Maass wave form

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In mathematics, Maass wave forms or Maass forms are studied in the theory of automorphic forms. Maass wave 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 wave forms need not be holomorphic. They were studied first by Hans Maass in 1949.

General remarks[edit]

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 wave forms[edit]

Definition of the hyperbolic Laplace operator[edit]

The hyperbolic Laplace operator on is defined as

Definition of a Maass wave form[edit]

A Maass wave form for the group is a complex-valued smooth function on so that

If

we call Maass cusp form.

Relation between Maass wave forms and Dirichlet series[edit]

Let be a Maass wave 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 wave form[edit]

Let be the eigenvalue of the Maass wave form f corresponding to. There is a which is 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 Waveforms : Let . Then i operates on all functions by . i commutes with the hyperbolic Laplacian. A Maass wave form is called even, if and odd if. If f is a Maass wave form, then is an even Maass wave form and an odd Maass wave form and it holds that .

Theorem: The L-Function of a Maass wave form[edit]

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 E[edit]

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 waveform[edit]

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 theorem of the Fourier-expansion of E.

We will first show the - invariance. Let

be the stabilizer group corresponding to the operation of on . Then the following Lemma holds :

Lemma: The map

is a bijection.

Proposition: E is - invariant[edit]

(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

by using the Lemma.

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 operator[edit]

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 E[edit]

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 wave forms of weight k[edit]

Congruence subgroups[edit]

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 wave forms of weight[edit]

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 wave form of weight for the group is a function , which is an Eigenform 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 aus is of moderate growth at , if is bounded by a polynomial in y for . Let be another cusp . Then there exists ein 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 cusp form.

Cuspidal Maass wave forms are called Maass cusp forms.

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

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

The spectral problem[edit]

Let be a congruence subgroup of .

and let be the vector space of all measurable functions with for all .

Furthermore, define the vector space

modulo functions with . The integral is well definded, since the function is - invariant.

a Hilbert space with scalar 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 selfadjoint 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 wave forms. (see Bump, C. 2.3)

Automorpic representations of the adele group[edit]

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 Cuspforms[edit]

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 group[edit]

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 group[edit]

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-Functions[edit]

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-Funktion has a meromorphic continuation on . This is possible, since , satisfies certain functional equations.

References[edit]