Automorphic form

An automorphic form is a function with arithmetic symmetries, together with analytic regularity and growth conditions. In a classical example, a modular form of integer weight $k$ on the modular group is a holomorphic function on the upper half-plane satisfyingand a holomorphy condition at the cusp. The transformation rule expresses compatibility with a group action, rather than ordinary periodicity alone: translation gives $f(z+1)=f(z)$, while inversion gives $f(-1/z)=z^{k}f(z)$.

The broader theory also permits nonholomorphic forms and groups beyond ${\mathrm{SL}}_2$. In the adelic formulation for ${\mathrm{GL}}_2$, automorphic forms are smooth, arithmetically invariant functions satisfying moderate growth and finiteness conditions under compact-group translations and central differential operators. The classical modular forms embed in this larger space. This connects number theory with representation theory and harmonic analysis.^[ref]

The constant term of an automorphic form is its average along the periodic or unipotent direction associated with a cusp. In the basic period-one upper-half-plane setting,It is the zeroth coefficient of the Fourier series in $x$. Despite the name, it may depend on $y$. For a cusp width $h$, use the normalized average $h^{-1}{\int }_0^h$ after moving the chosen cusp to infinity.^[ref]

For a weight-zero Maass form with $\Delta =-y^2({\partial }_x^2+{\partial }_y^2)$ and $\Delta f=\lambda f$, periodicity makes the averaged $x$-derivative term vanish. ThusIf $\lambda =s(1-s)$ and $s\ne 1/2$, the solutions are $A{y}^s+B{y}^{1-s}$. At $s=1/2$, the repeated root gives $y^{1/2}(A+B\log y)$. A constant Fourier mode need not therefore be a constant function on the upper half-plane.

For the constant automorphic function $f=1$, the constant term is one and $\lambda =0$. A Maass cusp form instead has identically zero constant term at every cusp. The condition concerns the whole coefficient function, rather than merely its value at one height.

For a finite-index subgroup $\Gamma$ of ${\mathrm{SL}}_2(\mathbb{Z})$, a cusp is a $\Gamma$-equivalence class of rational boundary points, including infinity:Geometrically it describes a noncompact end of the quotient of the upper half-plane. Adding these classes compactifies the corresponding modular curve. The added points describe boundary data of the quotient.^[ref]

The full modular group has one cusp: a reduced rational number $a/c$ is the image of infinity under a matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with determinant one. In contrast, ${\Gamma }_0(2)$ has inequivalent cusps represented by infinity and zero. A matrix sending infinity to zero must have $a=0$, hence $-bc=1$ and odd $c$, so it cannot belong to ${\Gamma }_0(2)$.

At infinity the cusp width is the least positive $h$ for which $z\mapsto z+h$ belongs to the effective stabilizer. The local coordinate $q=e^{2\pi iz/h}$ tends to zero as $y\to \infty$. Filling in $q=0$ adds one point. At other cusps one first changes coordinates by a matrix carrying infinity to the chosen rational boundary point. Growth and constant term of an automorphic form conditions must be checked at each inequivalent cusp.

In the weight-zero convention, a Maass form for a discrete subgroup $\Gamma \subseteq {\mathrm{SL}}_2(\mathbb{R})$ is a smooth function on the upper half-plane, invariant under $z\mapsto (az+b)/(cz+d)$, satisfying a hyperbolic Laplacian eigenvalue equation and specified growth conditions at the cusps:Additional conditions specify a cuspidal or square-integrable class. The sign of $\Delta$ is a convention and must be stated.^[ref]

The constant function $f=1$ is an invariant eigenfunction with $\lambda =0$ and moderate growth, but is not cuspidal: its average along a horizontal period is nonzero. A Maass cusp form has zero such constant term at each cusp. The eigenfunction condition allows nonholomorphic automorphic forms, connecting arithmetic symmetries with spectral geometry.

