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.
- 2003, Claudia Miller, The Frobenius endomorphism and homological dimensions, 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 ${\\displaystyle \\phi _{2}^{2}=-[2]P}$ .