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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.