Maass wave form#Maass cusp form

The group

:

operates on the upper half plane

:$\backslash mathcal\{H\}\; =\; \backslash \{\; z\; \backslash in\; \backslash Complex\; :\; \backslash operatorname\{Im\}(z)\; >\; 0\; \backslash \}$

by fractional linear transformations:

:

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

:

:

:= \lim_{\operatorname{Im}(z) \to \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}

The Radon measure

: $d\; \backslash mu(z)\; :=\; \backslash frac\{dx\; dy\}\{y^2\}$

defined on $\backslash mathcal\{H\}$ is invariant under the operation of $\backslash mathrm\{SL\}\_2(\backslash R)$.

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

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

A fundamental domain for the modular group $\backslash Gamma(1)\; :=\; \backslash mathrm\{SL\}\_\{2\}(\backslash Z\; )$ is given by

:

(see Modular form).

A function $f\; :\; \backslash mathcal\{H\}\; \backslash to\; \backslash Complex$ is called $\backslash Gamma$-invariant, if $f(\backslash gamma\; z)\; =\; f(z)$ holds for all $\backslash gamma\; \backslash in\; \backslash Gamma$ and all $z\; \backslash in\; \backslash mathcal\{H\}$.

For every measurable, $\backslash Gamma$-invariant function $f:\; \backslash mathcal\{H\}\; \backslash to\; \backslash Complex$ the equation

: $\backslash int\_F\; f(z)\; \backslash ,\; d\backslash mu(z)\; =\; \backslash int\_\{\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\}\}\; f(z)\; \backslash ,\; d\backslash mu(z),$

holds. Here the measure $d\backslash mu$ on the right side of the equation is the induced measure on the quotient $\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\}.$

The hyperbolic Laplace operator on $\backslash mathcal\{H\}$ is defined as

:$\backslash Delta\; :\; C^\backslash infty\; (\backslash mathcal\{H\})\; \backslash to\; C^\backslash infty(\backslash mathcal\{H\}),$

:$\backslash Delta\; =\; -y^2\; \backslash left(\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; y^2\}\; \backslash right)$

A Maass form for the group $\backslash Gamma\; (1)\; :=\; \backslash mathrm\{SL\}\_\{2\}(\backslash Z)$ is a complex-valued smooth function $f$ on $\backslash mathcal\{H\}$ satisfying

1. $f(\backslash gamma\; z)=f(z)\; \backslash text\{\; for\; all\; \}\; \backslash gamma\; \backslash in\; \backslash Gamma\; (1),\; \backslash qquad\; z\; \backslash in\; \backslash mathcal\{H\}$
2. $\backslash text\{there\; exists\; \}\; \backslash lambda\; \backslash in\; \backslash Complex\; \backslash text\{\; with\; \}\; \backslash Delta\; (f)\; =\; \backslash lambda\; f$
3. $\backslash text\{there\; exists\; \}\; N\; \backslash in\; \backslash N\; \backslash text\{\; with\; \}\; f(x+iy)\; =\; \backslash mathcal\{O\}\; (y^N)\; \backslash text\{\; for\; \}\; y\; \backslash ge\; 1$

If

: $\backslash int\_0^1\; f(z+t)\; dt\; =\; 0\; \backslash text\{\; for\; all\; \}\; z\; \backslash in\; \backslash mathcal\{H\}$

we call $f$ Maass cusp form.

Let $f$ be a Maass form. Since

:

we have:

:$\backslash forall\; z\; \backslash in\; \backslash mathcal\{H\}:\; \backslash qquad\; f(z)\; =\; f(\backslash gamma\; z)\; =\; f(z+1).$

Therefore $f$ has a Fourier expansion of the form

: $f(x+iy)\; =\; \backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}\; a\_\{n\}(y)e^\{2\backslash pi\; inx\},$

with coefficient functions $a\_\{n\},\; n\; \backslash in\; \backslash Z.$

It is easy to show that $f$ is Maass cusp form if and only if $a\_\{0\}(y)=0\; \backslash ;\backslash ;\; \backslash forall\; y\; >\; 0$.

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

Definition: The Bessel function $K\_v$ is defined as

: $K\_s(y)\; :=\; \backslash frac\{1\}\{2\}\; \backslash int\_0^\backslash infty\; e^\{-\backslash frac\{y(t+t^\{-1\})\}\{2\}\}t^s\backslash frac\{dt\}\{t\},$

\qquad s \in \Complex, y > 0.

The integral converges locally uniformly absolutely for $y\; >\; 0$ in $s\; \backslash in\; \backslash Complex$ and the inequality

: $K\_s(y)\; \backslash leq\; e^\{-\backslash frac\{y\}\{2\}\}K\_\{\backslash operatorname\{Re\}(s)\}(2)$

holds for all $y>4$.

