Siegel modular form
E871397
A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Siegel modular form canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10543859 — 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 modular form Context triple: [Carl Ludwig Siegel, notableWork, Siegel modular form]
-
A.
modular j-invariant
The modular j-invariant is a fundamental modular function that classifies complex elliptic curves up to isomorphism and plays a central role in number theory, complex analysis, and the theory of modular forms.
-
B.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
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C.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
D.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
-
E.
Shimura varieties
Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
- 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 modular form Target entity description: A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
-
A.
modular j-invariant
The modular j-invariant is a fundamental modular function that classifies complex elliptic curves up to isomorphism and plays a central role in number theory, complex analysis, and the theory of modular forms.
-
B.
Hecke operators
Hecke operators are algebraic operators acting on modular forms that play a central role in number theory, particularly in understanding congruences, L-functions, and the arithmetic of modular forms.
-
C.
Selberg trace formula
The Selberg trace formula is a fundamental result in analytic number theory and spectral theory that relates lengths of closed geodesics on a Riemannian manifold to the spectrum of its Laplace operator, serving as a non-abelian analogue of the Poisson summation formula.
-
D.
Hecke theory
Hecke theory is a branch of number theory centered on Hecke operators and modular forms, providing powerful tools to study arithmetic properties of modular forms and related objects.
-
E.
Shimura varieties
Shimura varieties are higher-dimensional algebraic varieties that generalize modular curves and play a central role in the Langlands program by connecting number theory, automorphic forms, and arithmetic geometry.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
automorphic form
ⓘ
complex analytic function ⓘ mathematical object ⓘ |
| appearsIn | theory of Siegel modular threefolds ⓘ |
| canBeInterpretedAs | section of line bundle on Siegel modular variety ⓘ |
| cuspFormCondition | vanishing at all cusps ⓘ |
| definedOn | Siegel upper half-space NERFINISHED ⓘ |
| degree | positive integer g ⓘ |
| degree1SpecialCase | elliptic modular form ⓘ |
| domain | space of symmetric complex matrices with positive definite imaginary part ⓘ |
| field |
algebraic geometry
ⓘ
number theory ⓘ |
| FourierCoefficientsIndexedBy | symmetric, semi-positive definite integral matrices ⓘ |
| generalizes |
Eisenstein series
NERFINISHED
ⓘ
classical modular form ⓘ cusp forms ⓘ |
| group | symplectic group Sp(2g,ℤ) NERFINISHED ⓘ |
| groupAction | fractional linear transformation on Siegel upper half-space ⓘ |
| hasCongruenceSubgroupVersion | Siegel modular form for congruence subgroup of Sp(2g,ℤ) GENERATED ⓘ |
| hasFourierExpansion | yes ⓘ |
| hasHeckeAction | Hecke operators for Sp(2g,ℤ) ⓘ |
| hasParameter |
character
ⓘ
degree ⓘ level ⓘ weight ⓘ |
| hasStructure | graded ring by weight ⓘ |
| hasVectorValuedVersion | vector-valued Siegel modular form ⓘ |
| namedAfter | Carl Ludwig Siegel NERFINISHED ⓘ |
| playsCentralRoleIn |
Langlands program
NERFINISHED
ⓘ
higher-dimensional modular form theory ⓘ |
| relatedTo |
Jacobi form
NERFINISHED
ⓘ
Siegel modular variety NERFINISHED ⓘ Siegel upper half-space NERFINISHED ⓘ automorphic representation ⓘ theta function ⓘ |
| specialCase |
Siegel Eisenstein series
NERFINISHED
ⓘ
Siegel cusp form ⓘ |
| studiedBy |
Carl Ludwig Siegel
NERFINISHED
ⓘ
Goro Shimura NERFINISHED ⓘ Jun-ichi Igusa NERFINISHED ⓘ |
| transformationProperty | invariance under symplectic group action ⓘ |
| usedIn |
arithmetic geometry
ⓘ
moduli of principally polarized abelian varieties ⓘ theory of L-functions ⓘ theory of abelian varieties ⓘ |
| usedToConstruct | Siegel modular L-function NERFINISHED ⓘ |
| usedToDefine | Hecke eigenforms for symplectic groups ⓘ |
| weightCondition | holomorphic with polynomial growth at infinity ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Siegel modular form Description of subject: A Siegel modular form is a type of complex analytic function defined on the Siegel upper half-space that generalizes classical modular forms to higher dimensions and plays a central role in number theory and algebraic geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.