Weil conjectures

E244835

result in arithmetic geometry set of mathematical conjectures

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.

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All labels observed (9)

Label	Occurrences
Weil conjectures canonical	11
La conjecture de Weil I	1
Riemann hypothesis for curves over finite fields	1
Riemann hypothesis over finite fields	1
Weil Conjectures	1
Weil bounds	1
Weil bounds for curves over finite fields	1
Weil conjectures for curves	1
Weil conjectures for varieties over finite fields	1

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 ⓘ

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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.

André Weil → notableWork → Weil conjectures ⓘ

Pierre Deligne → notableWork → Weil conjectures ⓘ

this entity surface form: La conjecture de Weil I

Hodge Conjecture → relatedTo → Weil conjectures ⓘ

this entity surface form: Weil Conjectures

Hasse–Weil zeta function → satisfies → Weil conjectures ⓘ

this entity surface form: Weil conjectures for varieties over finite fields

Hasse–Weil zeta function → relatedTo → Weil conjectures ⓘ

Hasse bound for elliptic curves → isSpecialCaseOf → Weil conjectures ⓘ

this entity surface form: Weil conjectures for curves

Diophantine geometry → relatedTo → Weil conjectures ⓘ

Weil → notableWork → Weil conjectures ⓘ

subject surface form: André Weil

Weil conjectures → hasPart → Weil conjectures self-linksurface differs ⓘ

this entity surface form: Riemann hypothesis over finite fields

Weil conjectures → implies → Weil conjectures self-linksurface differs ⓘ

this entity surface form: Weil bounds for curves over finite fields

Weil conjectures → relatedTo → Weil conjectures self-linksurface differs ⓘ

this entity surface form: Weil bounds

Sur les courbes algébriques et les variétés qui s’en déduisent → influenced → Weil conjectures ⓘ

Sur les courbes algébriques et les variétés qui s’en déduisent → relatedTo → Weil conjectures ⓘ

this entity surface form: Riemann hypothesis for curves over finite fields

Weil divisor → appearsIn → Weil conjectures ⓘ

Weil cohomology → developedInContextOf → Weil conjectures ⓘ

étale cohomology → developedInContextOf → Weil conjectures ⓘ

Grothendieck–Ogg–Shafarevich formula → isRelatedTo → Weil conjectures ⓘ

Ramanujan–Petersson conjecture → proofUses → Weil conjectures ⓘ