Therefore, $|K\_s|$ decreases exponentially for $y\; \backslash to\; \backslash infty$. Furthermore, we have $K\_\{-s\}(y)=K\_s(y)$ for all $s\; \backslash in\; \backslash Complex,\; y\; >\; 0$.

{{math theorem|name=Theorem (Fourier coefficients of Maass forms)|math_statement= Let $\backslash lambda\; \backslash in\; \backslash Complex$ be the eigenvalue of the Maass form $f$ corresponding to $\backslash Delta.$ There exist $\backslash nu\; \backslash in\; \backslash Complex$, unique up to sign, such that $\backslash lambda\; =\; \backslash frac\{1\}\{4\}\; -\; \backslash nu^2$. Then the Fourier coefficients of $f$ are

$$\backslash begin\{align\}$$

a_n(y) &= c_n\sqrt{y} K_\nu(2\pi |n|y) \quad c_n \in \Complex && n \neq 0 \\

a_0(y) &= c_0 y^{\frac{1}{2}-\nu} + d_0 y^{\frac{1}{2}+\nu} \quad c_{0}, d_0 \in \Complex && n=0

\end{align}}}

Proof: We have

:$\backslash Delta(f)=\backslash left\; (\backslash frac\{1\}\{4\}\; -\; \backslash nu^\{2\}\; \backslash right\; )f.$

By the definition of the Fourier coefficients we get

: $a\_n(y)\; =\; \backslash int\_0^1\; f(x+iy)e^\{-2\backslash pi\; inx\}\; dx$

for $n\; \backslash in\; \backslash Z.$

Together it follows that

:$\backslash begin\{align\}$

\left(\frac{1}{4} - \nu^2\right) a_n(y) &= \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 i n)^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 i n)^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{align}

for $n\; \backslash in\; \backslash Z.$

In (1) we used that the nth Fourier coefficient of $\backslash frac\{\backslash partial^\{2\}f\}\{\backslash partial\; x^\{2\}\}$ is $(2\backslash pi\; i\; n)^\{2\}a\_\{n\}(y)$ 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:

: $y^2\backslash frac\{\backslash partial^2\}\{\backslash partial\; y^2\}\; a\_n(y)\; +\; \backslash left(\; \backslash frac\{1\}\{4\}\; -\; \backslash nu^2-4\backslash pi\; n^2\; y^2\; \backslash right)a\_n(y)\; =\; 0$

For $n\; =\; 0$ one can show, that for every solution $f$ there exist unique coefficients $c\_0,\; d\_\{0\}\; \backslash in\; \backslash Complex$ with the property $a\_0(y)=c\_0\; y^\{\backslash frac\{1\}\{2\}\; -\; \backslash nu\}\; +\; d\_0\; y^\{\backslash frac\{1\}\{2\}\; +\; \backslash nu\}.$

For $n\backslash neq\; 0$ every solution $f$ has coefficients of the form

: $a\_n(y)\; =\; c\_n\backslash sqrt\{y\}K\_v(2\backslash pi|n|y)+\; d\_n\backslash sqrt\{y\}I\_v(2\backslash pi|n|y)$

for unique $c\_n,d\_n\; \backslash in\; \backslash Complex$. Here $K\_v(s)$ and $I\_v(s)$ are Bessel functions.

The Bessel functions $I\_v$ grow exponentially, while the Bessel functions $K\_v$ decrease exponentially. Together with the polynomial growth condition 3) we get $f\; :\; a\_\{n\}(y)=c\_\{n\}\backslash sqrt\{y\}K\_\{v\}(2\backslash pi|n|y)$ (also $d\_\{n\}\; =\; 0$) for a unique $c\_\{n\}\; \backslash in\; \backslash Complex$. Q.E.D.

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

Let

:$f(x+iy)=\backslash sum\_\{n\; \backslash neq\; 0\}\; c\_\{n\}\backslash sqrt\{y\}K\_\{\backslash nu\}(2\backslash pi|n|y)e^\{2\backslash pi\; inx\}$

be a Maass cusp form. We define the L-function of $f$ as

:$L(s,f)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; c\_n\; n^\{-s\}.$

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

If $f$ is even or odd we get

:$\backslash Lambda(s,f)\; :=\; \backslash pi^\{-s\}\backslash Gamma\; \backslash left(\; \backslash frac\{s+\backslash varepsilon+\backslash nu\}\{2\}\; \backslash right)\; \backslash Gamma\; \backslash left(\; \backslash frac\{s+\backslash varepsilon-\backslash nu\}\{2\}\; \backslash right)\; L(s,f).$

Here $\backslash varepsilon\; =\; 0$ if $f$ is even and $\backslash varepsilon\; =\; -1$ if $f$ is odd. Then $\backslash Lambda$ satisfies the functional equation

:$\backslash Lambda(s,f)=(-1)^\backslash varepsilon\; \backslash Lambda(1-s,f).$

The non-holomorphic Eisenstein-series is defined for $z\; =x+iy\; \backslash in\; \backslash mathcal\{H\}$ and $s\; \backslash in\; \backslash Complex$ as

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

where $\backslash Gamma(s)$ is the Gamma function.

