# Maass wave form

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 ${\displaystyle \Gamma }$ of ${\displaystyle Sl_{2}(\mathbb {R} )}$ as modular forms. They are Eigenforms of the hyperbolic Laplace Operator ${\displaystyle \Delta }$ defined on ${\displaystyle \mathbb {H} }$ and satisfy certain growth conditions at the cusps of a fundamental domain of ${\displaystyle \Gamma }$. 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

${\displaystyle G:=SL_{2}(\mathbb {R} )={\Bigg \{}{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\in M_{2}(\mathbb {R} ):ad-bc=1{\Bigg \}}}$

operates on the upper half plane

${\displaystyle \mathbb {H} =\{z\in \mathbb {C} :\operatorname {Im} (z)>0\}}$

by fractional linear transformations :

${\displaystyle {\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}.z:={\frac {az+b}{cz+d}}.}$

It can be extended to an operation on ${\displaystyle \mathbb {H} \cup \{\infty \}\cup \mathbb {\mathbb {R} } }$ by defining :

${\displaystyle {\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\cdot z:={\begin{cases}{\frac {az+b}{cz+d}},&{\text{if }}cz+d\neq 0,\\\infty ,&{\text{if }}cz+d=0,\end{cases}}}$
${\displaystyle {\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}.\infty :=\lim _{\operatorname {Im} (z)\rightarrow \infty }{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\cdot z={\begin{cases}{\frac {a}{c}},&{\text{if }}c\neq 0\\\infty ,&{\text{if }}c=0\end{cases}}}$

${\displaystyle d\mu :={\frac {dx\,dy}{y^{2}}}}$

defined on ${\displaystyle \mathbb {H} }$ is invariant under the operation of ${\displaystyle SL_{2}(\mathbb {R} )}$.

Let ${\displaystyle \Gamma }$ be a discrete subgroup of ${\displaystyle G}$. A fundamental domain for ${\displaystyle \Gamma }$ is an open set ${\displaystyle F\subset \mathbb {H} }$, so that there exists a system of representatives ${\displaystyle R}$ of ${\displaystyle \Gamma \setminus \mathbb {H} }$ with

${\displaystyle F\subset R\subset {\overline {F}}{\text{ and }}\mu ({\overline {F}}\setminus F)=0.}$

A fundamental domain for the modular group ${\displaystyle \Gamma (1):=SL_{2}(\mathbb {Z} )}$ is given by

${\displaystyle D:=\{z\in \mathbb {H} \mid \left|\operatorname {Re} (z)\right|<{\frac {1}{2}},\,|z|<1\}}$

(see Modular form).

A function ${\displaystyle f:\mathbb {H} \to \mathbb {C} }$ is called ${\displaystyle \Gamma }$-invariant, if ${\displaystyle f(\gamma \cdot z)=f(z)}$ holds for all ${\displaystyle \gamma \in \Gamma }$ and all ${\displaystyle z\in \mathbb {H} }$.

For every measurable, ${\displaystyle \Gamma }$-invariant function ${\displaystyle f\,:\,\mathbb {H} \to \mathbb {C} }$ the equation

${\displaystyle \int _{F}f\,d\mu =\int _{\Gamma \setminus \mathbb {H} }f\,d\mu ,}$

holds. Here the measure ${\displaystyle d\mu }$ on the right side of the equation is the induced measure on the quotient ${\displaystyle \Gamma \setminus \mathbb {H} .}$

## Classic Maass wave forms

### Definition of the hyperbolic Laplace operator

The hyperbolic Laplace operator on ${\displaystyle \mathbb {H} }$ is defined as

${\displaystyle \Delta :C^{\infty }(\mathbb {H} )\to C^{\infty }(\mathbb {H} ),}$
${\displaystyle \Delta (f)=-y^{2}\left({\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}\right)}$

### Definition of a Maass wave form

A Maass wave form for the group ${\displaystyle \Gamma (1):=SL_{2}(\mathbb {Z} )}$ is a complex-valued smooth function ${\displaystyle f}$ on ${\displaystyle \mathbb {H} }$ so that

${\displaystyle 1)\quad f(\gamma z)=f(z){\text{ for all }}\gamma \in \Gamma (1),\qquad z\in \mathbb {H} }$
${\displaystyle 2)\quad {\text{there exists }}\lambda \in \mathbb {C} {\text{ with }}\Delta (f)=\lambda f}$
${\displaystyle 3)\quad {\text{there exists }}N\in \mathbb {N} {\text{ with }}f(x+iy)={\mathcal {O}}(y^{N}){\text{ for }}y\geq 1}$

If

${\displaystyle \int _{0}^{1}f(z+t)\,dt=0{\text{ for all }}z\in \mathbb {H} }$

we call ${\displaystyle f}$ Maass cusp form.

### Relation between Maass wave forms and Dirichlet series

Let ${\displaystyle f}$ be a Maass wave form. Since ${\displaystyle \gamma :={\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}\in \Gamma (1)\,f(z)=f(\gamma .z)=f(z+1)\,\forall \,z\in \mathbb {H} }$, ${\displaystyle f}$ has a Fourier-expansion of the form

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f(x+iy)=\sum _{n=-\infty }^{\infty }a_{n}(y)e^{2\pi inx}}$ ,

with coefficient functions ${\displaystyle a_{n}\,,n\in \mathbb {N} }$.

It is easy to show that ${\displaystyle f}$ is Maass cusp form if and only if ${\displaystyle a_{0}(y)=0\,\,\,\forall y>0}$.

We can calculate the coefficient functions in a precise way. For this we need the Bessel function ${\displaystyle K_{v}}$.

Definition: The Bessel function ${\displaystyle K_{v}}$ is defined as

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,K_{s}(y):={\frac {1}{2}}\int _{0}^{\infty }e^{-y(t+t^{-1})/2}t^{s}{\frac {dt}{t}}\,\,,s\in \mathbb {C} \,,\,y>0}$.

