Hasse bound for elliptic curves
E207314
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Hasse bound for elliptic curves canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1862416 — 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 bound for elliptic curves Context triple: [Helmut Hasse, notableWork, Hasse bound for elliptic curves]
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A.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve cryptography.
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B.
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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C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
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D.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
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E.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
- 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 bound for elliptic curves Target entity description: 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.
-
A.
Koblitz curves
Koblitz curves are a special class of elliptic curves defined over binary fields that enable particularly efficient and fast implementations of elliptic curve cryptography.
-
B.
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.
-
C.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
-
D.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
E.
Fermat curve
A Fermat curve is an algebraic curve defined by an equation of the form \(x^n + y^n = 1\), studied in number theory and algebraic geometry for its rich arithmetic and geometric properties.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
result in arithmetic geometry
ⓘ
theorem in number theory ⓘ |
| appliesTo | elliptic curves over finite fields ⓘ |
| assumes | E is nonsingular (smooth) projective curve of genus 1 with a rational point ⓘ |
| classification | quantitative refinement of the fact that #E(F_q) is finite ⓘ |
| constrains | group order of elliptic curve over a given finite field ⓘ |
| curveCondition | elliptic curve E defined over F_q ⓘ |
| defines | a = q + 1 - N ⓘ |
| denotes | N = #E(F_q) ⓘ |
| equivalentFormulation |
eigenvalues of Frobenius have complex absolute value √q
ⓘ
|#E(F_q) - (q + 1)| ≤ 2√q ⓘ |
| expressesBoundOn | trace of Frobenius a of E over F_q ⓘ |
| fieldCondition | finite field F_q ⓘ |
| generalization |
Hasse–Weil bound for abelian varieties
ⓘ
Weil bounds for curves of higher genus ⓘ |
| givesLowerBound | #E(F_q) ≥ q + 1 - 2√q ⓘ |
| givesUpperBound | #E(F_q) ≤ q + 1 + 2√q ⓘ |
| historicalStatus | proved by Helmut Hasse in the 1930s ⓘ |
| implies |
#E(F_q) is approximately q + 1 with error term O(√q)
ⓘ
characteristic polynomial of Frobenius has roots of absolute value √q ⓘ possible group orders of E(F_q) lie in an interval of length about 4√q ⓘ q + 1 - 2√q ≤ #E(F_q) ≤ q + 1 + 2√q ⓘ |#E(F_q) - (q + 1)| is at most on the order of √q ⓘ |
| importance | fundamental in the arithmetic of elliptic curves over finite fields ⓘ |
| involvesQuantity | number of F_q-rational points on E ⓘ |
| isSpecialCaseOf |
Riemann hypothesis for curves over finite fields
ⓘ
Weil conjectures ⓘ
surface form:
Weil conjectures for curves
|
| isTight | yes, bounds are best possible in general ⓘ |
| language | usually stated over prime powers q = p^n ⓘ |
| mathematicalArea |
algebraic number theory
ⓘ
arithmetic geometry ⓘ finite field theory ⓘ |
| namedAfter | Helmut Hasse ⓘ |
| proofMethod | uses properties of L-functions and complex analysis in Hasse's original proof ⓘ |
| relatedConcept |
Hasse–Weil zeta function
ⓘ
surface form:
Hasse–Weil zeta function of an elliptic curve
Sato–Tate distribution (for families of elliptic curves) ⓘ trace of Frobenius of an elliptic curve ⓘ |
| sharpnessCondition | for many q there exist elliptic curves attaining equality |a| = 2√q when 2√q is integer ⓘ |
| statesInequality |
|a_q(E)| ≤ 2√q
ⓘ
|a| ≤ 2√q ⓘ |
| typicalNotation | a_q(E) = q + 1 - #E(F_q) ⓘ |
| usedIn |
CM method for generating elliptic curves
ⓘ
Schoof–Elkies–Atkin (SEA) point-counting algorithm ⓘ construction of elliptic curves with prescribed number of points ⓘ counting points on elliptic curves over finite fields ⓘ elliptic curve cryptography parameter selection ⓘ security analysis of elliptic curve cryptosystems ⓘ |
| yearProvedApprox | 1933 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hasse bound for elliptic curves Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.