---
tag: MXPTXV340K
kind: concept
name: Siegel modular form
msc: [11F46, 32N10]
sources:
  - url: https://en.wikipedia.org/wiki/Siegel_modular_form
same_as:
  wikidata: Q7510567
  wikipedia: Siegel modular form
---

## Summary

A Siegel modular form is a holomorphic function on the [[Siegel upper half-space|5PTYE97PF6]].
Under the Siegel modular group $Sp(2g, \Z)$ it transforms by a power of a determinant factor.
For $g = 1$ these are the classical [[modular forms|GWKN5VG78A]].
Siegel modular forms are sections of line bundles on the moduli space of principally polarized abelian varieties.
Their Fourier coefficients often encode arithmetic data, such as numbers of representations of one quadratic form by another.

## Explanations

### L1 human

A classical modular form is a function on the upper half-plane that changes in a prescribed way under the substitutions $z \mapsto (az + b)/(cz + d)$.
A Siegel modular form does the same in several variables: its argument is a symmetric matrix with positive definite imaginary part, and the substitutions are given by symplectic integer matrices.
Theta series of quadratic forms are typical examples.
As in one variable, the space of forms of a fixed weight is finite-dimensional.