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).

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, C). 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, C) = Sp(2, C), the Siegel upper half-space has only one metric up to scaling whose isometry group is Sp(2g, C). 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, C) are proportional to

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

See also

• 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
• Moduli of abelian varieties

References

• 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), "Hilbert geometry of the Siegel disk: The Siegel-Klein disk model", arXiv:2004.08160 [cs.CG]

• 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