Maass wave form


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.

General remarks

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
.
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 forms

Definition of the hyperbolic Laplace operator

The hyperbolic Laplace operator on is defined as

Definition of a Maass form

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 series

Let be a Maass form. Since
we have:
Therefore 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.
Proof: We have
By the definition of the Fourier coefficients we get
for
Together it follows that
for
In 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 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 form

Let
be a Maass cusp form. We define the L-function of as
Then the series converges for and we can continue it to a whole function on .
If 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

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

''E'' is a Maass form

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.
Proof. Define:
converges absolutely in for and
Since
we obtain
That proves the absolute convergence in for
Furthermore, it follows that
since the map
is a bijection follows.
We have for all.
For we get
Together with, is also invariant under.
We need the following Lemma :
Proof: The group is generated by the elements of the form
One calculates the claim for these generators and obtains the claim for all.
Since it is sufficient to show the differential equation for We have:
Furthermore, one has
Since the Laplace Operator commutes with the Operation of, we get
and so
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.

The Fourier-expansion of ''E''

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 ''k''

Congruence subgroups

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.

Maass forms of weight ''k''

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 is a function that is an eigenfunction of and is of moderate growth at the cusps.
The term moderate growth at cusps needs clarification. Infinity is a cusp for a function is of moderate growth at if is bounded by a polynomial in y as. Let be another cusp. Then there exists with. Let. Then, where is the congruence subgroup. We say is of moderate growth at the cusp, if is of moderate growth at.
Definition. If contains a principal congruence subgroup of level, we say that is cuspidal at infinity, if
We say that is cuspidal at the cusp if is cuspidal at infinity. If is cuspidal at every cusp, we call a cusp form.
We give a simple example of a Maass form of weight for the modular group:
Example. Let be a modular form of even weight for Then is a Maass form of weight for the group.

The spectral problem

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 defined, 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
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 non-holomorphic Eisenstein series, their meromorphic continuations and their residues..
If is a discrete 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 L^{2}(\Gamma \backslash G)

is a locally compact unimodular group with the topology of Let be a congruence subgroup. Since is discrete in, it is closed in as well. 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. This space decomposes into a Hilbert space direct sum:
where
and
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 subrepresentations. 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.

Maass cusp form

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.

Definition

Let k be an integer, s be a complex number, and Γ be a discrete subgroup of SL2. 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:
A weak Maass form is defined similarly but with the third condition replaced by "The function has at most linear exponential growth at cusps". Moreover, is said to be harmonic if it is annihilated by the Laplacian operator.

Major results

Let 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 dimensions

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

Automorphic representations of the adele group

The group \mathrm{GL}_2(\mathbb{A})

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 adeles and for a prime number let be the field of p-adic numbers. Furthermore, let be the ring of the p-adic integers. Define. Both and are locally compact unimodular groups if one equips them with the subspace topologies of respectively. Then:
The right side is the restricted 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
and is a locally compact group with the product topology, since and are both locally compact.
Let
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.. 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

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 approximation theorem", 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 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

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 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 interested 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

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

Automorphic L-functions

Let be an irreducible, admissible unitary representation of. By the tensor product theorem, is of the form
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.