Frobenius endomorphism

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Frobenius endomorphism

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Etymology

Named after German mathematician Ferdinand Georg Frobenius.

Noun

Frobenius endomorphism (plural Frobenius endomorphisms)

1. (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.
   • 2003, Claudia Miller, “The Frobenius endomorphism and homological dimensions”, in Luchezar L. Avramov, Marc Chardin, Marcel Morales, Claudia Polini, editors, Commutative Algebra: Interactions with Algebraic Geometry: International Conference, American Mathematical Society, page 208:

     Section 3 concerns what properties of the ring other than regularity are reflected by the homological properties of the Frobenius endomorphism.

   • 2005, Emmanuel Letellier, Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras, Springer, Lecture Notes in Mathematics 1859, page 11,

     Let ${\displaystyle k={\overline {\mathbb {F} }}_{p}}$, and let ${\displaystyle q}$ be a power of ${\displaystyle p}$ such that the group ${\displaystyle G}$ is defined over ${\displaystyle \mathbb {F} _{q}}$. We then denote by ${\displaystyle F:G\to G}$ the corresponding Frobenius endomorphism. The Lie algebra ${\displaystyle {\mathcal {G}}}$ and the adjoint action of ${\displaystyle G}$ on ${\displaystyle {\mathcal {G}}}$ are also defined over ${\displaystyle \mathbb {F} _{q}}$ and we still denote by ${\displaystyle F:{\mathcal {G}}\to {\mathcal {G}}}$ the Frobenius endomorphism on ${\displaystyle {\mathcal {G}}}$.

     Assume that ${\displaystyle H,X}$ and the action of ${\displaystyle H}$ over ${\displaystyle X}$ are all defined over ${\displaystyle \mathbb {F} _{q}}$. Let ${\displaystyle F:X\to X}$ and ${\displaystyle F:H\to H}$ 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

     ${\displaystyle E:y^{2}+y=x^{3}}$.

     In this case, the characteristic polynomial of the Frobenius endomorphism denoted by ${\displaystyle \phi _{2}}$ (cf. Example 4.87 and Section 13.1.8), which sends ${\displaystyle P_{\infty }}$ to itself and ${\displaystyle (x_{1},y_{1})}$ to ${\displaystyle (x_{1}^{2},y_{1}^{2})}$, is

     ${\displaystyle \chi _{E}(T)=T^{2}+2}$.

     Thus doubling is replaced by a twofold application of the Frobenius endomorphism and taking the negative as for all points ${\displaystyle P\in E(\mathbb {F} _{2^{d}})}$, we have ${\displaystyle \phi _{2}^{2}=-P}$.

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