The integral converges locally uniformly absolutely for ${\displaystyle y>0}$ in ${\displaystyle s\in \mathbb {C} }$ and the inequality

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,K_{s}(y)\leq e^{-y/2}K_{Re(s)}(2)}$ holds for all ${\displaystyle y>4}$.

Therefore, ${\displaystyle |K_{s}|}$ decreases exponentially for ${\displaystyle y\to \infty }$. Furthermore, we have ${\displaystyle K_{-s}(y)=K_{s}(y)}$ for all ${\displaystyle s\in \mathbb {C} }$ , ${\displaystyle y>0}$.

#### Theorem : The Fourier coefficients of a Maass wave form

Let ${\displaystyle \lambda \in \mathbb {C} }$ be the eigenvalue of the Maass wave form f corresponding to${\displaystyle \Delta }$. There is a ${\displaystyle \nu \in \mathbb {C} }$ which is unique up to sign such that ${\displaystyle \lambda ={\frac {1}{4}}-\nu ^{2}}$. Then the Fourier coefficients of ${\displaystyle f}$ are

${\displaystyle a_{n}(y)=c_{n}{\sqrt {y}}K_{\nu }(2\pi |n|y),\qquad c_{n}\in \mathbb {C} }$

if ${\displaystyle n\neq 0}$. If ${\displaystyle n=0}$ we get

${\displaystyle a_{0}(y)=c_{0}y^{{\frac {1}{2}}-\nu }+d_{0}y^{{\frac {1}{2}}+\nu }{\text{ with }}c_{0},\,d_{0}\in \mathbb {C} .}$

Proof: We have ${\displaystyle \Delta (f)=({\frac {1}{4}}-\nu ^{2})f}$. By the definition of the Fourier coefficients we get

${\displaystyle a_{n}=\int _{0}^{1}f(x+iy)e^{-2\pi inx}\,dx}$

for ${\displaystyle n\in \mathbb {Z} }$.

Together it follows that

{\displaystyle {\begin{aligned}&\left({\frac {1}{4}}-\nu ^{2}\right)a_{n}\\[4pt]={}&\int _{0}^{1}\left({\frac {1}{4}}-\nu ^{2}\right)f(x+iy)e^{-2\pi inx}\,dx\\[4pt]={}&\int _{0}^{1}(\Delta f)(x+iy)e^{-2\pi inx}\,dx\\[4pt]={}&-y^{2}\left(\int _{0}^{1}{\frac {\partial ^{2}f}{\partial x^{2}}}(x+iy)e^{-2\pi inx}\,dx+\int _{0}^{1}{\frac {\partial ^{2}f}{\partial y^{2}}}(x+iy)e^{-2\pi inx}\,dx\right)\\[4pt]{\overset {(1)}{=}}{}&-y^{2}(2\pi in)^{2}a_{n}(y)-y^{2}{\frac {\partial ^{2}}{\partial y^{2}}}\int _{0}^{1}f(x+iy)e^{-2\pi inx}\,dx\\[4pt]={}&-y^{2}(2\pi in)^{2}a_{n}(y)-y^{2}{\frac {\partial ^{2}}{\partial y^{2}}}a_{n}(y)\\[4pt]={}&4\pi ^{2}n^{2}y^{2}a_{n}(y)-y^{2}{\frac {\partial ^{2}}{\partial y^{2}}}a_{n}(y)\end{aligned}}}

for ${\displaystyle n\in \mathbb {Z} }$.

In (1) we used that the nth Fourier coefficient of ${\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}}$ is ${\displaystyle (2\pi in)^{2}a_{n}(y)}$ 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 :

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y^{2}{\frac {\partial ^{2}}{\partial y^{2}}}a_{n}(y)+({\frac {1}{4}}-\nu ^{2}-4\pi n^{2}y^{2})a_{n}(y)=0}$

For ${\displaystyle n=0}$ one can show, that for every solution ${\displaystyle f}$ there exist unique coefficients ${\displaystyle c_{0},d_{0}\in \mathbb {C} }$ with the property ${\displaystyle a_{0}(y)=c_{0}y^{{\frac {1}{2}}-\nu }+d_{0}y^{{\frac {1}{2}}+\nu }}$ .

For ${\displaystyle n\neq 0}$ every solution ${\displaystyle f}$ is of the form

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f(y)=c_{n}{\sqrt {y}}K_{v}(2\pi |n|y)+d_{n}{\sqrt {y}}I_{v}(2\pi |n|y)}$

for unique ${\displaystyle c_{n},d_{n}\in \mathbb {C} }$. Here ${\displaystyle K_{v}(s)}$ and ${\displaystyle I_{v}(s)}$ are Bessel functions.

The Bessel functions ${\displaystyle I_{v}}$ grow exponentially, while the Bessel functions ${\displaystyle K_{v}}$ decrease exponentially. Together with the polynomial growth condition 3) we get ${\displaystyle f\,\,\,}$ : ${\displaystyle \,\,\,\,\,\,a_{n}(y)=c_{n}{\sqrt {y}}K_{v}(2\pi |n|y)}$ (also ${\displaystyle d_{n}=0}$) for a unique ${\displaystyle c_{n}\in \mathbb {C} .\,\,\,\square }$

Even and odd Maass Waveforms : Let ${\displaystyle i(z):=-{\overline {z}}}$. Then i operates on all functions ${\displaystyle f:\mathbb {H} \to \mathbb {C} }$ by ${\displaystyle i(f):=f(i(z))}$. i commutes with the hyperbolic Laplacian. A Maass wave form ${\displaystyle f}$ is called even, if ${\displaystyle i(f)=f}$ and odd if${\displaystyle i(f)=-f}$. If f is a Maass wave form, then ${\displaystyle {\frac {1}{2}}(f+i(f))}$ is an even Maass wave form and ${\displaystyle {\frac {1}{2}}(f-i(f))}$ an odd Maass wave form and it holds that ${\displaystyle f={\frac {1}{2}}(f+i(f))+{\frac {1}{2}}(f-i(f))}$.

