on Hermitian symmetric domains, Journal ofNumber Theory 151 (2015) 230-262.

S. Yamana, L-Functions and theta correspondence for classicalgroups, Inventiones Mathematicae, (2014) 196: 651-732.

S. Yamana, On the Siegel-Weil formula for quaternionicunitary groups, Amer. J. Math., 135(5), 2013, 1383-1432.

C. Urtis, Special values of L-functions by aSiegel-Weil-Kudla-Rallis formula, J. Number Theory,125(1):149-181, 2007.

C. Urtis, Poles of L-Functions on Quaternion Groups Chin.Ann. Math., 35B(4), 2014, 519-526.


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