Weil conjectures
E244835
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
All labels observed (9)
How this entity was disambiguated
This entity first appeared as the object of triple T2228027 — 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: Weil conjectures Context triple: [André Weil, notableWork, Weil conjectures]
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A.
Hasse–Weil 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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B.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
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C.
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.
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D.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
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E.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
- 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: Weil conjectures Target entity description: The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
A.
Hasse–Weil 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.
-
B.
Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
-
C.
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.
-
D.
Hasse–Arf theorem
The Hasse–Arf theorem is a fundamental result in algebraic number theory that precisely characterizes the jumps in the ramification filtration of abelian extensions of local fields, showing they occur at integer values.
-
E.
Chebotarev density theorem
The Chebotarev density theorem is a fundamental result in algebraic number theory that generalizes the prime number theorem to describe how often primes in a number field have a given Frobenius conjugacy class in its Galois group.
- F. None of above. chosen
Statements (61)
| Predicate | Object |
|---|---|
| instanceOf |
result in arithmetic geometry
ⓘ
set of mathematical conjectures ⓘ |
| analogousTo |
Riemann hypothesis
ⓘ
surface form:
Riemann hypothesis for the Riemann zeta function
|
| appliesTo |
algebraic varieties over finite fields
ⓘ
smooth projective varieties over finite fields ⓘ |
| BettiNumbersPartProvedBy | Alexander Grothendieck ⓘ |
| concerns |
analogy with the Riemann zeta function
ⓘ
cohomology of algebraic varieties ⓘ counting points on varieties over finite fields ⓘ zeta function of a variety over a finite field ⓘ |
| field |
algebraic geometry
ⓘ
arithmetic geometry ⓘ number theory ⓘ |
| finalProofOfRiemannHypothesisPartBy | Pierre Deligne ⓘ |
| finalProofYearOfRiemannHypothesisPart | 1974 ⓘ |
| formulatedBy | André Weil ⓘ |
| formulationYear | 1949 ⓘ |
| functionalEquationPartProvedBy | Alexander Grothendieck ⓘ |
| hasPart |
Betti numbers
ⓘ
surface form:
Betti numbers conjecture
Weil conjectures self-linksurface differs ⓘ
surface form:
Riemann hypothesis over finite fields
functional equation conjecture ⓘ rationality conjecture ⓘ |
| implies |
Weil conjectures
self-linksurface differs
ⓘ
surface form:
Weil bounds for curves over finite fields
estimates for number of rational points on varieties over finite fields ⓘ |
| importance | central result in arithmetic geometry ⓘ |
| inspiredBy |
Hasse–Weil zeta function
ⓘ
Riemann hypothesis ⓘ |
| language | French ⓘ |
| mainTopic | zeta functions of algebraic varieties over finite fields ⓘ |
| motivatedDevelopmentOf |
Grothendieck’s standard conjectures on algebraic cycles
ⓘ
modern algebraic geometry ⓘ étale cohomology ⓘ ℓ-adic cohomology ⓘ |
| namedAfter | André Weil ⓘ |
| partialProofBy |
Alexander Grothendieck
ⓘ
Jean-Louis Verdier ⓘ Michael Artin ⓘ |
| provedBy |
Alexander Grothendieck
ⓘ
Jean-Louis Verdier ⓘ Michael Artin ⓘ Pierre Cartier ⓘ Pierre Deligne ⓘ |
| provedUsing |
Deligne’s theory of weights
ⓘ
Grothendieck’s theory of schemes ⓘ Grothendieck’s theory of weights ⓘ Lefschetz fixed-point theorem ⓘ
surface form:
Lefschetz trace formula
étale cohomology ⓘ ℓ-adic cohomology ⓘ |
| publishedIn |
Comptes rendus de l’Académie des sciences
ⓘ
surface form:
Comptes Rendus de l’Académie des Sciences
|
| rationalityPartProvedBy | Alexander Grothendieck ⓘ |
| relatedTo |
Hasse–Weil zeta function
ⓘ
Weil conjectures self-linksurface differs ⓘ
surface form:
Weil bounds
Weil conjectures on Tamagawa numbers ⓘ standard conjectures on algebraic cycles ⓘ |
| RiemannHypothesisPartProvedBy | Pierre Deligne ⓘ |
| status | proved ⓘ |
| usesConcept |
Frobenius endomorphism
ⓘ
Künneth formula ⓘ Poincaré duality ⓘ cohomological dimension ⓘ eigenvalues of Frobenius ⓘ |
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
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Input
Subject: Weil conjectures Description of subject: The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
Referenced by (19)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
La conjecture de Weil I
this entity surface form:
Weil Conjectures
this entity surface form:
Weil conjectures for varieties over finite fields
this entity surface form:
Weil conjectures for curves
subject surface form:
André Weil
this entity surface form:
Riemann hypothesis over finite fields
this entity surface form:
Weil bounds for curves over finite fields
this entity surface form:
Weil bounds
this entity surface form:
Riemann hypothesis for curves over finite fields