#### Theorem: The L-Function of a Maass wave form

Let ${\displaystyle f(x+iy)=\sum _{n\neq 0}c_{n}{\sqrt {y}}K_{\nu }(2\pi |n|y)e^{2\pi inx}}$ be a Maass cusp form. We define the L-function L of ${\displaystyle f}$ as

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L(s,f)=\sum _{n=1}^{\infty }c_{n}n^{-s}}$ .

Then the series ${\displaystyle L(s,f)}$ converges for ${\displaystyle Re(s)>{\frac {3}{2}}}$ and we can continue it to a whole function on ${\displaystyle \mathbb {C} }$ .

If f is even or odd we get

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Lambda (s,f):=\pi ^{-s}\Gamma ({\frac {s+\epsilon +\nu }{2}})\Gamma ({\frac {s+\epsilon -\nu }{2}})L(s,f)}$.

Here ${\displaystyle \epsilon =0}$ if ${\displaystyle f}$ is even and ${\displaystyle \epsilon =-1}$ if ${\displaystyle f}$ is odd. Then ${\displaystyle \Lambda }$ satisfies the functional equation

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Lambda (s,f)=(-1)^{\epsilon }\Lambda (1-s,f)}$ .

## Example: The non-holomorphic Eisenstein-series E

The non-holomorphic Eisenstein-series is defined for ${\displaystyle z\in \mathbb {H} }$ and ${\displaystyle s\in \mathbb {C} }$ as

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E(z,s):=\pi ^{-s}\Gamma (s){\frac {1}{2}}\sum _{(m,n)\neq (0,0)}{\frac {y^{s}}{|mz+n|^{2s}}}}$

where ${\displaystyle \Gamma (s)}$ is the Gamma function.

The series converges absolutely in ${\displaystyle z\in \mathbb {H} }$ for${\displaystyle Re(s)>1}$ and locally uniformly in ${\displaystyle \mathbb {H} \times \{Re(s)>1\}}$, since one can show, that the series ${\displaystyle S(z,s):=\sum _{(m,n)\neq (0,0)}{\frac {1}{|mz+n|^{s}}}}$ converges absolutely in ${\displaystyle z\in \mathbb {H} }$ , if ${\displaystyle Re(s)>2}$. More precisely it converges uniformly on every set ${\displaystyle K\times \{Re(s)\geq \alpha \}}$, for every compact set ${\displaystyle K\subset \mathbb {H} }$ and every ${\displaystyle \alpha >2}$.

### Theorem: E is a Maass waveform

We only show ${\displaystyle SL_{2}(\mathbb {Z} )}$ - 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 ${\displaystyle SL_{2}(\mathbb {Z} )}$ - invariance. Let

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma _{\infty }:=\pm {\begin{pmatrix}1&\mathbb {Z} \\0&1\\\end{pmatrix}}}$

be the stabilizer group ${\displaystyle \infty }$ corresponding to the operation of ${\displaystyle SL_{2}(\mathbb {Z} )}$ on ${\displaystyle \mathbb {H} \cup \{\infty \}}$. Then the following Lemma holds :

Lemma: The map

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma _{\infty }\backslash \Gamma \to \{\pm (x,y)\in \mathbb {Z} ^{2}/\{\pm 1\}:ggT(x,y)=1\}}$

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Gamma _{\infty }{\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\mapsto \pm (c,d)}$

is a bijection.

#### Proposition: E is ${\displaystyle \Gamma (1)}$ - invariant

(a) Let ${\displaystyle {\tilde {E}}(z,s):=\sum _{\gamma \in \Gamma _{\infty }\backslash \Gamma }Im(\gamma z)^{s}}$. Then ${\displaystyle {\tilde {E}}}$ converges absolutely in ${\displaystyle z\in \mathbb {H} }$ for ${\displaystyle Re(s)>1}$ and it holds that ${\displaystyle E(z,s)=\pi ^{-s}\Gamma (s)\zeta (2s){\tilde {E}}(z,s)}$ .

(b) We have ${\displaystyle E(\gamma z,s)=E(z,s)}$ for all ${\displaystyle \gamma \in \Gamma (1)}$.

Proof:

(a): For ${\displaystyle \gamma ={\begin{pmatrix}a&b\\c&d\\\end{pmatrix}}\in \Gamma (1)}$ it holds that ${\displaystyle Im(\gamma z)={\frac {Im(z)}{|cz+d|^{2}}}}$. Therefore, we obtain

${\displaystyle {\tilde {E}}(z,s)=\sum _{\gamma \in \Gamma _{\infty }\backslash \Gamma }Im(\gamma z)^{s}=\sum _{(c,d)=1\,mod\pm 1}{\frac {y^{s}}{|cz+d|^{2s}}}}$

by using the Lemma.

That proves the absolute convergence in ${\displaystyle z\in \mathbb {H} }$ for${\displaystyle Re(s)>1}$.

Furthermore, it follows that

${\displaystyle \zeta (2s){\tilde {E}}(z,s)=\sum _{n=1}^{\infty }n^{-s}\sum _{(c,d)=1\,mod\pm 1}{\frac {y^{s}}{|cz+d|^{2s}}}=\sum _{n=1}^{\infty }\sum _{(c,d)=1\,mod\pm 1}{\frac {y^{s}}{|ncz+nd|^{2s}}}=\sum _{(m,n)\neq (0,0)}{\frac {y^{s}}{|mz+n|^{2s}}},}$

since the map ${\displaystyle \mathbb {N} \times \{(x,y)\in \mathbb {Z} ^{2}-\{(0,0)\}:(x,y)=1\}\to \mathbb {Z} ^{2}-\{(0,0)\},(n,(x,y))\mapsto (nx,ny)}$ is a bijection.

(a) follows.