The series converges absolutely in $z\; \backslash in\; \backslash mathcal\{H\}$ for $\backslash Re(s)\; >\; 1$ and locally uniformly in $\backslash mathcal\{H\}\; \backslash times\; \backslash \{\backslash Re(s)\; >\; 1\backslash \}$, since one can show, that the series

:$S(z,s):=\backslash sum\_\{(m,n)\; \backslash neq\; (0,0)\}\backslash frac\{1\}\{|mz+n|^s\}$

converges absolutely in $z\backslash in\; \backslash mathcal\{H\}$, if $\backslash Re(s)\; >\; 2$. More precisely it converges uniformly on every set $K\; \backslash times\; \backslash \{\backslash Re(s)\; \backslash geq\; \backslash alpha\backslash \}$, for every compact set $K\; \backslash subset\; \backslash mathcal\{H\}$ and every $\backslash alpha\; >\; 2$.

We only show $\backslash mathrm\{SL\}\_\{2\}(\backslash Z)$-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 $\backslash mathrm\{SL\}\_\{2\}(\backslash Z)$-invariance. Let

:

be the stabilizer group $\backslash infty$ corresponding to the operation of $\backslash mathrm\{SL\}\_\{2\}(\backslash Z)$ on $\backslash mathcal\{H\}\; \backslash cup\; \backslash \{\backslash infty\backslash \}$.

:Proposition. E is $\backslash Gamma(1)$-invariant.

Proof. Define:

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

(a) $\backslash tilde\{E\}$ converges absolutely in $z\backslash in\; \backslash mathcal\{H\}$ for $\backslash Re(s)>1$ and $E(z,s)\; =\backslash pi^\{-s\}\; \backslash Gamma(s)\; \backslash zeta(2s)\; \backslash tilde\{E\}(z,s).$

Since

:

we obtain

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

That proves the absolute convergence in $z\; \backslash in\; \backslash mathcal\{H\}$ for $\backslash operatorname\{Re\}(s)>1.$

Furthermore, it follows that

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

since the map

:$\backslash begin\{cases\}$

\N \times \{(x,y) \in \Z ^{2} - \{(0,0)\} : (x,y)=1\} \to \Z ^{2} - \{(0,0)\} \\

(n,(x,y)) \mapsto (nx,ny)

\end{cases}

is a bijection (a) follows.

(b) We have $E(\backslash gamma\; z,s)=E(z,s)$ for all $\backslash gamma\; \backslash in\; \backslash Gamma(1)$.

For $\backslash tilde\{\backslash gamma\}\; \backslash in\; \backslash Gamma\; (1)$ we get

:$\backslash tilde\{E\}(\backslash tilde\{\backslash gamma\}\; z,s)$

= \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}(z,s).

Together with (a), $E$ is also invariant under $\backslash Gamma\; (1)$. Q.E.D.

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

We need the following Lemma:

:Lemma: $\backslash Delta$ commutes with the operation of $G$ on $C^\{\backslash infty\}(\backslash mathcal\{H\})$. More precisely for all $g\; \backslash in\; G$ we have: $L\_\{g\}\backslash Delta\; =\; \backslash Delta\; L\_\{g\}.$

Proof: The group $\backslash mathrm\{SL\}\_\{2\}(\backslash R)$ is generated by the elements of the form

:

One calculates the claim for these generators and obtains the claim for all $g\backslash in\; \backslash mathrm\{SL\}\_\{2\}(\backslash R)$. Q.E.D.

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

:$\backslash Delta\; \backslash tilde\{E\}(z,s)$

:= \Delta\sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \Im(\gamma z)^s

= \sum_{\gamma \in \Gamma_\infty \backslash \Gamma} \Delta \left (\Im(\gamma z)^s \right )

Furthermore, one has

:$\backslash Delta\; \backslash left\; (\backslash Im(z)^s\; \backslash right\; )=\backslash Delta(y^s)$

=-y^{2} \left (\frac{\partial^2 y^s}{\partial x^{2}} + \frac{\partial^{2}y^s}{\partial y^2} \right )

=s(1-s) y^s.

