Siegel upper half-space
In
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
The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure , on the underlying dimensional real vector space , that is, the set of such that and for all vectors .[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
- 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: MR 2409679
- Nielsen, Frank (2020), "The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain", Entropy, 22 (9): 1019, PMID 33286788
- S2CID 124337559