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In mathematics, Maass forms or Maass wave forms are studied in the theory of automorphic forms. Maass forms are complex-valued smooth functions of the upper half plane, which transform in a similar way under the operation of a discrete subgroup of as modular forms. They are Eigenforms of the hyperbolic Laplace Operator defined on and satisfy certain growth conditions at the cusps of a fundamental domain of . In contrast to the modular forms the Maass forms need not be holomorphic. They were studied first by Hans Maass in 1949.

Contents

General remarksEdit

The group

 

operates on the upper half plane

 

by fractional linear transformations :

 

It can be extended to an operation on   by defining :

 
 

The Radon measure

 

defined on   is invariant under the operation of  .

Let   be a discrete subgroup of  . A fundamental domain for   is an open set  , so that there exists a system of representatives   of   with

 

A fundamental domain for the modular group   is given by

 

(see Modular form).

A function   is called  -invariant, if   holds for all   and all  .

For every measurable,  -invariant function   the equation

 

holds. Here the measure   on the right side of the equation is the induced measure on the quotient  

Classic Maass formsEdit

Definition of the hyperbolic Laplace operatorEdit

The hyperbolic Laplace operator on   is defined as

 
 

Definition of a Maass formEdit

A Maass form for the group   is a complex-valued smooth function   on   satisfying

 
 
 

If

 

we call   Maass cusp form.

Relation between Maass forms and Dirichlet seriesEdit

Let   be a Maass form. Since  ,   has a Fourier expansion of the form

  ,

with coefficient functions  .

It is easy to show that   is Maass cusp form if and only if  .

We can calculate the coefficient functions in a precise way. For this we need the Bessel function  .

Definition: The Bessel function   is defined as

 .

The integral converges locally uniformly absolutely for   in   and the inequality

  holds for all  .

Therefore,   decreases exponentially for  . Furthermore, we have   for all   ,  .

Theorem : The Fourier coefficients of a Maass formEdit

Let   be the eigenvalue of the Maass form f corresponding to . There is a  , unique up to sign, such that  . Then the Fourier coefficients of   are

 

if  . If   we get

 

Proof: We have  . By the definition of the Fourier coefficients we get

 

for  .

Together it follows that

 

for  .

In (1) we used that the nth Fourier coefficient of   is   for the first summation term. In the second term we changed the order of integration and differentiation, which is allowed since f is smooth in y . We get a linear differential equation of second degree :

 

For   one can show, that for every solution   there exist unique coefficients   with the property   .

For   every solution   is of the form

 

for unique  . Here   and   are Bessel functions.

The Bessel functions   grow exponentially, while the Bessel functions   decrease exponentially. Together with the polynomial growth condition 3) we get   (also  ) for a unique  

Even and odd Maass forms : Let  . Then i operates on all functions   by   and commutes with the hyperbolic Laplacian. A Maass form   is called even, if   and odd if  . If f is a Maass form, then   is an even Maass form and   an odd Maass form and it holds that  .

Theorem: The L-Function of a Maass formEdit

Let   be a Maass cusp form. We define the L-function L of   as

  .

Then the series   converges for   and we can continue it to a whole function on   .

If f is even or odd we get

 .

Here   if   is even and   if   is odd. Then   satisfies the functional equation

  .

Example: The non-holomorphic Eisenstein-series EEdit

The non-holomorphic Eisenstein-series is defined for   and   as

 

where   is the Gamma function.

The series converges absolutely in   for  and locally uniformly in  , since one can show, that the series   converges absolutely in   , if  . More precisely it converges uniformly on every set  , for every compact set   and every  .

Theorem: E is a Maass formEdit

We only show   - invariance and the differential equation. A proof of the smoothness can be found in Deitmar or Bump. The growth condition follows from the Fourier expansion of the Eisenstein series.

We will first show the   - invariance. Let

 

be the stabilizer group   corresponding to the operation of   on  .

Proposition: E is   - invariantEdit

(a) Let  . Then   converges absolutely in   for   and it holds that   .