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

:$\backslash forall\; \backslash gamma\; \backslash in\; \backslash Gamma(1):\; \backslash quad\; \backslash Delta\; \backslash left\; (\backslash Im(\backslash gamma\; z)^s\; \backslash right\; )=s(1-s)\; \backslash Im(\backslash gamma\; z)^s$

and so

:$\backslash Delta\; \backslash tilde\{E\}(z,s)\; =\; s(1-s)\backslash tilde\{E\}(z,s).$

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

The nonholomorphic Eisenstein series has a Fourier expansion

:$E(z,s)=\backslash sum\_\{n=-\backslash infty\}^\backslash infty\; a\_n(y,s)e^\{2\backslash pi\; i\; nx\}$

where

:$\backslash begin\{align\}$

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} \\

a_n(y,s) &= 2|n|^{s-\frac{1}{2}}\sigma_{1-2s}(|n|)\sqrt{y}K_{s-\frac{1}{2}}(2\pi|n|y) && n \neq 0

\end{align}

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

The Eisenstein series satisfies the functional equation

:$E(z,s)=E(z,1-s)$

for all $z\backslash in\; \backslash mathcal\{H\}$.

Locally uniformly in $x\backslash in\; \backslash R$ the growth condition

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

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

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

For $N\; \backslash in\; \backslash N$ let $\backslash Gamma\; (N)$ be the kernel of the canonical projection

:$\backslash mathrm\{SL\}\_\{2\}(\backslash Z\; )\; \backslash to\; \backslash mathrm\{SL\}\_\{2\}(\backslash Z/N\; \backslash Z).$

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

Let

:$\backslash overline\{\backslash Gamma(1)\}:=\; \backslash Gamma(1)\; /\backslash \{\backslash pm\; 1\backslash \}.$

For a congruence subgroup $\backslash Gamma,$ let $\backslash overline\{\backslash Gamma\}$ be the image of $\backslash Gamma$ in $\backslash overline\{\backslash Gamma(1)\}$. If S is a system of representatives of $\backslash overline\{\backslash Gamma\}\; \backslash backslash\; \backslash overline\{\backslash Gamma(1)\}$, then

:$SD\; =\; \backslash bigcup\_\{\backslash gamma\; \backslash in\; S\}\; \backslash gamma\; D$

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

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

For every cusp $c$ there exists $\backslash sigma\; \backslash in\; \backslash Gamma(1)$ with $\backslash sigma\; \backslash infty\; =\; c$.

Let $\backslash Gamma$ be a congruence subgroup and $k\; \backslash in\; \backslash Z.$

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

:$\backslash Delta\_\{k\}\; :\; C^\{\backslash infty\; \}(\backslash mathcal\{H\})\; \backslash to\; C^\{\backslash infty\}(\backslash mathcal\{H\}),$

:$\backslash Delta\_\{k\}\; =\; -y^\{2\}\; \backslash left\; (\backslash frac\{\backslash partial^\{2\}\}\{\backslash partial\; x^\{2\}\}\; +\; \backslash frac\{\backslash partial^\{2\}\}\{\backslash partial\; y^\{2\}\}\; \backslash right\; )\; +\; iky\backslash frac\{\backslash partial\}\{\backslash partial\; x\}.$

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

We define an operation of $\backslash mathrm\{SL\}\_\{2\}(\backslash R)$ on $C^\{\backslash infty\}(\backslash mathcal\{H\})$ by

:$f\_$

where

:

It can be shown that

:$(\backslash Delta\_\{k\}f)\_\{||k\}g\; =\; \backslash Delta\_\{k\}(f\_\{||k\}g)$

holds for all $f\; \backslash in\; C^\{\backslash infty\}(\backslash mathcal\{H\}),\; k\; \backslash in\; \backslash Z$ and every $g\; \backslash in\; \backslash mathrm\{SL\}\_\{2\}(\backslash R)$.

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

:$C^\{\backslash infty\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k)\; :=\; \backslash \{\; f\; \backslash in\; C^\{\backslash infty\}(\backslash mathcal\{H\})\; :\; f\_\{||k\}\backslash gamma\; =\; f\; \backslash forall\; \backslash gamma\; \backslash in\; \backslash Gamma\backslash \}$.

Definition. A Maass form of weight $k\backslash in\; \backslash Z$ for $\backslash Gamma$ is a function $f\; \backslash in\; C^\{\backslash infty\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},\; k)$ that is an eigenfunction of $\backslash Delta\_\{k\}$ and is of moderate growth at the cusps.