(b): For ${\displaystyle {\tilde {\gamma }}\in \Gamma (1)}$ we get ${\displaystyle {\tilde {E}}({\tilde {\gamma }}z,s)=\sum _{\gamma \in \Gamma _{\infty }\backslash \Gamma }Im({\tilde {\gamma }}\gamma z)^{s}=\sum _{\gamma \in \Gamma _{\infty }\backslash \Gamma }Im(\gamma z)^{s}={\tilde {E}}(\gamma z,s)}$.

Together with (a), ${\displaystyle E}$ is also invariant under ${\displaystyle \Gamma (1)}$. ${\displaystyle \,\,\,\,\square }$

#### Proposition : E is an eigenform of the hyperbolic Laplace operator

We need the following Lemma :

Lemma: ${\displaystyle \Delta }$ commutes with the operation of ${\displaystyle G}$ on ${\displaystyle C^{\infty }(\mathbb {H} )}$. More precisely ${\displaystyle g\in G}$

${\displaystyle L_{g}\Delta =\Delta L_{g}}$

holds for all ${\displaystyle g\in G}$

Proof: The group ${\displaystyle SL_{2}(\mathbb {R} )}$ is generated by the elements of the form ${\displaystyle {\begin{pmatrix}a&0\\0&{\frac {1}{a}}\\\end{pmatrix}}}$ with ${\displaystyle a\in \mathbb {R} ^{\times }}$ and ${\displaystyle {\begin{pmatrix}1&x\\0&1\\\end{pmatrix}}}$ with ${\displaystyle x\in \mathbb {R} }$ and ${\displaystyle S={\begin{pmatrix}0&-1\\1&0\\\end{pmatrix}}}$. One calculates the claim for these generators and obtains the claim for all ${\displaystyle g\in SL_{2}(\mathbb {R} )}$. ${\displaystyle \,\,\,\square }$

Since ${\displaystyle E(z,s)=\pi ^{-s}\Gamma (s)\zeta (2s){\tilde {E}}(z,s)}$ it is sufficient to show the differential equation for ${\displaystyle {\tilde {E}}}$.

It holds that

${\displaystyle \Delta {\tilde {E}}(z,s):=\Delta \sum _{\gamma \in \Gamma _{\infty }\backslash \Gamma }Im(\gamma z)^{s}=\sum _{\gamma \in \Gamma _{\infty }\backslash \Gamma }\Delta Im(\gamma z)^{s}}$

Furthermore, one has

${\displaystyle \Delta (Im(z)^{s})=\Delta (y^{s})=-y^{2}({\frac {\partial ^{2}y^{s}}{\partial x^{2}}}+{\frac {\partial ^{2}y^{s}}{\partial y^{2}}})=s(1-s)y^{s}}$.

Since the Laplace Operator commutes with the Operation of ${\displaystyle \Gamma (1)}$, we get

${\displaystyle \Delta (Im(\gamma z)^{s})=s(1-s)Im(\gamma z)^{s}}$ and so ${\displaystyle \Delta {\tilde {E}}(z,s)=s(1-s){\tilde {E}}(z,s)}$.

for all ${\displaystyle \gamma \in \Gamma (1)}$.

Therefore, the differential equation holds for E in ${\displaystyle Re(s)>3}$. In order to obtain the claim for all ${\displaystyle s\in \mathbb {C} }$, consider the function ${\displaystyle \Delta E(z,s)-s(1-s)E(z,s)}$. By explicitly calculating the Fourier expansion of this function, we get that it is meromorphic. Since it vanishes for ${\displaystyle Re(s)>3}$, it must be the zero function by the Identity theorem.

#### Theorem : The Fourier-expansion of E

The nonholomorphic Eisenstein series has a Fourier expansion

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E(z,s)=\sum _{n=-\infty }^{\infty }a_{n}(y,s)e^{2\pi inx}}$

where

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{0}(y,s)=\pi ^{-s}\Gamma (s)\zeta (2s)y^{s}+\pi ^{s-1}\Gamma (1-s)\zeta (2(1-s))y^{1-s}}$

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a_{n}(y,s)=2|n|^{2-{\frac {1}{2}}}\sigma _{1-2s}(|n|){\sqrt {y}}K_{s-{\frac {1}{2}}}(2\pi |n|y)\,,\,n\neq 0}$.

If ${\displaystyle z\in \mathbb {H} }$, ${\displaystyle E(z,s)}$ has a meromorphic continuation on ${\displaystyle \mathbb {C} }$. It is holomorphic except for simple poles at ${\displaystyle s=0\,,\,1}$.

The Eisenstein series satisfies the functional equation

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E(z,s)=E(z,1-s)}$

for all ${\displaystyle z\in \mathbb {H} }$.

Locally uniformly in ${\displaystyle x\in \mathbb {R} }$ the growth condition

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,E(x+iy,s)={\mathcal {0}}(y^{\sigma })}$

holds, where ${\displaystyle \sigma =\max(\operatorname {Re} (s),1-\operatorname {Re} (s)).}$

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 ${\displaystyle N\,\in \,\mathbb {N} }$ let ${\displaystyle \Gamma (N)}$ be the kernel of the canonical projection

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,SL_{2}(\mathbb {Z} )\to SL_{2}(\mathbb {Z} \,/\,N\mathbb {Z} )}$.

We call ${\displaystyle \Gamma (N)}$ principal congruence subgroup of level ${\displaystyle N}$. A subgroup ${\displaystyle \Gamma \,\subseteq \,SL_{2}(\mathbb {Z} )}$ is called congruence subgroup, if there exists ${\displaystyle N\,\in \,\mathbb {N} }$, so that ${\displaystyle \Gamma (N)\subseteq \Gamma }$. All congruence subgroups are discrete.

Let ${\displaystyle {\overline {\Gamma (1)}}:=\Gamma (1)\,/\,\pm \,1}$. For a congruence subgroup ${\displaystyle \Gamma }$ ,let ${\displaystyle {\overline {\Gamma }}}$ be the image of ${\displaystyle \Gamma }$ in ${\displaystyle {\overline {\Gamma (1)}}}$. If S is a system of representatives of ${\displaystyle {\overline {\Gamma }}\setminus {\overline {\Gamma (1)}}}$, then

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,SD=\bigcup _{\gamma \in S}\gamma D}$

is a fundamental domain for ${\displaystyle \Gamma }$. The set ${\displaystyle S}$ is uniquely determined by the fundamental domain ${\displaystyle SD}$. Furthermore, ${\displaystyle S}$ is finite.

