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 (3)
| Label | Occurrences |
|---|---|
| Minkowski–Siegel formula | 1 |
| Siegel mass formula canonical | 1 |
| Smith–Minkowski–Siegel mass formula | 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
ⓘ
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 ⓘ |
| provenBy | Carl Ludwig Siegel ⓘ |
| relatedTo |
Minkowski–Siegel formula
ⓘ
linked to:
Siegel mass formula
Smith–Minkowski–Siegel mass formula ⓘ
linked to:
Siegel mass formula
Tamagawa measures ⓘ theta correspondence ⓘ |
| relates |
masses of quadratic forms
ⓘ
special values of L-functions ⓘ special values of zeta functions ⓘ |
| usesConcept |
Eisenstein series
ⓘ
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
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 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 (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Siegel mass formula
linked to: Siegel mass formula