The term moderate growth at cusps needs clarification. Infinity is a cusp for $\backslash Gamma,$ a function $f\; \backslash in\; C^\{\backslash infty\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},\; k)$ is of moderate growth at $\backslash infty$ if $f(x+iy)$ is bounded by a polynomial in y as $y\; \backslash to\; \backslash infty$. Let $c\; \backslash in\; \backslash Q$ be another cusp. Then there exists $\backslash theta\; \backslash in\; \backslash mathrm\{SL\}\_\{2\}(\backslash Z)$ with $\backslash theta\; (\backslash infty)=c$. Let $f\text{'}:=f\_\{||k\}\backslash theta$. Then $f\text{'}\; \backslash in\; C^\{\backslash infty\}(\backslash Gamma\text{'}\; \backslash backslash\; \backslash mathcal\{H\},\; k)$, where $\backslash Gamma\text{'}$ is the congruence subgroup $\backslash theta^\{-1\}\backslash Gamma\backslash theta$. We say $f$ is of moderate growth at the cusp $c$, if $f\text{'}$ is of moderate growth at $\backslash infty$.

Definition. If $\backslash Gamma$ contains a principal congruence subgroup of level $N$, we say that $f$ is cuspidal at infinity, if

:$\backslash forall\; z\; \backslash in\; \backslash mathcal\{H\}:\; \backslash quad\; \backslash int\_\{0\}^\{N\}\; f(z+u)\; du\; =\; 0.$

We say that $f$ is cuspidal at the cusp $c$ if $f\text{'}$ is cuspidal at infinity. If $f$ is cuspidal at every cusp, we call $f$ a cusp form.

We give a simple example of a Maass form of weight $k\; >\; 1$ for the modular group:

Example. Let $g\; :\; \backslash mathcal\{H\}\; \backslash to\; \backslash Complex$ be a modular form of even weight $k$ for $\backslash Gamma\; (1).$ Then $f(z):=\; y^\{\backslash frac\{k\}\{2\}\}g(z)$ is a Maass form of weight $k$ for the group $\backslash Gamma\; (1)$.

Let $\backslash Gamma$ be a congruence subgroup of $\backslash mathrm\{SL\}\_\{2\}(\backslash R)$ and let $L^\{2\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k)$ be the vector space of all measurable functions $f\; :\backslash mathcal\{H\}\; \backslash to\; \backslash Complex$ with $f\_\{||k\}\backslash gamma\; =\; f$ for all $\backslash gamma\; \backslash in\; \backslash Gamma$ satisfying

:

modulo functions with $\backslash |f\backslash |\; =\; 0.$ The integral is well defined, since the function $|f(z)|^2$ is $\backslash Gamma$-invariant. This is a Hilbert space with inner product

:$\backslash langle\; f,g\backslash rangle\; =\; \backslash int\_\{\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\}\}\; f(z)\backslash overline\{g(z)\}\; d\backslash mu(z).$

The operator $\backslash Delta\_k$ can be defined in a vector space $B\; \backslash subset\; L^2(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k)\; \backslash cap\; C^\{\backslash infty\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k)$ which is dense in $L^2(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k)$. There $\backslash Delta\_k$ is a positive semidefinite symmetric operator. It can be shown, that there exists a unique self-adjoint continuation on $L^2(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k).$

Define $C(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k)$ as the space of all cusp forms $L^\{2\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\}\; ,k)\; \backslash cap\; C^\{\backslash infty\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\}\; ,k).$ Then $\backslash Delta\_\{k\}$ operates on $C(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},k)$ 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 $\backslash Gamma$ is a discrete (torsion free) subgroup of $\backslash mathrm\{SL\}\_\{2\}(\backslash R)$, so that the quotient $\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\}$ is compact, the spectral problem simplifies. This is because a discrete cocompact subgroup has no cusps. Here all of the space $L^\{2\}(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\}\; ,k)$ is a sum of eigenspaces.

$G\; =\; \backslash mathrm\{SL\}\_\{2\}(\backslash R)$ is a locally compact unimodular group with the topology of $\backslash R^4.$ Let $\backslash Gamma$ be a congruence subgroup. Since $\backslash Gamma$ is discrete in $G$, it is closed in $G$ as well. The group $G$ is unimodular and since the counting measure is a Haar-measure on the discrete group $\backslash Gamma$, $\backslash Gamma$ is also unimodular. By the Quotient Integral Formula there exists a $G$-right-invariant Radon measure $dx$ on the locally compact space $\backslash Gamma\; \backslash backslash\; G$. Let $L^\{2\}(\backslash Gamma\; \backslash backslash\; G)$ be the corresponding $L^2$-space. This space decomposes into a Hilbert space direct sum:

