Siegel upper half-space

In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939). It is the symmetric space associated to the symplectic group Sp(2g, R).

The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case g = 1. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group Sp(2g, R). Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group SL(2, R) = Sp(2, R), the Siegel upper half-space has only one metric up to scaling whose isometry group is Sp(2g, R). Writing a generic matrix Z in the Siegel upper half-space in terms of its real and imaginary parts as Z = X + iY, all metrics with isometry group Sp(2g, R) are proportional to

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

The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure ${\displaystyle \omega }$, on the underlying ${\displaystyle 2n}$ dimensional real vector space ${\displaystyle V}$, that is, the set of ${\displaystyle J\in \mathrm {Hom} (V)}$ such that ${\displaystyle J^{2}=-1}$ and ${\displaystyle \omega (Jv,v)>0}$ for all vectors ${\displaystyle v\neq 0}$.^[1]

As a symmetric space of non-compact type, the Siegel upper half space ${\displaystyle {\mathcal {H}}_{g}}$ is the quotient

${\displaystyle {\mathcal {H}}_{g}=\mathrm {Sp} (2g,\mathbb {R} )/\mathrm {U} (n),}$

where we used that ${\displaystyle \mathrm {U} (n)=\mathrm {Sp} (2g,\mathbb {R} )\cap \mathrm {GL} (g,\mathbb {C} )}$ is the maximal torus. Since the isometry group of a symmetric space ${\displaystyle G/K}$ is ${\displaystyle G}$, we recover that the isometry group of ${\displaystyle {\mathcal {H}}_{g}}$ is ${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )}$. An isometry acts via a generalized Möbius transformation

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

The quotient space ${\displaystyle {\mathcal {H}}_{g}/\mathrm {Sp} (2g,\mathbb {Z} )}$ is the moduli space of principally polarized abelian varieties of dimension ${\displaystyle g}$.

See also

• Moduli of abelian varieties
• Paramodular group, a generalization of the Siegel modular group
• Siegel domain, a generalization of the Siegel upper half space
• 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. Bowman

• 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

• Nielsen, Frank (2020), "The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain", Entropy, 22 (9): 1019, arXiv:2004.08160, doi:10.3390/e22091019, PMC 7597112, PMID 33286788

• 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