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Wikipedia / 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 as modular forms. They are eigenforms defined on the upper half plane and satisfy certain growth conditions at the cusps of a fundamental domain. In contrast to modular forms, 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:

by defining:

The Radon measure

.

with

is given by

(see Modular form).

.

For every measurable the equation

Classic Maass forms

Definition of the hyperbolic Laplace operator

The hyperbolic Laplace operator is defined as

Definition of a Maass form

A Maass form satisfying

If

Maass cusp form.

Relation between Maass forms and Dirichlet series

be a Maass form. Since

we have:

has a Fourier expansion of the form

.

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

Definition: is defined as

and the inequality

.

.

Proof: We have

By the definition of the Fourier coefficients we get

Together it follows that

In (1) we used that the n 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:

has coefficients of the form

are Bessel functions.

. Q.E.D.

Even and odd Maass forms:. Then i.

Theorem: The L-function of a Maass form

Let

as

.

is even or odd we get

satisfies the functional equation

Example: The non-holomorphic Eisenstein-series E

as

is the Gamma function.

, since one can show, that the series

.

E is a Maass form

-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.

-invariance. Let

.

  1. Proposition. E-invariant.

Proof. Define:

Since

we obtain

That proves the absolute convergence

Furthermore, it follows that

since the map

is a bijection (a) follows.

.

we get

. Q.E.D.

  1. Proposition. E is an eigenform of the hyperbolic Laplace operator

We need the following Lemma:

  1. Lemma:

Proof: is generated by the elements of the form

. Q.E.D.

. We have:

Furthermore, one has

, we get

and so

Therefore, the differential equation holds for E. By explicitly calculating the Fourier expansion of this function, we get that it is meromorphic, it must be the zero function by the identity theorem.

The Fourier-expansion of E

The nonholomorphic Eisenstein series has a Fourier expansion

where

The Eisenstein series satisfies the functional equation

.

the growth condition

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

Maass forms of weight k

Congruence subgroups

be the kernel of the canonical projection

principal congruence subgroup. All congruence subgroups are discrete.

Let

. If S, then

is finite.

.

.

Maass forms of weight k

as

.

by

where

It can be shown that

.

operates on the vector space

  1. .

Definition. A Maass form and is of moderate growth at the cusps.

is bounded by a polynomial in y.

Definition. is cuspidal at infinity, if

a cusp form.

for the modular group:

Example..

The spectral problem

satisfying

-invariant. This is a Hilbert space with inner product

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).

is a sum of eigenspaces.

Embedding into the space L2(Γ \ G)

is a locally compact unimodular group-space. This space decomposes into a Hilbert space direct sum:

where

and

. The isometry is given by the map

.

, the so-called right regular representation:

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 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(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:

  • is the weight k

  • is of at most polynomial growth at cusps.

A weak Maass form is said to be harmonic if it is annihilated by the Laplacian operator.

Major results

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(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 group

The group GL2(A)

be the field of p be the ring of the p-adic integers (see Adele ring. Then:

locally compact group, if we equip it with the restricted product topology.

is isomorphic to

are both locally compact.

Let

The subgroup

.

.

.

.

with each other.

is not compact but has finite Haar-measure.

is unimodular.

Adelisation of cuspforms

. This can be achieved with the "strong approximation theorem", which states that the map

-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

.

the space of automorphic forms of the adele group.

Cusp forms of the adele group

. This group is isomorphic to the additive group of R.

cusp form, if

into irreducible closed subspaces.

We have the following theorem:

:

is one of the most important and most difficult problems in the theory of automorphic forms.

Cuspidal representations of the adele group

.

.

One can show, that every cuspidal representation is admissible.

is isomorphic to an infinite tensor product

. Almost all of them need to be umramified.

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

(see cuspidal representations of the adele group)

as

. A construction of local L-functions can be found in Deitmar C. 8.2.

, satisfies certain functional equations.

See also

References