: $L^2(\backslash Gamma\; \backslash backslash\; G)\; =\; \backslash bigoplus\_\{k\; \backslash in\; \backslash Z\; \}L^2(\backslash Gamma\; \backslash backslash\; G,k)$

where

:$L^2(\backslash Gamma\; \backslash backslash\; G,\; k):=\; \backslash left\; \backslash \{\backslash phi\; \backslash in\; L^\{2\}(\backslash Gamma\; \backslash backslash\; G)\backslash mid\; \backslash phi\; (xk\_\backslash theta)\; =\; e^\{ik\; \backslash theta\}\; F(x)\; \backslash forall\; x\; \backslash in\; \backslash Gamma\; \backslash backslash\; G\; \backslash forall\; \backslash theta\; \backslash in\; \backslash R\; \backslash right\; \backslash \}$

and

:

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

:$\backslash begin\{cases\}\; \backslash psi\_k\; :\; L^2(\backslash Gamma\; \backslash backslash\; \backslash mathcal\{H\},\; k)\; \backslash to\; L^2(\backslash Gamma\; \backslash backslash\; G,\; k)\; \backslash \backslash \; \backslash psi\_\{k\}(f)(g)\; :=\; f\_\{||k\}\backslash gamma(i)\; \backslash end\{cases\}$

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

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

:$R\_g\backslash phi\; :=\; \backslash phi(xg),\; \backslash text\{\; where\; \}\; x\; \backslash in\; \backslash Gamma\; \backslash backslash\; G\; \backslash text\{\; and\; \}\; \backslash phi\; \backslash in\; L^2(\backslash Gamma\; \backslash backslash\; G).$

One can easily show, that $R$ is a unitary representation of $G$ on the Hilbert space $L^\{2\}(\backslash Gamma\; \backslash backslash\; G)$. One is interested in a decomposition into irreducible subrepresentations. This is only possible if $\backslash Gamma$ 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 forms. (see Bump, C. 2.3)

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 {{harvtxt|Maass|1949}}.

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 $z\; \backslash in\; \backslash mathcal\{H\}$, we have $f\backslash left(\backslash frac\{az+b\}\{cz+d\}\backslash right)\; =\; \backslash left(\backslash frac\{cz+d\}$

  cz+d

  \right)^k f(z).
• We have $\backslash Delta\_k\; f\; =\; sf$, where $\backslash Delta\_k$ is the weight k hyperbolic Laplacian defined as $$\backslash Delta\_k\; =\; -y^2\; \backslash left(\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; y^2\}\backslash right)\; +\; i\; k\; y\; \backslash frac\; \backslash partial\; \{\backslash partial\; x\}.$$
• The function $f$ is of at most polynomial growth at cusps.

A weak Maass form is defined similarly but with the third condition replaced by "The function $f$ has at most linear exponential growth at cusps". Moreover, $f$ is said to be harmonic if it is annihilated by the Laplacian operator.

Let $f$ be a weight 0 Maass cusp form. Its normalized Fourier coefficient at a prime p is bounded by p^7/64 + p^−7/64. This theorem is due to Henry Kim and Peter Sarnak. It is an approximation toward Ramanujan-Petersson conjecture.

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.

Let $R$ be a commutative ring with unit and let $G\_R:=\backslash mathrm\{GL\}\_2(R)$ be the group of $2\; \backslash times\; 2$ matrices with entries in $R$ and invertible determinant. Let $\backslash mathbb\{A\}=\backslash mathbb\{A\}\_\backslash Q$ be the ring of rational adeles, $\backslash mathbb\{A\}\_\backslash text\{fin\}$ the ring of the finite (rational) adeles and for a prime number $p\; \backslash in\; \backslash N$ let $\backslash Q\_p$ be the field of p-adic numbers. Furthermore, let $\backslash Z\_p$ be the ring of the p-adic integers (see Adele ring). Define $G\_p:=G\_\{\backslash Q\_p\}$. Both $G\_p$ and $G\_\{\backslash R\}$ are locally compact unimodular groups if one equips them with the subspace topologies of $\backslash Q\_p^4$ respectively $\backslash R^4$. Then:

:

The right side is the restricted product, concerning the compact, open subgroups $K\_p:=G\_\{\backslash Z\_p\}$ of $G\_p$. Then $G\_\backslash text\{fin\}$ locally compact group, if we equip it with the restricted product topology.

The group $G\_\{\backslash mathbb\{A\}\}$ is isomorphic to

: $G\_\backslash text\{fin\}\; \backslash times\; G\_\{\backslash R\}$

and is a locally compact group with the product topology, since $G\_\backslash text\{fin\}$ and $G\_\{\backslash R\}$ are both locally compact.

