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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Siegel upper half-space canonical | 4 |
| Siegel upper half-space of degree g | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10543864 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Siegel upper half-space Context triple: [Carl Ludwig Siegel, notableWork, Siegel upper half-space]
-
A.
Poincaré upper half-plane model
The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
-
B.
Hurwitz quaternions
Hurwitz quaternions are a specific lattice of quaternions with integer and half-integer components that form a maximal order in the quaternion algebra and provide a natural algebraic framework for understanding representations of integers as sums of four squares.
-
C.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
D.
Blaschke
Blaschke is a German surname most notably associated with Wilhelm Blaschke, a prominent mathematician known for his contributions to differential and convex geometry.
-
E.
Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Siegel upper half-space Target entity description: 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.
-
A.
Poincaré upper half-plane model
The Poincaré upper half-plane model is a standard representation of the hyperbolic plane using the complex numbers with positive imaginary part, equipped with a specific metric that makes geodesics appear as semicircles and vertical lines.
-
B.
Hurwitz quaternions
Hurwitz quaternions are a specific lattice of quaternions with integer and half-integer components that form a maximal order in the quaternion algebra and provide a natural algebraic framework for understanding representations of integers as sums of four squares.
-
C.
Klein quartic
The Klein quartic is a highly symmetric algebraic curve of genus 3 that plays a central role in complex geometry, group theory, and the study of Riemann surfaces.
-
D.
Blaschke
Blaschke is a German surname most notably associated with Wilhelm Blaschke, a prominent mathematician known for his contributions to differential and convex geometry.
-
E.
Gaussian rationals ℚ(i)
Gaussian rationals ℚ(i) are the field of complex numbers whose real and imaginary parts are rational, formed by adjoining the imaginary unit i to the rational numbers.
- F. None of above. chosen
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) ⓘ |
| hasStructure |
Kähler manifold structure
ⓘ
Riemannian symmetric space structure ⓘ complex analytic structure ⓘ |
| isHomogeneousSpaceOf |
real symplectic group Sp(2g,R)
ⓘ
linked to:
symplectic group Sp(2n,R)
|
| isomorphicTo | Sp(2g,R)/U(g) as a symmetric space ⓘ |
| metric | invariant Kähler metric induced by the symplectic group ⓘ |
| namedAfter | Carl Ludwig Siegel ⓘ |
| 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
ⓘ
linked to:
Siegel domain
Siegel modular form ⓘ Siegel modular group Sp(2g,Z) ⓘ Siegel modular variety ⓘ
linked to:
Siegel modular varieties
|
| 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Siegel upper half-space Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Siegel upper half-space