The points ${\displaystyle \gamma \infty }$ for ${\displaystyle \gamma \in S}$ are called cusps of the fundamental domain ${\displaystyle SD}$. They are a subset of ${\displaystyle \mathbb {Q} \cup \{\infty \}}$.

For every cusp ${\displaystyle c}$ there exists ${\displaystyle \sigma \in \Gamma (1)}$ with ${\displaystyle \sigma \infty =c}$.

### Definition: Maass wave forms of weight

Let ${\displaystyle \Gamma }$ be a congruence subgroup and ${\displaystyle k\in \mathbb {Z} }$.

We define the hyperbolic Laplace operator ${\displaystyle \Delta _{k}}$ of weight ${\displaystyle k}$ as

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Delta _{k}:C^{\infty }(\mathbb {H} )\to C^{\infty }(\mathbb {H} )}$ ,

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Delta _{k}(f)=-y^{2}({\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}})+iky{\frac {\partial f}{\partial x}}}$

This is a generalization of the hyperbolic Laplace operator ${\displaystyle \Delta _{0}=\Delta }$.

We define an operation of ${\displaystyle SL_{2}(\mathbb {R} )}$ on ${\displaystyle C^{\infty }(\mathbb {H} )}$ by

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,f_{||k}g(z):=\left({\frac {cz+d}{|cz+d|}}\right)^{-k}f(g.z)}$

where ${\displaystyle z\in \mathbb {H} \,\,,\,\,g={\begin{pmatrix}\ast &\ast \\c&d\\\end{pmatrix}}\in SL_{2}(\mathbb {R} )\,\,,\,\,f\in C^{\infty }(\mathbb {H} )}$.

It can be shown, that

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\Delta _{k}f)_{||k}g=\Delta _{k}(f_{||k}g)}$

holds for all ${\displaystyle f\in C^{\infty }(\mathbb {H} )}$, ${\displaystyle k\in \mathbb {Z} }$ and every ${\displaystyle g\in SL_{2}(\mathbb {R} )}$.

Therefore, ${\displaystyle \Delta _{k}}$ operates on the vector space

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,C^{\infty }(\Gamma \setminus \mathbb {H} ,k):=\{f\in C^{\infty }(\mathbb {H} ):f_{||k}\gamma =f\,\,\,\forall \gamma \in \Gamma \}}$.

Definition : A Maass wave form of weight ${\displaystyle k\in \mathbb {Z} }$ for the group ${\displaystyle \Gamma }$ is a function ${\displaystyle f\in C^{\infty }(\Gamma \setminus \mathbb {H} ,k)}$, which is an Eigenform of ${\displaystyle \Delta _{k}}$ and is of moderate growth at the cusps.

The term moderate growth at cusps will need clarification :

Let ${\displaystyle \Gamma }$ be a congruence subgroup. Then ${\displaystyle \infty }$ is a cusp and we say that a function ${\displaystyle f}$ aus ${\displaystyle C^{\infty }(\Gamma \setminus \mathbb {H} ,k)}$is of moderate growth at ${\displaystyle \infty }$, if ${\displaystyle f(x+iy)}$ is bounded by a polynomial in y for ${\displaystyle y\to \infty }$. Let ${\displaystyle c\in \mathbb {Q} }$ be another cusp . Then there exists ein ${\displaystyle \theta \in SL_{2}(\mathbb {Z} )}$ with ${\displaystyle \theta (\infty )=c}$. Let ${\displaystyle f':=f_{||k}\theta }$. One calculates, that ${\displaystyle f'}$ is an element of${\displaystyle C^{\infty }(\Gamma '\setminus \mathbb {H} ,k)}$, where ${\displaystyle \Gamma '}$ is the congruence subgroup ${\displaystyle \theta ^{-1}\Gamma \theta }$. We say ${\displaystyle f}$ is of moderate growth at the cusp ${\displaystyle c}$, if ${\displaystyle f'}$ is of moderate growth at ${\displaystyle \infty }$.

If${\displaystyle \Gamma }$ contains a principal congruence subgroup of level ${\displaystyle N}$, we say that ${\displaystyle f}$ is cuspidal at infinity, if

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int _{0}^{N}\,f(z+u)\,du=0}$ holds for all ${\displaystyle z\in \mathbb {H} }$ .

We say that ${\displaystyle f}$ is cuspidal at the cusp ${\displaystyle c}$, if ${\displaystyle f'}$ is cuspidal at infinity.

If ${\displaystyle f}$ is cuspidal at every cusp, we call ${\displaystyle f}$ cusp form.

Cuspidal Maass wave forms are called Maass cusp forms.

We give a simple example of a Maass wave form of weight ${\displaystyle k>1}$ for the modular group :

Example : Let ${\displaystyle g:\mathbb {H} \to \mathbb {C} }$ be a modular form of weight ${\displaystyle k\in 2\mathbb {N} }$ for the group ${\displaystyle \Gamma (1)}$. Then ${\displaystyle f(z):=y^{\frac {k}{2}}g(z)}$ is a Maass wave form of weight ${\displaystyle k}$ for the group ${\displaystyle \Gamma (1)}$.

### The spectral problem

Let ${\displaystyle \Gamma }$ be a congruence subgroup of ${\displaystyle SL_{2}(\mathbb {R} )}$.

and let ${\displaystyle L^{2}(\Gamma \setminus \mathbb {H} ,k)}$ be the vector space of all measurable functions ${\displaystyle f:\mathbb {H} \to \mathbb {C} }$ with ${\displaystyle f_{||k}\gamma =f}$ for all ${\displaystyle \gamma \in \Gamma }$.

