Siegel mass formula
E871399
The Siegel mass formula is a fundamental result in number theory that relates the weighted count (mass) of quadratic forms in a given genus to special values of zeta and L-functions, providing deep connections between arithmetic and geometry.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Siegel mass formula canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10543863 — 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 mass formula Context triple: [Carl Ludwig Siegel, notableWork, Siegel mass formula]
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A.
Higher composition laws I–IV
Higher composition laws I–IV is a landmark four-part series of papers by Manjul Bhargava that generalizes Gauss’s composition of binary quadratic forms and develops new structures and methods in algebraic number theory.
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B.
Rational Quadratic Forms
Rational Quadratic Forms is a classic monograph in number theory that systematically develops the arithmetic theory of quadratic forms over the rational numbers.
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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.
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D.
Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
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E.
Siegel's theorem on integral points
Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
- 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 mass formula Target entity description: The Siegel mass formula is a fundamental result in number theory that relates the weighted count (mass) of quadratic forms in a given genus to special values of zeta and L-functions, providing deep connections between arithmetic and geometry.
-
A.
Higher composition laws I–IV
Higher composition laws I–IV is a landmark four-part series of papers by Manjul Bhargava that generalizes Gauss’s composition of binary quadratic forms and develops new structures and methods in algebraic number theory.
-
B.
Rational Quadratic Forms
Rational Quadratic Forms is a classic monograph in number theory that systematically develops the arithmetic theory of quadratic forms over the rational numbers.
-
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.
Siegel’s theorem on zeros of L-functions
Siegel’s theorem on zeros of L-functions is a result in analytic number theory that gives strong bounds on how close nontrivial zeros of Dirichlet L-functions can approach 1, with deep implications for the distribution of primes in arithmetic progressions.
-
E.
Siegel's theorem on integral points
Siegel's theorem on integral points is a fundamental result in number theory and Diophantine geometry stating that certain algebraic curves, notably those of genus at least one, have only finitely many integral points.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
result in number theory ⓘ |
| appliesTo |
genera of quadratic forms
ⓘ
lattices ⓘ quadratic forms ⓘ |
| characterizes | mass of a genus of quadratic forms ⓘ |
| concerns |
indefinite quadratic forms
ⓘ
integral quadratic forms ⓘ positive definite quadratic forms ⓘ rational quadratic forms ⓘ |
| connects |
arithmetic of quadratic forms and analytic number theory
ⓘ
global arithmetic invariants ⓘ local representation theory of quadratic forms ⓘ |
| field |
arithmetic geometry
ⓘ
arithmetic of quadratic forms ⓘ number theory ⓘ |
| formalism |
adelic orthogonal groups
ⓘ
orthogonal group of a quadratic space ⓘ |
| gives |
expression for mass as product of local factors
ⓘ
expression involving Dedekind zeta functions ⓘ expression involving Dirichlet L-functions ⓘ product formula for the mass of a genus ⓘ |
| hasGeneralization |
Tamagawa number formula for orthogonal groups
NERFINISHED
ⓘ
mass formula for hermitian forms ⓘ mass formula for quadratic lattices over number fields ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| influenced |
development of the theory of Tamagawa numbers
ⓘ
modern theory of automorphic forms ⓘ |
| involves |
product of local densities at all primes
ⓘ
volume of quotient of orthogonal group ⓘ |
| isToolFor |
classification of quadratic forms
ⓘ
computation of class numbers of quadratic forms ⓘ study of lattices in Euclidean space ⓘ study of representation numbers of quadratic forms ⓘ |
| namedAfter | Carl Ludwig Siegel NERFINISHED ⓘ |
| provenBy | Carl Ludwig Siegel NERFINISHED ⓘ |
| relatedTo |
Minkowski–Siegel formula
NERFINISHED
ⓘ
Smith–Minkowski–Siegel mass formula NERFINISHED ⓘ Tamagawa measures NERFINISHED ⓘ theta correspondence ⓘ |
| relates |
masses of quadratic forms
ⓘ
special values of L-functions ⓘ special values of zeta functions ⓘ |
| usesConcept |
Eisenstein series
NERFINISHED
ⓘ
adelic methods ⓘ automorphism group of a quadratic form ⓘ equivalence classes of quadratic forms ⓘ genus of quadratic forms ⓘ local densities ⓘ theta series ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Siegel mass formula Description of subject: The Siegel mass formula is a fundamental result in number theory that relates the weighted count (mass) of quadratic forms in a given genus to special values of zeta and L-functions, providing deep connections between arithmetic and geometry.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.