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 ${\displaystyle J\in Hom(V)}$ such that ${\displaystyle J^{2}=-1}$ and ${\displaystyle \omega (Jv,v)>0}$ for all vectors ${\displaystyle v\neq 0}$. ^[1]

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

• 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

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