Furthermore, define the vector space

${\displaystyle \|f\|^{2}:=\int _{\Gamma \setminus \mathbb {H} }|f(z)|^{2}\,{\frac {dx\,dy}{y^{2}}}<\infty }$

modulo functions with ${\displaystyle ||f||=0}$. The integral is well definded, since the function ${\displaystyle |f(z)|^{2}}$ is ${\displaystyle \Gamma }$ - invariant.

${\displaystyle L^{2}(\Gamma \setminus \mathbb {H} ,k)}$

a Hilbert space with scalar product

${\displaystyle \langle f,g\rangle =\int _{\Gamma \setminus \mathbb {H} }f(z){\overline {g(z)}}\,{\frac {dx\,dy}{y^{2}}}.}$

The operator ${\displaystyle \Delta _{k}}$ can be defined in a vector space ${\displaystyle B\subset L^{2}(\Gamma \setminus \mathbb {H} ,k)\cap C^{\infty }(\Gamma \setminus \mathbb {H} ,k)}$ which is dense in ${\displaystyle L^{2}(\Gamma \setminus \mathbb {H} ,k)}$. There ${\displaystyle \Delta _{k}}$ is a positive semidefinite symmetric operator. It can be shown, that there exists a unique selfadjoint continuation on ${\displaystyle L^{2}(\Gamma \setminus \mathbb {H} ,k)}$ .

We define ${\displaystyle C(\Gamma \setminus \mathbb {H} ,k)}$ as the space of all cusp forms ${\displaystyle L^{2}(\Gamma \setminus \mathbb {H} ,k)\cap C^{\infty }(\Gamma \setminus \mathbb {H} ,k)}$.

Then ${\displaystyle \Delta _{k}}$ operates on ${\displaystyle C(\Gamma \setminus \mathbb {H} ,k)}$ 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 ${\displaystyle \Gamma }$ is a discrete (torsionfree) subgroup of ${\displaystyle SL_{2}(\mathbb {R} )}$, so that the quotient ${\displaystyle \Gamma \setminus \mathbb {H} }$ is compact, the spectral problem simplifies. This is because a discrete cocompact subgroup has no cusps. Here all of the space ${\displaystyle L^{2}(\Gamma \setminus \mathbb {H} ,k)}$ is a sum of eigenspaces.

### Embedding into the space ${\displaystyle L^{2}(\Gamma \setminus G)}$

${\displaystyle G=SL_{2}(\mathbb {R} )}$ is a unimodular locally compact group with the topology of ${\displaystyle \mathbb {R} ^{4}}$.

Let ${\displaystyle \Gamma }$ be a congruence subgroup. Since ${\displaystyle \Gamma }$ is discrete in ${\displaystyle G}$, ${\displaystyle \Gamma }$ is closed in ${\displaystyle G}$. The group ${\displaystyle G}$ is unimodular and since the counting measure is a Haar-measure on the discrete group ${\displaystyle \Gamma }$, ${\displaystyle \Gamma }$ is also unimodular. By the Quotient Integral Formula there exists a ${\displaystyle G}$ - right-invariant Radon measure ${\displaystyle dx}$ on the locally compact space ${\displaystyle \Gamma \setminus G}$. Let ${\displaystyle L^{2}(\Gamma \setminus G)}$ be the corresponding ${\displaystyle L^{2}}$ - space.

The space ${\displaystyle L^{2}(\Gamma \setminus G)}$ decomposes into a Hilbert space direct sum

${\displaystyle L^{2}(\Gamma \setminus G)=\bigoplus _{k\in \mathbb {Z} }L^{2}(\Gamma \setminus G,k)}$

where ${\displaystyle L^{2}(\Gamma \setminus G\,,\,k):=\{\phi \in L^{2}(\Gamma \setminus G)\mid \phi (xk_{\theta })=e^{ik\theta }F(x)\,\,\,\forall \,x\in \Gamma \setminus G\,\,\,\forall \,\theta \in \mathbb {R} \}{\text{ and }}k_{\theta }={\begin{pmatrix}\cos(\theta )&-\sin(\theta )\\\sin(\theta )&\cos(\theta )\\\end{pmatrix}}\,\,\in SO(2){\text{ for }}\theta \in \mathbb {R} }$.

The Hilbert-space ${\displaystyle L^{2}(\Gamma \setminus \mathbb {H} \,,\,k)}$ can be embedded isometrically into the Hilbert space ${\displaystyle L^{2}(\Gamma \setminus G\,,\,k)}$. The isometry is given by the map

${\displaystyle \psi _{k}\,:\,L^{2}(\Gamma \setminus \mathbb {H} \,,\,k)\to L^{2}(\Gamma \setminus G\,,\,k)\,,\,\psi _{k}(f)(g):=f_{||k}\gamma (i)}$

Therefore, all Maass cusp forms for the congruence group ${\displaystyle \Gamma }$ can be thought of as elements of ${\displaystyle L^{2}(\Gamma \setminus G)}$.

${\displaystyle L^{2}(\Gamma \setminus G)}$ is a Hilbert space carrying an operation of the group ${\displaystyle G}$, the so-called right regular representation :

${\displaystyle R_{g}\phi :=\phi (xg),{\text{ where }}x\in \Gamma \setminus G{\text{ and }}\phi \in L^{2}(\Gamma \setminus G).}$

One can easily show, that ${\displaystyle R}$ is a unitary representation of ${\displaystyle G}$ on the Hilbert space ${\displaystyle L^{2}(\Gamma \setminus G)}$. One is interested in a decomposition into irreducible suprepresentations. This is only possible if ${\displaystyle \Gamma }$ 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

