- General remarks
- Classic Maass wave forms Definition of the hyperbolic Laplace operator Definition of a Maass wave form Relation between Maass wave forms and Dirichlet series Theorem : The Fourier coefficients of a Maass wave form Theorem: The L-Function of a Maass wave form
- Example: The non-holomorphic Eisenstein-series E Theorem: E is a Maass waveform Proposition: E is
- invariant Proposition : E is an eigenform of the hyperbolic Laplace operator Theorem : The Fourier-expansion of E - Maas wave forms of weight k Congruence subgroups Definition: Maass wave forms of weight The spectral problem Embedding into the space
- Maass cusp form Definition Major results Higher dimensions
- Automorphic representations of the adele group The Group
Adelisation of Cuspforms Cusp forms of the adele group Cuspial representations of the adele group Automorphic L-Functions - See also
- References
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
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
Definition of the hyperbolic Laplace operator
The hyperbolic Laplace operator on is defined as
Definition of a Maass wave form
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
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
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
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
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
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
(a) Let 
. Then 
converges absolutely in for and it holds that
.
(b) We have 
for all 
.
(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
We need the following Lemma :
Lemma: commutes with the operation of on 
. More precisely 
holds for all
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
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
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 
.
Definition: Maass wave forms of weight
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
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
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)
Maass cusp form
A Maass cusp form, a subset of Maass wave 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 {{harvtxt|Maass|1949}}.
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:
- 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 results
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 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
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
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
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
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
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.
See also
- Harmonic Maass form
- Mock modular form
- Real analytic Eisenstein series
- Automorphic form
- Modular form
- Voronoi formula
References
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- {{Citation | last1=Bump | first1=Daniel | authorlink = Daniel Bump | title=Automorphic forms and representations | publisher=Cambridge University Press | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-55098-7 |mr=1431508 | year=1997 | volume=55 | doi=10.1017/CBO9780511609572}}
- Anton Deitmar: Automorphe Formen. Springer, Berlin/ Heidelberg u. a. 2010, {{ISBN|978-3-642-12389-4}}.
- {{citation|last1=Duke|first1=W.|author1-link=William Duke (mathematician)|last2=Friedlander|first2=J. B.|author2-link=John Friedlander|last3=Iwaniec|first3=H.|author3-link=Henryk Iwaniec|doi=10.1007/s002220200223|issue=3|journal=Inventiones Mathematicae|mr=1923476|pages=489–577|title=The subconvexity problem for Artin {{mvar|L}}-functions|volume=149|year=2002}}
- Henryk Iwaniec : Spectral Methods of Automorphic Forms (Graduate Studies in Mathematics). American Mathematical Society; Auflage: 2. (November 2002), {{ISBN|978-0821831601}}.
- {{Citation | last1=Maass | first1=Hans | authorlink=Hans Maass|title=Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen | doi=10.1007/BF01329622 |mr=0031519 | year=1949 | journal=Mathematische Annalen | volume=121 | pages=141–183}}
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