Frobenius endomorphism

E860096

algebraic concept endomorphism field endomorphism group endomorphism ring endomorphism

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.

Try in SPARQL Jump to: Statements Referenced by

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 NERFINISHED ⓘ 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 NERFINISHED ⓘ geometric Frobenius NERFINISHED ⓘ
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 NERFINISHED ⓘ
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 NERFINISHED ⓘ Weil group elements ⓘ
usedToDefine	Frobenius eigenvalues on cohomology ⓘ L-functions in arithmetic geometry ⓘ Weil numbers NERFINISHED ⓘ zeta function of a variety over a finite field ⓘ

Referenced by (2)

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

Weil conjectures → usesConcept → Frobenius endomorphism ⓘ

Deligne–Lusztig theory → uses → Frobenius endomorphism ⓘ