### The Group ${\displaystyle Gl_{2}(\mathbb {A} )}$

Let ${\displaystyle R}$ be a commutative ring with unit and let ${\displaystyle G_{R}:=Gl_{2}(R)}$ be the group of ${\displaystyle 2\times 2}$ matrices with entries in ${\displaystyle R}$ and invertible determinant . Let ${\displaystyle \mathbb {A} =\mathbb {A} _{\mathbb {Q} }}$ be the ring of rational adeles, ${\displaystyle \mathbb {A} _{fin}}$ the ring of the finite (rational) adeles and for a prime number ${\displaystyle p\in \mathbb {N} }$ let ${\displaystyle \mathbb {Q} _{p}}$ be the field of p-adic numbers. Furthermore, let ${\displaystyle \mathbb {Z} _{p}}$ be the ring of the p-adic integers (see Adele ring). Define ${\displaystyle G_{p}:=G_{\mathbb {Q} _{p}}}$. Both ${\displaystyle G_{p}}$ and ${\displaystyle G_{\mathbb {R} }}$ are locally compact unimodular groups if one equips them with the subspace topologies of ${\displaystyle \mathbb {Q} _{p}^{4}}$ respectively ${\displaystyle \mathbb {R} ^{4}}$. The group ${\displaystyle G_{fin}:=G_{\mathbb {A} _{fin}}}$ is isomorphic to the group ${\displaystyle {\widehat {\prod _{p<\infty }}}^{K_{p}}G_{p}}$. Here the product is the resctricted product ${\displaystyle G_{p}}$, concerning the compact, open subgroups ${\displaystyle K_{p}:=G_{\mathbb {Z} _{p}}}$ of ${\displaystyle G_{p}}$. Then ${\displaystyle G_{fin}}$ locally compact group, if we equip it with the restricted product topology.

The group ${\displaystyle G_{\mathbb {A} }}$ is isomorphic to the group

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G_{fin}\times G_{\mathbb {R} }}$

and is a locally compact group with the product topology, since ${\displaystyle G_{fin}}$ and ${\displaystyle G_{\mathbb {R} }}$ are both locally compact.

Let ${\displaystyle {\widehat {\mathbb {Z} }}}$ be the ring ${\displaystyle \prod _{p<\infty }\mathbb {Z} _{p}}$. The subgroup

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G_{\widehat {\mathbb {Z} }}:=\prod _{p<\infty }K_{p}}$

is a maximal compact, open subgroup of ${\displaystyle G_{fin}}$ and can be thought of as a subgroup of ${\displaystyle G_{\mathbb {A} }}$, when we consider the embedding ${\displaystyle x_{fin}\mapsto (x_{fin},1_{\infty })}$.

We define ${\displaystyle Z_{\mathbb {R} }}$ as the center of ${\displaystyle G_{\infty }}$, that means ${\displaystyle Z_{\mathbb {R} }}$ is the group of all diagonal matrices of the form ${\displaystyle {\begin{pmatrix}\lambda &\\&\lambda \\\end{pmatrix}}}$, where ${\displaystyle \lambda \in \mathbb {R} ^{\times }}$. We think of ${\displaystyle Z_{\mathbb {R} }}$ as a subgroup of ${\displaystyle G_{\mathbb {A} }}$ since we can embed the group by ${\displaystyle z\mapsto (1_{G_{fin}},z)}$.

The group ${\displaystyle G_{\mathbb {Q} }}$ is embedded diagonally in ${\displaystyle G_{\mathbb {A} }}$, which is possible, since all four entries of a ${\displaystyle x\in G_{\mathbb {Q} }}$ can only have finite amount of prime divisors and therefore ${\displaystyle x\in K_{p}}$ for all but finitely many prime numbers ${\displaystyle p\in \mathbb {N} }$.

Let ${\displaystyle G_{\mathbb {A} }^{1}}$ be the group of all ${\displaystyle x\in G_{\mathbb {A} }}$ with ${\displaystyle |det(x)|=1}$. (see Adele Ring for a definition of the absolute value of an Idele). One can easily calculate, that ${\displaystyle G_{\mathbb {Q} }}$ is a subgroup of ${\displaystyle G_{\mathbb {A} }^{1}}$.

With the one-to-one map ${\displaystyle G_{\mathbb {A} }^{1}\hookrightarrow G_{\mathbb {A} }}$ we can identify the groups ${\displaystyle G_{\mathbb {Q} }\setminus G_{\mathbb {A} }^{1}}$ and ${\displaystyle G_{\mathbb {Q} }Z_{\mathbb {R} }\setminus G_{\mathbb {A} }}$ with each other.

The group ${\displaystyle G_{\mathbb {Q} }}$ is dense in ${\displaystyle G_{fin}}$ and discrete in ${\displaystyle G_{\mathbb {A} }}$. The Quotient ${\displaystyle G_{\mathbb {Q} }Z_{\mathbb {R} }\setminus G_{\mathbb {A} }=G_{\mathbb {Q} }\setminus G_{\mathbb {A} }^{1}}$ is not compact but has finite Haar-measure.

Therefore, ${\displaystyle G_{\mathbb {Q} }}$ is a lattice of ${\displaystyle G_{\mathbb {A} }^{1}}$ , similar to the classical case of the modular group and ${\displaystyle Sl_{2}(\mathbb {R} )}$. By harmonic analysis one also gets that ${\displaystyle G_{\mathbb {A} }^{1}}$ is unimodular.

We now want to embed the classical Maass cusp forms of weight 0 for the modular group into ${\displaystyle Z_{\mathbb {R} }\,\,G_{\mathbb {Q} }\setminus G_{\mathbb {A} }}$. This can be achieved with the "strong approximationtheorem", which states, that the map

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\psi \,:\,G_{\mathbb {Z} }x_{\infty }\mapsto G_{\mathbb {Q} }(1,x_{\infty })G_{\widehat {\mathbb {Z} }}}$

is a ${\displaystyle G_{\mathbb {R} }}$ - equivariant homeomorphism. So we get

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G_{\mathbb {Z} }\setminus G_{\mathbb {R} }\,\,{\tilde {\to }}\,\,G_{\mathbb {Q} }\setminus G_{\mathbb {A} }\,\,/\,\,G_{\widehat {\mathbb {Z} }}}$

and furthermore

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,G_{\mathbb {Z} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {R} }\,\,{\tilde {\to }}\,\,G_{\mathbb {Q} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {A} }\,\,/\,\,G_{\widehat {\mathbb {Z} }}}$.

