Siegel upper half-space

Short description: Space of complex matrices with positive definite imaginary part

In mathematics, given a positive integer $g$, the Siegel upper half-space ${{\mathscr{H}}}_g$ of degree $g$ is the set of $g\times g$ symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939). The space ${{\mathscr{H}}}_g$ is the symmetric space associated to the symplectic group $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$. When $g=1$ one recovers the Poincaré upper half-plane.

The space ${{\mathscr{H}}}_g$ is sometimes called the Siegel upper half-plane.

Definitions

As a complex domain

The space ${{\mathscr{H}}}_g$ is the subset of $M_g(\mathbb{ℂ})$ defined by :

${{\mathscr{H}}}_g=\{X+iY:X,Y\in M_g(\mathbb{ℝ}),X^t=X,Y^t=Y,Y\text{ is definite positive}\}.$

It is an open subset in the space of $g\times g$ complex symmetric matrices, hence it is a complex manifold of complex dimension $\frac{g(g+1)}{2}$.

This is a special case of a Siegel domain.

As a symmetric space

The symplectic group $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$ can be defined as the following matrix group:

$\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})=\{\left(\begin{array}{cc}A& B\\ C& D\end{array}\right):A,B,C,D\in M_g(\mathbb{ℝ}),A{B}^t-B{A}^t=0,C{D}^t-D{C}^t=0,A{D}^t-B{C}^t=1_g\}.$

It acts on ${{\mathscr{H}}}_g$ as follows:

$Z\mapsto (AZ+B)(CZ+D{)}^{-1}\text{ where }Z\in {{\mathscr{H}}}_g,\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in {\mathrm{S}\mathrm{p}}_{2g}(\mathbb{ℝ}).$

This action is continuous, faithful and transitive. The stabiliser of the point $i{1}_g\in {{\mathscr{H}}}_g$ for this action is the unitary subgroup $U(g)$, which is a maximal compact subgroup of $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$. Hence ${{\mathscr{H}}}_g$ is diffeomorphic to the symmetric space of $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$.

An invariant Riemannian metric on ${{\mathscr{H}}}_g$ can be given in coordinates as follows:

$d{s}^2=\text{tr}(Y^{-1}dZ{Y}^{-1}d\overline{Z}),Z=X+iY.$

Relation with moduli spaces of Abelian varieties

Siegel modular group

The Siegel modular group is the arithmetic subgroup ${\mathrm{\Gamma }}_g=\mathrm{S}\mathrm{p}(2g,\mathbb{\mathbb{Z}})$ of $\mathrm{S}\mathrm{p}(2g,\mathbb{ℝ})$.

Moduli spaces

Main article: Moduli of abelian varieties

The quotient of ${{\mathscr{H}}}_g$ by ${\mathrm{\Gamma }}_g$ can be interpreted as the moduli space of $g$-dimensional principally polarised complex Abelian varieties as follows. If $\tau =X+iY\in {{\mathscr{H}}}_g$ then the positive definite Hermitian form $H$ on ${\mathbb{ℂ}}^g$ defined by $H(z,w)=w^{*}{Y}^{-1}z$ takes integral values on the lattice ${\mathbb{\mathbb{Z}}}^g+{\mathbb{\mathbb{Z}}}^g\tau$We view elements of ${\mathbb{\mathbb{Z}}}^g$ as row vectors hence the left-multiplication. Thus the complex torus ${\mathbb{ℂ}}^g/{\mathbb{\mathbb{Z}}}^g+{\mathbb{\mathbb{Z}}}^g\tau$ is a Abelian variety and $H$ is a polarisation of it. The form $H$ is unimodular which means that the polarisation is principal. This construction can be reversed, hence the quotient space ${\mathrm{\Gamma }}_g\mathrm{\setminus }{{\mathscr{H}}}_g$ parametrises principally polarised Abelian varieties.

See also

• Paramodular group, a generalization of the Siegel modular group
• Siegel modular form, a type of automorphic form defined on the Siegel upper half-space
• Siegel modular variety, a moduli space constructed as a quotient of the Siegel upper half-space

References

• Bowman, Joshua P.. "Some Elementary Results on the Siegel Half-plane". https://pi.math.cornell.edu/~bowman/siegel.pdf. .

• van der Geer, Gerard (2008), "Siegel modular forms and their applications", in Ranestad, Kristian, The 1-2-3 of modular forms, Universitext, Berlin: Springer-Verlag, pp. 181–245, doi:10.1007/978-3-540-74119-0, ISBN 978-3-540-74117-6 

• Siegel, Carl Ludwig (1939), "Einführung in die Theorie der Modulfunktionen n-ten Grades", Mathematische Annalen 116: 617–657, doi:10.1007/BF01597381, ISSN 0025-5831