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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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 NERFINISHED ⓘ Weil’s proof of the Riemann hypothesis for curves over finite fields ⓘ
domain	finite fields ⓘ
expressedInTermsOf	Weil polynomial NERFINISHED ⓘ characteristic polynomial of Frobenius ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ arithmetic of abelian varieties ⓘ
generalizes	Hasse bound for elliptic curves NERFINISHED ⓘ
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 NERFINISHED ⓘ Helmut Hasse NERFINISHED ⓘ
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 NERFINISHED ⓘ
relatedTo	L-functions of abelian varieties ⓘ zeta function of an abelian variety ⓘ
specialCase	Hasse–Weil bound for curves via Jacobians NERFINISHED ⓘ
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 NERFINISHED ⓘ Riemann hypothesis for varieties over finite fields NERFINISHED ⓘ Weil conjectures NERFINISHED ⓘ Weil numbers NERFINISHED ⓘ eigenvalues of Frobenius ⓘ ℓ-adic cohomology NERFINISHED ⓘ

Referenced by (1)

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 ⓘ