(b) We have   for all  .

Proof:

(a): For   it holds that  . Therefore, we obtain

 

That proves the absolute convergence in   for .

Furthermore, it follows that

 

since the map   is a bijection.

(a) follows.

(b): For   we get  .

Together with (a),   is also invariant under  .  

Proposition : E is an eigenform of the hyperbolic Laplace operatorEdit

We need the following Lemma :

Lemma:   commutes with the operation of   on  . More precisely  

 

holds for all  

Proof: The group   is generated by the elements of the form   with   and   with   and  . One calculates the claim for these generators and obtains the claim for all  .  

Since   it is sufficient to show the differential equation for  .

It holds that

 

Furthermore, one has

 .

Since the Laplace Operator commutes with the Operation of  , we get

  and so  .

for all  .

Therefore, the differential equation holds for E in  . In order to obtain the claim for all  , consider the function  . By explicitly calculating the Fourier expansion of this function, we get that it is meromorphic. Since it vanishes for  , it must be the zero function by the Identity theorem.

Theorem : The Fourier-expansion of EEdit

The nonholomorphic Eisenstein series has a Fourier expansion

 

where

 

 .

If  ,   has a meromorphic continuation on  . It is holomorphic except for simple poles at  .

The Eisenstein series satisfies the functional equation

 

for all  .

Locally uniformly in   the growth condition

 

holds, where  

The meromorphic continuation of E is very important in the spectral theory of the hyperbolic Laplace operator.

Maas forms of weight kEdit

Congruence subgroupsEdit

For   let   be the kernel of the canonical projection

 .

We call   principal congruence subgroup of level  . A subgroup   is called congruence subgroup, if there exists  , so that  . All congruence subgroups are discrete.

Let  . For a congruence subgroup  , let   be the image of   in  . If S is a system of representatives of  , then

 

is a fundamental domain for  . The set   is uniquely determined by the fundamental domain  . Furthermore,   is finite.

The points   for   are called cusps of the fundamental domain  . They are a subset of  .

For every cusp   there exists   with  .

Definition: Maass forms of weight kEdit

Let   be a congruence subgroup and  .

We define the hyperbolic Laplace operator   of weight   as

  ,

 

This is a generalization of the hyperbolic Laplace operator  .

We define an operation of   on   by

 

where  .

It can be shown that

 

holds for all  ,   and every  .

Therefore,   operates on the vector space

 .

Definition : A Maass form of weight   for the group   is a function   that is an eigenfunction of   and is of moderate growth at the cusps.

The term moderate growth at cusps will need clarification :

Let   be a congruence subgroup. Then   is a cusp and we say that a function   is of moderate growth at  , if   is bounded by a polynomial in y for  . Let   be another cusp . Then there exists   with  . Let  . One calculates that   is an element of  , where   is the congruence subgroup  . We say   is of moderate growth at the cusp  , if   is of moderate growth at  .

If   contains a principal congruence subgroup of level  , we say that   is cuspidal at infinity, if

  holds for all   .

We say that   is cuspidal at the cusp   if   is cuspidal at infinity.

If   is cuspidal at every cusp, we call   a cusp form.

Cuspidal Maass forms are called Maass cusp forms.

We give a simple example of a Maass form of weight   for the modular group :

Example : Let   be a modular form of weight   for the group  . Then   is a Maass form of weight   for the group  .

The spectral problemEdit

Let   be a congruence subgroup of   and let   be the vector space of all measurable functions   with   for all   satisfying

 

modulo functions with  . The integral is well definded, since the function   is   - invariant. This is a Hilbert space with inner product

 

The operator   can be defined in a vector space   which is dense in  . There   is a positive semidefinite symmetric operator. It can be shown, that there exists a unique self-adjoint continuation on   .

We define   as the space of all cusp forms  .

Then   operates on   and has a discrete spectrum. The spectrum belonging to the orthogonal complement has a continuous part and can be described with the help of (modified) non-holomorphic Eisenstein series, their meromorphic continuations and their residues. (See Bump or Iwaniec) .