Let

:

The subgroup

:

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

We define $Z\_\{\backslash R\}$ as the center of $G\_\backslash infty$, that means $Z\_\{\backslash R\}$ is the group of all diagonal matrices of the form , where $\backslash lambda\; \backslash in\; \backslash R^\backslash times$. We think of $Z\_\{\backslash R\}$ as a subgroup of $G\_\{\backslash mathbb\{A\}\}$ since we can embed the group by $z\; \backslash mapsto\; (1\_\{G\_\backslash text\{fin\}\},z)$.

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

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

With the one-to-one map $G^1\_\{\backslash mathbb\{A\}\}\; \backslash hookrightarrow\; G\_\{\backslash mathbb\{A\}\}$ we can identify the groups $G\_\backslash Q\; \backslash backslash\; G^1\_\{\backslash mathbb\{A\}\}$ and $G\_\backslash Q\; Z\_\{\backslash R\}\; \backslash backslash\; G\_\{\backslash mathbb\{A\}\}$ with each other.

The group $G\_\backslash Q$ is dense in $G\_\backslash text\{fin\}$ and discrete in $G\_\{\backslash mathbb\{A\}\}$. The quotient $G\_\backslash Q\; Z\_\{\backslash R\}\; \backslash backslash\; G\_\{\backslash mathbb\{A\}\}\; =\; G\_\backslash Q\; \backslash backslash\; G^1\_\{\backslash mathbb\{A\}\}$ is not compact but has finite Haar-measure.

Therefore, $G\_\backslash Q$ is a lattice of $G^1\_\{\backslash mathbb\{A\}\},$ similar to the classical case of the modular group and $\backslash mathrm\{SL\}\_2(\backslash R)$. By harmonic analysis one also gets that $G^1\_\{\backslash mathbb\{A\}\}$ is unimodular.

We now want to embed the classical Maass cusp forms of weight 0 for the modular group into $Z\_\{\backslash R\}\; G\_\backslash Q\; \backslash backslash\; G\_\{\backslash mathbb\{A\}\}$. This can be achieved with the "strong approximation theorem", which states that the map

:$\backslash psi\; :\; G\_\{\backslash Z\}x\_\backslash infty\; \backslash mapsto\; G\_\backslash Q(1,x\_\{\backslash infty\})\; G\_\{\backslash widehat\{\backslash Z\}\}$

is a $G\_\{\backslash R\}$-equivariant homeomorphism. So we get

:$G\_\{\backslash Z\}\; \backslash backslash\; G\_\{\backslash R\}\; \backslash overset\backslash sim\backslash to\; G\_\backslash Q\; \backslash backslash\; G\_\{\backslash mathbb\{A\}\}\; /\; G\_\{\backslash widehat\{\backslash Z\}\}$

and furthermore

:$G\_\{\backslash Z\}\; Z\_\{\backslash R\}\; \backslash backslash\; G\_\{\backslash R\}\; \backslash overset\backslash sim\backslash to\; G\_\backslash Q\; Z\_\{\backslash R\}\backslash backslash\; G\_\{\backslash mathbb\{A\}\}\; /\; G\_\{\backslash widehat\{\backslash Z\}\}.$

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

:$L^2(\backslash mathrm\{SL\}\_2(\backslash Z)\; \backslash backslash\; \backslash mathrm\{SL\}\_2(\backslash R))\; \backslash cong\; L^2(\backslash mathrm\{GL\}\_2\; (\backslash Z)\; Z\_\{\backslash R\}\; \backslash backslash\; \backslash mathrm\{GL\}\_2(\backslash R)).$

By the strong approximation theorem this space is unitary isomorphic to

:$L^2(G\_\{\backslash Q\}\; Z\_\{\backslash R\}\backslash backslash\; G\_\{\backslash mathbb\{A\}\}\; /\; G\_\{\backslash widehat\{\backslash Z\}\})\; \backslash cong\; L^2(G\_\{\backslash Q\}\; Z\_\{\backslash R\}\backslash backslash\; G\_\{\backslash mathbb\{A\}\})^\{G\_\{\backslash widehat\{\backslash Z\}\}\}$

which is a subspace of $L^2(G\_\backslash Q\; Z\_\{\backslash R\}\backslash backslash\; G\_\{\backslash mathbb\{A\}\}).$

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 $\backslash Gamma$ in $L^2(G\_\{\backslash Q\}\; Z\_\{\backslash R\}\backslash backslash\; G\_\{\backslash mathbb\{A\}\})$.

We call $L^2(G\_\{\backslash Q\}\; Z\_\{\backslash R\}\backslash backslash\; G\_\{\backslash mathbb\{A\}\})$ the space of automorphic forms of the adele group.

Let $R$ be a Ring and let $N\_R$ be the group of all where $r\; \backslash in\; R$. This group is isomorphic to the additive group of R.

