Hasse–Weil bound for abelian varieties

E753157

result in number theory theorem in arithmetic geometry

The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.

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All labels observed (3)

Label	Occurrences
Hasse–Weil bound for abelian varieties canonical	1
Hasse–Weil bound for curves via Jacobians	1
Serre–Tate results on abelian varieties over finite fields	1

How this entity was disambiguated

This entity first appeared as the object of triple T8733523 — 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 bound for abelian varieties
Context triple: [Hasse bound for elliptic curves, generalization, Hasse–Weil bound for abelian varieties]

A. Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
B. Weil 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.
C. 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.
D. Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
E. Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
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 bound for abelian varieties
Target entity description: The Hasse–Weil bound for abelian varieties is a fundamental result in arithmetic geometry that gives sharp estimates for the number of rational points on abelian varieties over finite fields in terms of their dimension and the field size.

A. Hasse bound for elliptic curves
The Hasse bound for elliptic curves is a fundamental result in number theory that gives tight limits on how far the number of points on an elliptic curve over a finite field can deviate from the size of the field plus one.
B. Weil 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.
C. 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.
D. Faltings' theorem
Faltings' theorem is a landmark result in arithmetic geometry that proves every algebraic curve of genus greater than one over a number field has only finitely many rational points.
E. Hurwitz bound on automorphism groups of curves
The Hurwitz bound on automorphism groups of curves is a classical result in algebraic geometry stating that a compact Riemann surface of genus at least 2 has at most 84(g − 1) automorphisms.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	result in number theory ⓘ theorem in arithmetic geometry ⓘ
appliesTo	abelian varieties over finite fields ⓘ
concerns	distribution of rational points on abelian varieties ⓘ number of rational points over finite fields ⓘ
connectedTo	Honda–Tate theory ⓘ Weil’s proof of the Riemann hypothesis for curves over finite fields ⓘ
domain	finite fields ⓘ
expressedInTermsOf	Weil polynomial ⓘ characteristic polynomial of Frobenius ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ arithmetic of abelian varieties ⓘ
generalizes	Hasse bound for elliptic curves ⓘ
gives	estimate for number of F_q-rational points ⓘ sharp upper and lower bounds for point counts ⓘ
hasConsequence	constraints on possible zeta functions of abelian varieties over finite fields ⓘ inequalities for traces of Frobenius ⓘ
historicalContext	developed in the mid 20th century ⓘ
implies	absolute values of Frobenius eigenvalues equal q^{1/2} ⓘ finiteness of isogeny classes with fixed dimension and field ⓘ point counts are close to q^g for dimension g ⓘ
isToolFor	bounding error terms in point count estimates ⓘ explicit point counting algorithms ⓘ
mathematicalSubjectClassification	11G10 ⓘ 14G15 ⓘ
namedAfter	André Weil ⓘ Helmut Hasse ⓘ
parameterDependsOn	dimension of the abelian variety ⓘ eigenvalues of Frobenius acting on Tate module ⓘ size of the finite field ⓘ
refinedBy	Serre–Tate results on abelian varieties over finite fields ⓘ linked to: Hasse–Weil bound for abelian varieties
relatedTo	L-functions of abelian varieties ⓘ zeta function of an abelian variety ⓘ
specialCase	Hasse–Weil bound for curves via Jacobians ⓘ linked to: Hasse–Weil bound for abelian varieties
typeOf	Weil-type estimate ⓘ
usedIn	classification of abelian varieties over finite fields ⓘ coding theory ⓘ construction of algebraic geometric codes ⓘ cryptography based on abelian varieties ⓘ estimates for rational points on curves ⓘ study of isogeny classes of abelian varieties ⓘ
usesConcept	Frobenius endomorphism ⓘ Riemann hypothesis for varieties over finite fields ⓘ linked to: Weil conjectures Weil conjectures ⓘ Weil numbers ⓘ eigenvalues of Frobenius ⓘ ℓ-adic cohomology ⓘ linked to: étale cohomology

How these facts were elicited

Referenced by (3)

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

Hasse bound for elliptic curves → generalization → Hasse–Weil bound for abelian varieties ⓘ

Hasse–Weil bound for abelian varieties → specialCase → Hasse–Weil bound for curves via Jacobians ⓘ

linked to: Hasse–Weil bound for abelian varieties

Hasse–Weil bound for abelian varieties → refinedBy → Serre–Tate results on abelian varieties over finite fields ⓘ