If   is a discrete (torsionfree) subgroup of  , so that the quotient   is compact, the spectral problem simplifies. This is because a discrete cocompact subgroup has no cusps. Here all of the space   is a sum of eigenspaces.

Embedding into the space  Edit

  is a unimodular locally compact group with the topology of  .

Let   be a congruence subgroup. Since   is discrete in  ,   is closed in  . The group   is unimodular and since the counting measure is a Haar-measure on the discrete group  ,   is also unimodular. By the Quotient Integral Formula there exists a   - right-invariant Radon measure   on the locally compact space  . Let   be the corresponding   - space.

The space   decomposes into a Hilbert space direct sum

 

where  .

The Hilbert-space   can be embedded isometrically into the Hilbert space  . The isometry is given by the map

 

Therefore, all Maass cusp forms for the congruence group   can be thought of as elements of  .

  is a Hilbert space carrying an operation of the group  , the so-called right regular representation :

 

One can easily show, that   is a unitary representation of   on the Hilbert space  . One is interested in a decomposition into irreducible suprepresentations. This is only possible if   is cocompact. If not, there is also a continuous Hilbert-integral part. The interesting part is, that the solution of this problem also solves the spectral problem of Maass forms. (see Bump, C. 2.3)

Maass cusp formEdit

A Maass cusp form, a subset of Maass forms, is a function on the upper half-plane that transforms like a modular form but need not be holomorphic. They were first studied by Hans Maass in Maass (1949).

DefinitionEdit

Let k be an integer, s be a complex number, and Γ be a discrete subgroup of SL2(R). A Maass form of weight k for Γ with Laplace eigenvalue s is a smooth function from the upper half-plane to the complex numbers satisfying the following conditions:

  • For all   and all  , we have  .
  • We have  , where   is the weight k hyperbolic Laplacian defined as
 
  • The function ƒ is of at most polynomial growth at cusps.

A weak Maass form is defined similarly but with the third condition replaced by "The function ƒ has at most linear exponential growth at cusps". Moreover, ƒ is said to be harmonic if it is annihilated by the Laplacian operator.

Major resultsEdit

Let ƒ be a weight 0 Maass cusp form. Its normalized Fourier coefficient at a prime p is bounded by p7/64 + p−7/64. This theorem is due to Henry Kim and Peter Sarnak. It is an approximation toward Ramanujan-Petersson conjecture.

Higher dimensionsEdit

Maass cusp forms can be regarded as automorphic forms on GL(2). It is natural to define Maass cusp forms on GL(n) as spherical automorphic forms on GL(n) over the rational number field. Their existence is proved by Miller, Mueller, etc.

Automorphic representations of the adele groupEdit

The Group  Edit

Let   be a commutative ring with unit and let   be the group of   matrices with entries in   and invertible determinant . Let   be the ring of rational adeles,   the ring of the finite (rational) adeles and for a prime number   let   be the field of p-adic numbers. Furthermore, let   be the ring of the p-adic integers (see Adele ring). Define  . Both   and   are locally compact unimodular groups if one equips them with the subspace topologies of   respectively  . The group   is isomorphic to the group  . Here the product is the resctricted product  , concerning the compact, open subgroups   of  . Then   locally compact group, if we equip it with the restricted product topology.

The group   is isomorphic to the group

 

and is a locally compact group with the product topology, since   and   are both locally compact.

Let   be the ring  . The subgroup

 

is a maximal compact, open subgroup of   and can be thought of as a subgroup of  , when we consider the embedding  .

We define   as the center of  , that means   is the group of all diagonal matrices of the form  , where  . We think of   as a subgroup of   since we can embed the group by  .

The group   is embedded diagonally in  , which is possible, since all four entries of a   can only have finite amount of prime divisors and therefore   for all but finitely many prime numbers  .

Let   be the group of all   with  . (see Adele Ring for a definition of the absolute value of an Idele). One can easily calculate, that   is a subgroup of  .

