Siegel upper half-space
E871400
The Siegel upper half-space is a higher-dimensional generalization of the complex upper half-plane that serves as a fundamental domain in the theory of Siegel modular forms and symplectic geometry.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
Hermitian symmetric domain
ⓘ
bounded symmetric domain (up to biholomorphism) ⓘ mathematical object ⓘ |
| admitsAction | Sp(2g,R) by fractional linear transformations ⓘ |
| appearsIn |
Arakelov geometry and moduli of abelian varieties
ⓘ
study of period matrices of Riemann surfaces ⓘ theory of theta functions ⓘ |
| boundaryStructure | has Satake and Baily–Borel compactification boundaries via arithmetic quotients ⓘ |
| compactDual | Lagrangian Grassmannian of a complex symplectic vector space ⓘ |
| coordinateDescription | H_g = { Z in M_g(C) | Z^T = Z, Im(Z) > 0 } ⓘ |
| definedAs | set of complex symmetric g×g matrices with positive definite imaginary part ⓘ |
| dimensionFormula | g(g+1)/2 as a complex manifold ⓘ |
| field |
algebraic geometry
ⓘ
automorphic forms ⓘ complex analysis ⓘ number theory ⓘ symplectic geometry ⓘ |
| generalizationOf | complex upper half-plane ⓘ |
| hasIsotropyGroup | unitary group U(g) NERFINISHED ⓘ |
| hasStructure |
Kähler manifold structure
ⓘ
Riemannian symmetric space structure ⓘ complex analytic structure ⓘ |
| isHomogeneousSpaceOf | real symplectic group Sp(2g,R) NERFINISHED ⓘ |
| isomorphicTo | Sp(2g,R)/U(g) as a symmetric space ⓘ |
| metric | invariant Kähler metric induced by the symplectic group ⓘ |
| namedAfter | Carl Ludwig Siegel NERFINISHED ⓘ |
| parameter | positive integer g (genus) ⓘ |
| property |
contractible
ⓘ
negatively curved (in the sense of its symmetric space metric) ⓘ non-compact ⓘ simply connected ⓘ |
| relatedConcept |
Siegel domain of the first kind
NERFINISHED
ⓘ
Siegel modular form ⓘ Siegel modular group Sp(2g,Z) NERFINISHED ⓘ Siegel modular variety NERFINISHED ⓘ |
| relatedTo | moduli space of curves via Torelli map ⓘ |
| role |
parameter space for principally polarized abelian varieties (via quotient by arithmetic groups)
ⓘ
universal covering space of Siegel modular varieties ⓘ |
| servesAs |
domain of definition for Siegel modular forms
ⓘ
period domain for polarized Hodge structures of weight 1 ⓘ |
| specialCase | H_1 is the complex upper half-plane ⓘ |
| symbol | H_g ⓘ |
| usedIn |
moduli theory
ⓘ
representation theory of symplectic groups ⓘ theory of Siegel modular forms ⓘ theory of abelian varieties ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.