A shifted polyharmonic Maass form is a smooth automorphic function that satisfies a generalized eigenfunction equation for a hyperbolic Laplacian. In weight zero, choose the convention $\Delta =-y^2({\partial }_x^2+{\partial }_y^2)$ and requirefor an integer $m\ge 1$, together with automorphy and the specified growth conditions at each cusp. Moderate growth means at most polynomial growth in cusp coordinates. The weight-$k$ version replaces $\Delta$ by the corresponding Weight-k hyperbolic Laplacian and uses the weight-$k$ transformation law.^[ref]

The least possible $m$ is the exact depth. For $m=1$ this is an ordinary eigenfunction condition; higher depth allows a chain of generalized eigenfunctions. It is not merely an equation with a different eigenvalue. The cited paper uses the opposite Laplacian sign, so its eigenvalue parameter is the negative of the one used here.

For an illuminating construction, let a nonholomorphic Eisenstein series $E(z,s)$ satisfy $\Delta E=s(1-s)E$. At a regular value $s=2$, differentiate in $s$. The function $F={\partial }_{s}E{|}_{s=2}$ obeysThe first right-hand side is nonzero, so this gives depth two. Parameter differentiation introduces a generalized eigenfunction while preserving automorphy.^[ref]

In the integer-depth convention, a polyharmonic Maass form is a smooth automorphic function annihilated by a positive integer power of its weight-appropriate hyperbolic Laplacian, with specified growth at the cusps. In weight zero, using $\Delta =-y^2({\partial }_x^2+{\partial }_y^2)$, the equation is ${\Delta }^{m}F=0$. It is the zero-shift case of a shifted polyharmonic Maass form. Exact depth is the least such integer $m$. Some refined conventions also use half-integer depths, so the convention must be stated.^[ref]

A global depth-two example comes from the finite part of a completed nonholomorphic Eisenstein series. Near $s=1$, writewhere $c$ is a nonzero constant and $\Delta E^{\ast }=s(1-s)E^{\ast }$. Comparing constant Laurent coefficients gives $\Delta F=-c$, hence ${\Delta }^{2}F=0$ but $\Delta F\ne 0$. The finite part is automorphic and has moderate growth.^[ref]

This demonstrates why higher depth is substantive: the space contains functions that are not harmonic but become harmonic after one application of the Laplacian. Merely solving a local differential equation is insufficient; automorphy and cusp growth remain part of the definition.

A Maass cusp form is a nonzero Maass form with vanishing constant term of an automorphic form at every cusp. For weight zero on the modular group $\Gamma ={\mathrm{SL}}_2(\mathbb{Z})$, it is a smooth, $\Gamma$-invariant function on the upper half-plane, square-integrable on a fundamental domain with measure $dxdy/y^2$, satisfyingThe integral condition describes the cusp at infinity; other groups require the corresponding condition in coordinates at each cusp.^[ref]

Translation invariance gives a Fourier series. When $\lambda =1/4+r^2$, its decaying modes have the formwhere $K_{ir}$ is a Modified Bessel function of the second kind. The absent $n=0$ term expresses cuspidality. The Fourier coefficients must also satisfy the full automorphic symmetry.^[ref]

The constant function $f=1$ is a useful test: it satisfies the symmetry and has eigenvalue zero, but its horizontal average is $1$, so it is not a Maass cusp form. The extra condition isolates oscillating eigenfunctions connecting arithmetic symmetry with spectral geometry.

A smooth automorphic form is an automorphic form whose archimedean variables are infinitely differentiable, without necessarily being holomorphic. Conventions differ over whether compact-group finiteness is included or one takes a larger smooth representation space. In the classical finite-vector convention on $G={\mathrm{SL}}_2(\mathbb{R})$, one asks for left invariance under an arithmetic subgroup $\Gamma$, moderate growth, finite-dimensional span under right translations by $K=\mathrm{SO}(2)$, and finiteness under the centre of the universal enveloping algebra.^[ref]

The constant function on $\Gamma \backslash G$ is an example: it is smooth, bounded, $K$-invariant and annihilated by positive-order invariant differentiation. More interesting examples come from Maass forms, whose Laplacian eigenfunction condition replaces holomorphy. Smoothness is therefore a regularity requirement, not a claim that the Cauchy-Riemann equations hold.