Siegel upper half-space

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

The space ${\displaystyle {\mathcal {H}}_{g}}$ is sometimes called the Siegel upper half-plane.^[1]

Definitions

As a complex domain

The space ${\displaystyle {\mathcal {H}}_{g}}$ is the subset of ${\displaystyle M_{g}(\mathbb {C} )}$ defined by :

${\displaystyle {\mathcal {H}}_{g}=\{X+iY:X,Y\in M_{g}(\mathbb {R} ),X^{t}=X,\,Y^{t}=Y,\,Y{\text{ is definite positive}}\}.}$

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

This is a special case of a Siegel domain.

As a symmetric space

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

${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )=\left\{{\begin{pmatrix}A&B\\C&D\end{pmatrix}}:A,B,C,D\in M_{g}(\mathbb {R} ),\,AB^{t}-BA^{t}=0,CD^{t}-DC^{t}=0,AD^{t}-BC^{t}=1_{g}\right\}.}$

It acts on ${\displaystyle {\mathcal {H}}_{g}}$ as follows:

${\displaystyle Z\mapsto (AZ+B)(CZ+D)^{-1}{\text{ where }}Z\in {\mathcal {H}}_{g},{\begin{pmatrix}A&B\\C&D\end{pmatrix}}\in \mathrm {Sp} _{2g}(\mathbb {R} ).}$

This action is continuous, faithful and transitive. The stabiliser of the point ${\displaystyle i1_{g}\in {\mathcal {H}}_{g}}$ for this action is the unitary subgroup ${\displaystyle U(g)}$, which is a maximal compact subgroup of ${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )}$.^[2] Hence ${\displaystyle {\mathcal {H}}_{g}}$ is diffeomorphic to the symmetric space of ${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )}$.

An invariant Riemannian metric on ${\displaystyle {\mathcal {H}}_{g}}$ can be given in coordinates as follows:

${\displaystyle ds^{2}={\text{tr}}(Y^{-1}dZY^{-1}d{\bar {Z}}),\,Z=X+iY.}$

Relation with moduli spaces of Abelian varieties

Siegel modular group

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

Moduli spaces

Main article: Moduli of abelian varieties

The quotient of ${\displaystyle {\mathcal {H}}_{g}}$ by ${\displaystyle \Gamma _{g}}$ can be interpreted as the moduli space of ${\displaystyle g}$-dimensional principally polarised complex Abelian varieties as follows.^[3] If ${\displaystyle \tau =X+iY\in {\mathcal {H}}_{g}}$ then the positive definite Hermitian form ${\displaystyle H}$ on ${\displaystyle \mathbb {C} ^{g}}$ defined by ${\displaystyle H(z,w)=w^{*}Y^{-1}z}$ takes integral values on the lattice ${\displaystyle \mathbb {Z} ^{g}+\mathbb {Z} ^{g}\tau }$We view elements of ${\displaystyle \mathbb {Z} ^{g}}$ as row vectors hence the left-multiplication. Thus the complex torus ${\displaystyle \mathbb {C} ^{g}/\mathbb {Z} ^{g}+\mathbb {Z} ^{g}\tau }$ is a Abelian variety and ${\displaystyle H}$ is a polarisation of it. The form ${\displaystyle H}$ is unimodular which means that the polarisation is principal. This construction can be reversed, hence the quotient space ${\displaystyle \Gamma _{g}\backslash {\mathcal {H}}_{g}}$ 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

1. Friedland, Shmuel; Freitas, Pedro J. (2004). "Revisiting the Siegel upper half plane. I". Linear Algebra Appl. 376: 19–44. doi:10.1016/S0024-3795(03)00662-1.
2. van der Geer 2008, p. 185.
3. van der Geer 2008, Section 10.

• Bowman, Joshua P. "Some Elementary Results on the Siegel Half-plane" (PDF)..

• van der Geer, Gerard (2008), "Siegel modular forms and their applications", in Ranestad, Kristian (ed.), 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, MR 2409679

• 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, MR 0001251, S2CID 124337559