With the one-to-one map   we can identify the groups   and   with each other.

The group   is dense in   and discrete in  . The Quotient   is not compact but has finite Haar-measure.

Therefore,   is a lattice of   , similar to the classical case of the modular group and  . By harmonic analysis one also gets that   is unimodular.

Adelisation of CuspformsEdit

We now want to embed the classical Maass cusp forms of weight 0 for the modular group into  . This can be achieved with the "strong approximationtheorem", which states, that the map

 

is a   - equivariant homeomorphism. So we get

 

and furthermore

 .

Maass cuspforms of weight 0 for modular group can be embedded into

 .

By the strong approximation theorem this space is unitary isomorphic to

 

which is a subspace of  .

In the same way one can embed the classical holomorphic cusp forms. With a small generalization of the approximation theorem, one can embed all Maass cusp forms (as well as the holomorphic cuspforms) of any weight for any congruence subgroup   in  .

We call   the space of automorphic forms of the adele group.

Cusp forms of the adele groupEdit

Let   be a Ring and let   be the group of all  , where  . This group is isomorphic to the additive group of R.

We call a function   cusp form, if

 

holds for almost all . Let   (or just  ) be the vectorspace of these cusp forms.   is a closed subspace of   and it is invariant under the right regular representation of  .

One is again intereseted in a decomposition of   into irreducible closed subspaces .

We have the following theorem :

The space   decomposes in a direct sum of irreducible Hilbert-spaces with finite multiplicities   :

 

The calculation of these multiplicities   is one of the most important and most difficult problems in the theory of automorphic forms.

Cuspial representations of the adele groupEdit

An irreducible representation   of the group   is called cuspidal, if it is isomorphic to a subrepresentation of   ist.

An irreducible representation   of the group   is called admissible if there exists a compact subgroup   of  , so that   for all  .

One can show, that every cuspidal representation is admissible.

The admissibility is needed to proof the so-called Tensorprodukt-Theorem anzuwenden, which says, that every irreducible, unitary and admissible representation of the group   is isomorphic to an infinite tensor product

 . The   are irreducible representations of the group  . Almost all of them need to be umramified.

(A representation   of the group     is called unramified, if the vector space

 

is not the zero space.)

A construction of an infinite tensor product can be found in Deitmar,C.7.

Automorphic L-FunctionsEdit

Let   be an irreducible, admissible unitary representation of  . By the tensor product theorem,   is of the form   (see cuspidal representations of the adele group)

Let   be a finite set of places containing   and all ramified places . One defines the global Hecke - function of   as

 

where   is a so-called local L - function of the local representation  . A construction of local L - functions can be found in Deitmar C. 8.2.

If   is a cuspidal representation, the L-function   has a meromorphic continuation on  . This is possible, since  , satisfies certain functional equations.

See alsoEdit

ReferencesEdit

  • Bringmann, Kathrin; Folsom, Amanda (2014), "Almost harmonic Maass forms and Kac–Wakimoto characters", Journal für die Reine und Angewandte Mathematik, 694: 179–202, arXiv:1112.4726, doi:10.1515/crelle-2012-0102, MR 3259042
  • Bump, Daniel (1997), Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, 55, Cambridge University Press, doi:10.1017/CBO9780511609572, ISBN 978-0-521-55098-7, MR 1431508
  • Anton Deitmar: Automorphe Formen. Springer, Berlin/ Heidelberg u. a. 2010, ISBN 978-3-642-12389-4.
  • Duke, W.; Friedlander, J. B.; Iwaniec, H. (2002), "The subconvexity problem for Artin L-functions", Inventiones Mathematicae, 149 (3): 489–577, doi:10.1007/s002220200223, MR 1923476
  • Henryk Iwaniec : Spectral Methods of Automorphic Forms (Graduate Studies in Mathematics). American Mathematical Society; Auflage: 2. (November 2002), ISBN 978-0821831601.
  • Maass, Hans (1949), "Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen", Mathematische Annalen, 121: 141–183, doi:10.1007/BF01329622, MR 0031519