Frobenius endomorphism

== English == === Etymology === Named after German mathematician Ferdinand Georg Frobenius. === Noun === Frobenius endomorphism (plural Frobenius endomorphisms) (algebra, commutative algebra, field theory) Given a commutative ring R with prime characteristic p, the endomorphism that maps x → x p for all x ∈ R. 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11, Let k = F ¯ p {\displaystyle k={\overline {\mathbb {F} }}_{p}} , and let q {\displaystyle q} be a power of p {\displaystyle p} such that the group G {\displaystyle G} is defined over F q {\displaystyle \mathbb {F} _{q}} . We then denote by F : G → G {\displaystyle F:G\to G} the corresponding Frobenius endomorphism. The Lie algebra G {\displaystyle {\mathcal {G}}} and the adjoint action of G {\displaystyle G} on G {\displaystyle {\mathcal {G}}} are also defined over F q {\displaystyle \mathbb {F} _{q}} and we still denote by F : G → G {\displaystyle F:{\mathcal {G}}\to {\mathcal {G}}} the Frobenius endomorphism on G {\displaystyle {\mathcal {G}}} . […] Assume that H , X {\displaystyle H,X} and the action of H {\displaystyle H} over X {\displaystyle X} are all defined over F q {\displaystyle \mathbb {F} _{q}} . Let F : X → X {\displaystyle F:X\to X} and F : H → H {\displaystyle F:H\to H} be the corresponding Frobenius endomorphisms. 2006, Christophe Doche, Tanja Lange, Chapter 15: Arithmetic of Special Curves, Henri Cohen, Gerhard Frey, Roberto Avanzi, Christophe Doche, Tanja Lange, Kim Nguyen, Frederik Vercauteren (editors), Handbook of Elliptic and Hyperelliptic Curve Cryptography, Taylor & Francis (Chapman & Hall / CRC Press), page 356, The first attempt to use the Frobenius endomorphism to compute scalar multiples was made by Menezes and Vanstone (MEVA 1900) using the curve E : y 2 + y = x 3 {\displaystyle E:y^{2}+y=x^{3}} . In this case, the characteristic polynomial of the Frobenius endomorphism denoted by ϕ 2 {\displaystyle \phi _{2}} (cf. Example 4.87 and Section 13.1.8), which sends P ∞ {\displaystyle P_{\infty }} to itself and ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} to ( x 1 2 , y 1 2 ) {\displaystyle (x_{1}^{2},y_{1}^{2})} , is χ E ( T ) = T 2 + 2 {\displaystyle \chi _{E}(T)=T^{2}+2} . Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points P ∈ E ( F 2 d ) {\displaystyle P\in E(\mathbb {F} _{2^{d}})} , we have ϕ 2 2 = − [ 2 ] P {\displaystyle \phi _{2}^{2}=-[2]P} . ==== Synonyms ==== (particular endomorphism on a commutative ring with prime characteristic): Frobenius homomorphism ==== Related terms ==== Frobenius automorphism Frobenius closure Frobenius element Frobenius morphism ==== Translations ==== === Further reading === Characteristic (algebra) on Wikipedia.Wikipedia Frobenioid on Wikipedia.Wikipedia Perfect field on Wikipedia.Wikipedia Frobenius endomorphism on Encyclopedia of Mathematics Frobenius morphism on nLab