Math 2.0

Number theory Definition

Maass wave form

Builds on Upper half-planeModular groupCongruence subgroupSmooth functionEigenvalue

Definition

Let ℍ={z=x+iy:y>0}\mathbb{H} = \set{z = x + iy : y > 0} be the upper half-plane. Let Γ\Gamma be a congruence subgroup of SL2(ℤ)\mathrm{SL}_2(\Z), acting on ℍ\mathbb{H} by γz=az+bcz+d\gamma z = \frac{az + b}{cz + d} for γ=(abcd)\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}. Let Δ=−y2(∂2∂x2+∂2∂y2)\Delta = -y^2 \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) be the hyperbolic Laplace operator. A Maass wave form for Γ\Gamma is a smooth function f ⁣:ℍ→ℂf \colon \mathbb{H} \to \C such that:

  1. f(γz)=f(z)f(\gamma z) = f(z) for all γ∈Γ\gamma \in \Gamma and z∈ℍz \in \mathbb{H}.
  2. There is λ∈ℂ\lambda \in \C with Δf=λf\Delta f = \lambda f.
  3. ff has moderate growth at every cusp: for every σ∈SL2(ℤ)\sigma \in \mathrm{SL}_2(\Z) there is N>0N > 0 with f(σz)=O(yN)f(\sigma z) = O(y^N) as y→∞y \to \infty, uniformly in xx.

The number λ\lambda is the eigenvalue of ff. A Maass cusp form is a Maass form whose constant term ∫01f(σ(x+iy)) dx\int_0^1 f(\sigma(x + iy)) \, dx vanishes at every cusp, that is, for every σ∈SL2(ℤ)\sigma \in \mathrm{SL}_2(\Z).

Remarks

Example: for Γ=SL2(ℤ)\Gamma = \mathrm{SL}_2(\Z) and Re(s)>1\mathrm{Re}(s) > 1, the real-analytic Eisenstein series E(z,s)=12∑(c,d)∈ℤ2gcd⁡(c,d)=1ys∣cz+d∣2sE(z, s) = \frac{1}{2} \sum_{\substack{(c, d) \in \Z^2 \\ \gcd(c, d) = 1}} \frac{y^s}{\abs{cz + d}^{2s}} is a Maass form with eigenvalue s(1−s)s(1 - s). Each term is (Im γz)s(\mathrm{Im}\, \gamma z)^s for a matrix γ\gamma with bottom row (c,d)(c, d), so the sum is Γ\Gamma-invariant, and Δys=−y2s(s−1)ys−2=s(1−s)ys\Delta y^s = -y^2 s(s-1) y^{s-2} = s(1-s) y^s term by term. It is not a cusp form, since its constant term is ys+φ(s)y1−sy^s + \varphi(s) y^{1-s} for some function φ\varphi.

Non-example: f(z)=ysf(z) = y^s satisfies clauses 2 and 3 and is invariant under z↦z+1z \mapsto z + 1, but it is not invariant under z↦−1/zz \mapsto -1/z, so clause 1 fails.

A cusp form decays rapidly at the cusps and is square-integrable on Γ\ℍ\Gamma \backslash \mathbb{H}; the eigenvalue of a nonzero Maass cusp form is then real and positive. One often writes λ=14+r2\lambda = \frac{1}{4} + r^2 or λ=s(1−s)\lambda = s(1 - s). Some authors also allow a weight or a character in clause 1.

Reading path

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Complex numbers, object, no evidence levelComplex numbersobjectCongruence subgroup, concept, no evidence levelCongruence subgroupconceptEigenvalue, concept, no evidence levelEigenvalueconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMöbius transformation, concept, no evidence levelMöbius transformationconceptQuotient group, concept, no evidence levelQuotient groupconceptSmooth function, concept, no evidence levelSmooth functionconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectUpper half-plane, object, no evidence levelUpper half-planeobjectModular group, object, no evidence levelModular groupobjectMaass wave form, definition, no evidence levelMaass wave formdefinition
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  1. Complex numbers object
  2. Congruence subgroup concept
  3. Eigenvalue concept
  4. General linear group object
  5. Identity matrix object
  6. Matrix definition
  7. Möbius transformation concept
  8. Quotient group concept
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  10. Vector space definition
  11. Square matrix definition
  12. Linear map definition
  13. Multilinear map definition
  14. Alternating multilinear map definition
  15. Determinant definition
  16. Special linear group object
  17. Upper half-plane object
  18. Modular group object
  19. Maass wave form definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Maass_wave_form
About this page
Tag4GM75FC4PP
URIm2://4GM75FC4PP
Kinddefinition
WritingW0 stub
MSC11F37, 11F12, 11F72
DefinesMaass wave form
UsesUpper half-plane, Modular group, Congruence subgroup, Smooth function, Eigenvalue
Changed2 November 2026

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