Maass wave form

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

and satisfy certain growth conditions at the cusps of a fundamental domain of

is defined as The integral converges locally uniformly absolutely for

In the second term we changed the order of integration and differentiation, which is allowed since f is smooth in y .

satisfies the functional equation The non-holomorphic Eisenstein-series is defined for

The growth condition follows from the Fourier expansion of the Eisenstein series.

By explicitly calculating the Fourier expansion of this function, we get that it is meromorphic.

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

The term moderate growth at cusps needs clarification.

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.

is unimodular and since the counting measure is a Haar-measure on the discrete group

The isometry is given by the map Therefore, all Maass cusp forms for the congruence group

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: A weak Maass form is defined similarly but with the third condition replaced by "The function

This theorem is due to Henry Kim and Peter Sarnak.

It is natural to define Maass cusp forms on GL(n) as spherical automorphic forms on GL(n) over the rational number field.

the ring of the finite (rational) adeles and for a prime number

are locally compact unimodular groups if one equips them with the subspace topologies of

Then: The right side is the restricted product, concerning the compact, open subgroups

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

is isomorphic to and is a locally compact group with the product topology, since

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

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

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

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

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

is called admissible if there exists a compact subgroup

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

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