Maass cuspforms of weight 0 for modular group can be embedded into

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L^{2}(Sl_{2}(\mathbb {Z} )\setminus Sl_{2}(\mathbb {R} ))\,\,{\tilde {=}}\,\,L^{2}(Gl_{2}(\mathbb {Z} )\,Z_{\mathbb {R} }\setminus Gl_{2}(\mathbb {R} ))}$.

By the strong approximation theorem this space is unitary isomorphic to

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L^{2}(G_{\mathbb {Q} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {A} }\,\,/\,\,G_{\widehat {\mathbb {Z} }})\,\,{\tilde {=}}\,\,L^{2}(G_{\mathbb {Q} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {A} })^{G_{\widehat {\mathbb {Z} }}}}$

which is a subspace of ${\displaystyle L^{2}(G_{\mathbb {Q} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {A} })}$.

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 ${\displaystyle \Gamma }$ in ${\displaystyle L^{2}(G_{\mathbb {Q} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {A} })}$.

We call ${\displaystyle L^{2}(G_{\mathbb {Q} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {A} })}$ the space of automorphic forms of the adele group.

### Cusp forms of the adele group

Let ${\displaystyle R}$ be a Ring and let ${\displaystyle N_{R}}$ be the group of all ${\displaystyle {\begin{pmatrix}1&r\\&1\\\end{pmatrix}}}$, where ${\displaystyle \,r\,\in \,R}$. This group is isomorphic to the additive group of R.

We call a function ${\displaystyle f\,\in \,L^{2}(G_{\mathbb {Q} }\setminus G_{\mathbb {A} }^{1})}$ cusp form, if

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\int _{N_{\mathbb {Q} }\setminus N_{\mathbb {A} }}\,f(nx)\,dn=0}$

holds for almost all${\displaystyle x\in G_{\mathbb {Q} }\setminus G_{\mathbb {A} }^{1}}$. Let ${\displaystyle L_{cusp}^{2}(G_{\mathbb {Q} }\setminus G_{\mathbb {A} }^{1})}$ (or just ${\displaystyle L_{cusp}^{2}}$) be the vectorspace of these cusp forms. ${\displaystyle L_{cusp}^{2}}$ is a closed subspace of ${\displaystyle L^{2}(G_{\mathbb {Q} }\,Z_{\mathbb {R} }\setminus G_{\mathbb {A} })}$ and it is invariant under the right regular representation of ${\displaystyle G_{\mathbb {A} }^{1}}$.

One is again intereseted in a decomposition of ${\displaystyle L_{cusp}^{2}}$ into irreducible closed subspaces .

We have the following theorem :

The space ${\displaystyle L_{cusp}^{2}}$ decomposes in a direct sum of irreducible Hilbert-spaces with finite multiplicities ${\displaystyle N_{cusp}(\pi )\in \mathbb {N} _{0}}$ :

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L_{cusp}^{2}={\widehat {\bigoplus }}_{\pi \in {\widehat {G_{\mathbb {A} }}}}N_{cusp}(\pi )\pi }$

The calculation of these multiplicities ${\displaystyle N_{cusp}(\pi )}$ 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 ${\displaystyle \pi }$ of the group ${\displaystyle G_{\mathbb {A} }}$ is called cuspidal, if it is isomorphic to a subrepresentation of ${\displaystyle L_{cusp}^{2}}$ ist.

An irreducible representation ${\displaystyle \pi }$ of the group ${\displaystyle G_{\mathbb {A} }}$ is called admissible if there exists a compact subgroup ${\displaystyle K}$ of ${\displaystyle K\subset G_{\mathbb {A} }}$, so that ${\displaystyle dim_{K}(V_{\pi },V_{\tau })<\infty }$ for all ${\displaystyle \tau \in \in {\widehat {G_{\mathbb {A} }}}}$.

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 ${\displaystyle G_{\mathbb {A} }}$ is isomorphic to an infinite tensor product

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\bigotimes _{p\leq \infty }\,\pi _{p}}$. The ${\displaystyle \pi _{p}}$ are irreducible representations of the group ${\displaystyle G_{p}}$. Almost all of them need to be umramified.

(A representation ${\displaystyle \pi _{p}}$ of the group ${\displaystyle G_{p}}$ ${\displaystyle (p<\infty )}$ is called unramified, if the vector space

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,V_{\pi _{p}}^{K_{p}}=\{v\in V_{\pi _{p}}\,|\,\pi _{p}(k)v=v\,\,\,\forall \,k\in K_{p}\}}$

is not the zero space.)

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

### Automorphic L-Functions

Let ${\displaystyle \pi }$ be an irreducible, admissible unitary representation of ${\displaystyle G_{\mathbb {A} }}$. By the tensor product theorem, ${\displaystyle \pi }$ is of the form ${\displaystyle \pi =\bigotimes _{p\leq \infty }\,\pi _{p}}$ (see cuspidal representations of the adele group)

Let ${\displaystyle F}$ be a finite set of places containing ${\displaystyle \infty }$ and all ramified places . One defines the global Hecke - function of ${\displaystyle \pi }$ as

${\displaystyle \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,L^{F}(\pi ,s):=\prod _{p\notin F}L(\pi _{p},s)}$

where ${\displaystyle L(\pi _{p},s)}$ is a so-called local L - function of the local representation ${\displaystyle \pi _{p}}$. A construction of local L - functions can be found in Deitmar C. 8.2.

If ${\displaystyle \pi }$ is a cuspidal representation, the L-Funktion ${\displaystyle L^{F}(\pi ,s)}$ has a meromorphic continuation on ${\displaystyle \mathbb {C} }$. This is possible, since ${\displaystyle L^{F}(\pi ,s)}$, satisfies certain functional equations.