We call a function $f\; \backslash in\; L^\{2\}(G\_\{\backslash Q\}\backslash backslash\; G^\{1\}\_\{\backslash mathbb\{A\}\})$ cusp form, if

:$\backslash int\_\{N\_\{\backslash Q\}\; \backslash backslash\; N\_\{\backslash mathbb\{A\}\}\}\; f(nx)\; dn\; =\; 0$

holds for almost all$x\; \backslash in\; G\_\{\backslash Q\}\backslash backslash\; G^\{1\}\_\{\backslash mathbb\{A\}\}$. Let $L^2\_\backslash text\{cusp\}(G\_\backslash Q\backslash backslash\; G^1\_\{\backslash mathbb\{A\}\})$ (or just $L^\{2\}\_\backslash text\{cusp\}$) be the vector space of these cusp forms. $L^2\_\backslash text\{cusp\}$ is a closed subspace of $L^2(G\_\{\backslash Q\}\; Z\_\{\backslash R\}\; \backslash backslash\; G\_\{\backslash mathbb\{A\}\})$ and it is invariant under the right regular representation of $G^1\_\{\backslash mathbb\{A\}\}.$

One is again interested in a decomposition of $L^2\_\backslash text\{cusp\}$ into irreducible closed subspaces.

We have the following theorem:

The space $L^2\_\backslash text\{cusp\}$ decomposes in a direct sum of irreducible Hilbert-spaces with finite multiplicities $N\_\backslash text\{cusp\}(\backslash pi)\; \backslash in\; \backslash N\; \_0$:

: $L^2\_\backslash text\{cusp\}\; =\; \backslash widehat\{\backslash bigoplus\_\{\backslash pi\; \backslash in\; \backslash widehat\{G\}\_\{\backslash mathbb\{A\}\}\}\}\; N\_\backslash text\{cusp\}(\backslash pi)\backslash pi$

The calculation of these multiplicities $N\_\backslash text\{cusp\}(\backslash pi)$ is one of the most important and most difficult problems in the theory of automorphic forms.

An irreducible representation $\backslash pi$ of the group $G\_\{\backslash mathbb\{A\}\}$ is called cuspidal, if it is isomorphic to a subrepresentation of $L^2\_\backslash text\{cusp\}$.

An irreducible representation $\backslash pi$ of the group $G\_\{\backslash mathbb\{A\}\}$ is called admissible if there exists a compact subgroup $K$ of $K\; \backslash subset\; G\_\{\backslash mathbb\{A\}\}$, so that for all $\backslash tau\; \backslash in\; \backslash widehat\{G\}\_\{\backslash mathbb\{A\}\}$.

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 $G\_\{\backslash mathbb\{A\}\}$ is isomorphic to an infinite tensor product

: $\backslash bigotimes\_\{p\backslash leq\; \backslash infty\}\; \backslash pi\_p.$

The $\backslash pi\_p$ are irreducible representations of the group $G\_p$. Almost all of them need to be umramified.

(A representation $\backslash pi\_p$ of the group $G\_p$ is called unramified, if the vector space

: $V\_\{\backslash pi\_p\}^\{K\_p\}\; =\; \backslash left\backslash \{\; v\; \backslash in\; V\_\{\backslash pi\_p\}\backslash mid\; \backslash pi\_p(k)v\; =\; v\; \backslash forall\; k\; \backslash in\; K\_p\; \backslash right\; \backslash \}$

is not the zero space.)

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

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

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

:$L^S(s,\backslash pi)\; :=\backslash prod\_\{p\; \backslash notin\; S\}\; L(s,\backslash pi\_\{p\})$

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

If $\backslash pi$ is a cuspidal representation, the L-function $L^S(s,\backslash pi)$ has a meromorphic continuation on $\backslash Complex$. This is possible, since $L^S(s,\backslash pi)$, satisfies certain functional equations.

• Harmonic Maass form
• Mock modular form
• Real analytic Eisenstein series
• Automorphic form
• Modular form
• Voronoi formula

{{Refbegin}}

• {{Citation|last1=Bringmann|first1=Kathrin|author1-link=Kathrin Bringmann|last2=Folsom|first2=Amanda|author2-link=Amanda Folsom| doi=10.1515/crelle-2012-0102|journal=Journal für die Reine und Angewandte Mathematik| mr=3259042| pages=179–202|title=Almost harmonic Maass forms and Kac–Wakimoto characters|volume=2014|year=2014|issue=694 |arxiv=1112.4726|s2cid=54896147 }}
• {{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|bibcode=2002InMat.149..489D |s2cid=121720199 }}
• 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| s2cid=119494842 }}

{{Refend}}

Category:Automorphic forms