Artin–Schreier curve

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In mathematics, an Artin–Schreier curve is a plane curve defined over an algebraically closed field of characteristic by an equation

for some rational function over that field.

One of the most important examples of such curves is hyperelliptic curves in characteristic 2, whose Jacobian varieties have been suggested for use in cryptography.[1] It is common to write these curves in the form

for some polynomials and .

Definition[edit]

More generally, an Artin-Schreier curve defined over an algebraically closed field of characteristic is a branched covering

of the projective line of degree . Such a cover is necessarily cyclic, that is, the Galois group of the corresponding algebraic function field extension is the cyclic group . In other words, is an Artin–Schreier extension.

The fundamental theorem of Artin–Schreier theory implies that such a curve defined over a field has an affine model

for some rational function that is not equal for for any other rational function . In other words, if we define polynomial , then we require that .

Ramification[edit]

Let be an Artin–Schreier curve. Rational function over an algebraically closed field has partial fraction decomposition

for some finite set of elements of and corresponding non-constant polynomials defined over , and (possibly constant) polynomial . After a change of coordinates, can be chosen so that the above polynomials have degrees coprime to , and the same either holds for or it is zero. If that is the case, we define

Then the set is precisely the set of branch points of the covering .

For example, Artin–Schreier curve , where is a polynomial, is ramified at a single point over the projective line.

Since the degree of the cover is a prime number, over each branching point lies a single ramification point with corresponding ramification index equal to

Genus[edit]

Since, does not divide , ramification indices are not divisible by either. Therefore, Riemann-Roch theorem may be used to compute that genus of an Artin–Schreier curve is given by

For example, for a hyperelliptic curve defined over a field of characteristic by equation with decomposing as above, we have

Generalizations[edit]

Artin–Schreier curves are a particular case of plane curves defined over an algebraically closed field of characteristic by an equation

for some separable polynomial and rational function . Mapping yields a covering map from the curve to the projective line . Separability of defining polynomial ensures separability of the corresponding function field extension . If , a change of variables can be found so that and . It has been shown [2] that such curves can be built via a sequence of Artin-Schreier extension, that is, there exists a sequence of cyclic coverings of curves

each of degree , starting with the projective line.

See also[edit]

References[edit]

  1. ^ Koblitz, Neal (1989). "Hyperelliptic cryptosystems". Journal of Cryptology. 1: 139–150. doi:10.1007/BF02252872. 
  2. ^ Sullivan, Francis J. (1975). "p-Torsion in the class group of curves with too many automorphisms". Archiv der Mathematik. Springer. 26 (1): 253–261. doi:10.1007/BF01229737. 
  • Farnell, Shawn; Pries, Rachel (2014). "Families of Artin-Schreier curves with Cartier-Manin matrix of constant rank". Linear Algebra and its Applications. 439 (7): 2158–2166. doi:10.1016/j.laa.2013.06.012.