Frobenius endomorphism
E860096
The Frobenius endomorphism is a fundamental map in algebra and arithmetic geometry that raises elements to their p-th power in characteristic p, playing a central role in the study of varieties over finite fields and their zeta functions.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Frobenius endomorphism canonical | 9 |
| Frobenius automorphism | 2 |
| the Frobenius endomorphism | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10388800 — 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: Frobenius endomorphism Context triple: [Weil conjectures, usesConcept, Frobenius endomorphism]
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A.
Frobenius element
The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
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B.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
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C.
Hensel’s lemma
Hensel’s lemma is a fundamental result in number theory and p-adic analysis that allows one to lift solutions of polynomial congruences modulo a prime power to higher powers, analogous to Newton’s method in the p-adic setting.
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D.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
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E.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
- 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: Frobenius endomorphism Target entity description: The Frobenius endomorphism is a fundamental map in algebra and arithmetic geometry that raises elements to their p-th power in characteristic p, playing a central role in the study of varieties over finite fields and their zeta functions.
-
A.
Frobenius element
The Frobenius element is a distinguished element in a Galois group associated to an unramified prime, encoding how that prime splits in a field extension and playing a central role in algebraic number theory and arithmetic geometry.
-
B.
Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
-
C.
Hensel’s lemma
Hensel’s lemma is a fundamental result in number theory and p-adic analysis that allows one to lift solutions of polynomial congruences modulo a prime power to higher powers, analogous to Newton’s method in the p-adic setting.
-
D.
Levine-Fricke Field
Levine-Fricke Field is the home softball stadium of the University of California, Berkeley Golden Bears, located on the university’s campus in Berkeley, California.
-
E.
Artin–Schreier theory
Artin–Schreier theory is a branch of algebraic number theory and field theory that characterizes cyclic extensions of prime degree in fields of characteristic p using additive polynomials.
- F. None of above. chosen
Statements (52)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic concept
ⓘ
endomorphism ⓘ field endomorphism ⓘ group endomorphism ⓘ ring endomorphism ⓘ |
| actsBy | raising elements to their p-th power ⓘ |
| appearsIn |
study of p-curvature and differential equations in characteristic p
ⓘ
theory of F-singularities ⓘ theory of tight closure in commutative algebra ⓘ |
| centralRoleIn |
Weil conjectures
ⓘ
arithmetic geometry ⓘ theory of varieties over finite fields ⓘ zeta functions of varieties over finite fields ⓘ étale cohomology ⓘ |
| commutesWith | base change to algebraic closures in characteristic p ⓘ |
| definedOn |
fields of characteristic p
ⓘ
rings of characteristic p ⓘ schemes over fields of characteristic p ⓘ varieties over finite fields ⓘ |
| generalizedBy |
arithmetic Frobenius
ⓘ
linked to:
Frobenius element
geometric Frobenius ⓘ
linked to:
Frobenius element
|
| hasActionOn |
cohomology groups of varieties over finite fields
ⓘ
crystalline cohomology ⓘ l-adic cohomology ⓘ étale cohomology groups ⓘ |
| hasDefinition | map that sends x to x^p in characteristic p ⓘ |
| hasFormulationIn |
field theory
ⓘ
group theory ⓘ ring theory ⓘ scheme theory ⓘ |
| hasKeyFeature | nonlinear over the base field structure when viewed as a map of schemes ⓘ |
| hasProperty |
automorphism of finite fields
ⓘ
bijective on finite fields ⓘ generates the Galois group of a finite field extension over its prime field ⓘ injective on reduced rings of characteristic p ⓘ iterates form a semigroup under composition ⓘ p-th iterate equals identity on finite field of size p ⓘ ring homomorphism in characteristic p ⓘ |
| hasVariant |
absolute Frobenius morphism of schemes
ⓘ
relative Frobenius morphism of schemes ⓘ |
| isFunctorialWithRespectTo | morphisms of schemes over fields of characteristic p ⓘ |
| namedAfter | Ferdinand Georg Frobenius ⓘ |
| playsRoleIn |
counting rational points on varieties over finite fields
ⓘ
definition of weights in l-adic cohomology ⓘ proof of the Weil conjectures by Deligne ⓘ |
| relatedTo |
Frobenius elements in Galois groups
ⓘ
Galois representations ⓘ Weil group elements ⓘ |
| usedToDefine |
Frobenius eigenvalues on cohomology
ⓘ
L-functions in arithmetic geometry ⓘ Weil numbers ⓘ zeta function of a variety over a finite field ⓘ |
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: Frobenius endomorphism Description of subject: The Frobenius endomorphism is a fundamental map in algebra and arithmetic geometry that raises elements to their p-th power in characteristic p, playing a central role in the study of varieties over finite fields and their zeta functions.
Referenced by (12)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Frobenius endomorphism
Sato–Tate distribution (for families of elliptic curves)
→
associatedWith
→
Frobenius endomorphism
ⓘ
linked to: Frobenius endomorphism
linked to: Frobenius endomorphism