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 (3)
| Label | Occurrences |
|---|---|
| Hasse bound for elliptic curves canonical | 3 |
| Hasse bound |a_p(E)| ≤ 2√p | 1 |
| Hasse’s theorem on elliptic curves | 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 for curves ⓘ
linked to:
Weil conjectures
|
| 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 of an elliptic curve
ⓘ
linked to:
Hasse–Weil zeta function
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hasse bound for elliptic curves
linked to: Hasse bound for elliptic curves