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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.

The hyperbolic Laplacian is the Laplace-Beltrami operator for the metric of hyperbolic space. In upper-half-space coordinates $(x_1,\dots ,x_{n-1},y)$ with $y>0$,Using the nonnegative convention $\Delta =-\mathrm{div}\mathrm{grad}$ givesIndeed, insert $g^{ij}=y^2{\delta }^{ij}$ and $\sqrt{\det g}=y^{-n}$ into the coordinate formula $-{\left(\det g\right)}^{-1/2}{\partial }_i(\sqrt{\det g}{g}^{ij}{\partial }_j)$. The choice of overall sign must be specified when comparing spectral formulas.^[ref]

In dimension two the first-derivative term disappears. For $f(y)=y^s$, direct differentiation gives $\Delta f=s(n-1-s)f$, and in particular $s(1-s)f$ in the hyperbolic plane. This is the eigenvalue expression underlying many Maass form conventions.

Despite its name, this is an elliptic differential operator on $y>0$: its leading symbol in the chosen sign convention is $y^2|\xi {|}^2$. The adjective hyperbolic refers to the geometry, not to the classification of a hyperbolic partial differential equation.

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.

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.

Robert C. Rhoades coauthored work extending the study of Maass forms beyond a single eigenfunction equation. With Jeffrey C. Lagarias, he studied polyharmonic Maass forms: functions with the required arithmetic symmetry and growth conditions that are annihilated by a power of a weight-dependent hyperbolic Laplacian, ${\Delta }_k^{m}f=0$. Their paper constructs bases using derivatives of Eisenstein series with respect to a parameter.^[ref]

An elementary analogy explains why taking a power changes the problem. For the derivative operator $D=d/dt$, the function $f(t)=t$ satisfies $D^{2}f=0$ but $Df=1\ne 0$. Thus requiring two applications to give zero allows more functions than requiring one. The arithmetic setting adds essential symmetry and growth restrictions; the analogy alone does not produce a Maass form. The paper shows how these restrictions lead to explicit finite-dimensional spaces, rather than the unrestricted solution sets of a differential equation.^[ref]

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.

The Weight-k hyperbolic Laplacian is a differential operator adapted to weight-$k$ transformation laws on the upper half-plane $z=x+iy$, $y>0$. Using the sign consistent with the nonnegative weight-zero hyperbolic Laplacian, and the holomorphic-weight convention, defineHere ${\partial }_z=\frac{1}{2}({\partial }_x-i{\partial }_y)$ and ${\partial }_{\bar{z}}=\frac{1}{2}({\partial }_x+i{\partial }_y)$ are Wirtinger derivatives. The cited paper uses the opposite overall sign, so its eigenvalues must also have their signs reversed.^[ref]

The correction terms let the operator preserve the transformation law $f((az+b)/(cz+d))=(cz+d{)}^{k}f(z)$ for determinant-one integer matrices, as used for modular forms and Maass forms. At $k=0$ it reduces to the ordinary hyperbolic Laplacian. Every holomorphic function is locally annihilated because ${\partial }_{\bar{z}}f=0$; being holomorphic alone does not supply modularity or growth conditions.

For a direct calculation, take $f(z)=y^s$. Its $x$ derivatives vanish, givingThus both $1$ and $y^{1-k}$ are annihilated; when $k=1$ these coincide and a second independent solution depending only on $y$ is $\log y$. These are local operator examples, not claims that all satisfy a modular transformation law. Conventions using a unit-modulus weight multiplier have a different first-derivative correction, so the displayed formula matters as much as the name.