---
tag: 3R959YSNYP
kind: concept
name: Maass wave form
msc: [11F37, 11F12, 11F72]
specializes: [3C73HND6XA]
sources:
  - url: https://en.wikipedia.org/wiki/Maass_wave_form
same_as:
  wikidata: Q6721246
  wikipedia: Maass wave form
---

## Summary

A Maass wave form is a smooth function on the [[upper half-plane|X6XGGH5FPW]] that is invariant under a [[congruence subgroup|1H6R856P3C]] of the [[modular group|VEKV1EM77B]].
It is also an eigenfunction of the hyperbolic [[Laplace–Beltrami operator|F4NWZNTRMD]].
It is the non-holomorphic counterpart of a [[modular form|GWKN5VG78A]].
Maass forms are basic examples of [[automorphic forms|3C73HND6XA]], and their eigenvalues are the subject of spectral theory and trace formulas.

## Explanations

### L1 human

Think of the upper half-plane as a drum whose shape is folded up by the modular group.
A Maass form is a standing wave on this drum: a smooth wave that repeats under the symmetries and has a pure "frequency", its Laplace eigenvalue.
Holomorphic modular forms are rigid objects built from complex analysis; Maass forms are the wave-like analogues that need no complex differentiability.
A simple example is the real-analytic Eisenstein series $E(z, s)$, an average of the function $y^s$ over the modular group.
Square-integrable Maass forms for $\mathrm{SL}_2(\Z)$ exist, but no one knows a closed formula for any of them.