Hasse–Weil zeta function

E207313

analytic function arithmetical function object in arithmetic geometry zeta function

The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.

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All labels observed (4)

Label	Occurrences
Hasse–Weil zeta function canonical	4
Hasse–Weil L-function	2
Hasse–Weil L-functions	1
Hasse–Weil zeta function of an elliptic curve	1

How this entity was disambiguated

This entity first appeared as the object of triple T1862415 — 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: Hasse–Weil zeta function
Context triple: [Helmut Hasse, notableWork, Hasse–Weil zeta function]

A. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
B. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
C. Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
D. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
E. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
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: Hasse–Weil zeta function
Target entity description: The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.

A. Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
B. Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
C. Euler product formula for the Riemann zeta function
The Euler product formula for the Riemann zeta function is a fundamental identity in analytic number theory that expresses the zeta function as an infinite product over all prime numbers, revealing a deep connection between primes and the distribution of integers.
D. Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
E. Birch and Swinnerton-Dyer Conjecture
The Birch and Swinnerton-Dyer Conjecture is a central unsolved problem in number theory that predicts a deep connection between the arithmetic of rational points on an elliptic curve and the behavior of its associated L-function at a specific value.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	analytic function ⓘ arithmetical function ⓘ object in arithmetic geometry ⓘ zeta function ⓘ
associatedWith	algebraic variety over a number field ⓘ scheme of finite type over Spec of a number field ⓘ
canBeExpressedAs	product of local factors at finite and infinite places ⓘ
constructedFrom	Euler product over primes of a number field ⓘ local zeta factors at all places of a number field ⓘ
definedFor	algebraic varieties over global fields ⓘ algebraic varieties over number fields ⓘ
dependsOn	numbers of points over finite field extensions ⓘ
encodes	Frobenius eigenvalues on étale cohomology ⓘ arithmetic information of algebraic varieties ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ number theory ⓘ
generalizes	Riemann zeta function ⓘ
hasConjecturedProperty	meromorphic continuation to the whole complex plane ⓘ satisfies a functional equation relating s and 1 − s up to normalization ⓘ
hasDomain	complex plane ⓘ
hasProperty	admits an Euler product factorization ⓘ expected to satisfy a functional equation ⓘ expected to satisfy analytic continuation ⓘ
hasVariable	complex variable s ⓘ
localFactorAt	archimedean place of the base number field ⓘ finite prime of the base number field ⓘ
namedAfter	André Weil ⓘ Helmut Hasse ⓘ
playsRoleIn	global class field theory ⓘ modern arithmetic geometry ⓘ
relatedTo	Birch and Swinnerton-Dyer Conjecture ⓘ surface form: Birch–Swinnerton-Dyer conjecture Galois representations ⓘ L-functions ⓘ Taniyama–Shimura–Weil conjecture ⓘ Weil conjectures ⓘ Galois representations ⓘ surface form: Weil–Deligne representations automorphic L-functions ⓘ étale cohomology ⓘ
satisfies	Weil conjectures ⓘ surface form: Weil conjectures for varieties over finite fields
specialCase	Dedekind zeta functions ⓘ surface form: Dedekind zeta function for number fields Riemann zeta function for the projective line over the integers ⓘ
specialCaseOf	motivic L-function ⓘ
studiedIn	Langlands program ⓘ
usedIn	formulation of the Birch–Swinnerton-Dyer conjecture for elliptic curves ⓘ formulation of the Tate conjecture ⓘ proofs and formulations of modularity theorems ⓘ study of rational points on varieties ⓘ

How these facts were elicited

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

Helmut Hasse → notableWork → Hasse–Weil zeta function ⓘ

Riemann zeta function → generalization → Hasse–Weil zeta function ⓘ

this entity surface form: Hasse–Weil L-functions

Birch and Swinnerton-Dyer Conjecture → relatesConcept → Hasse–Weil zeta function ⓘ

this entity surface form: Hasse–Weil L-function

Hasse bound for elliptic curves → relatedConcept → Hasse–Weil zeta function ⓘ

this entity surface form: Hasse–Weil zeta function of an elliptic curve

Weil conjectures → inspiredBy → Hasse–Weil zeta function ⓘ

Weil conjectures → relatedTo → Hasse–Weil zeta function ⓘ

Grothendieck–Ogg–Shafarevich formula → isRelatedTo → Hasse–Weil zeta function ⓘ

L-functions → hasSpecialCase → Hasse–Weil zeta function ⓘ

subject surface form: L-function