Artin–Schreier curve explained

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

p by an equationy ^{p} - y = f(x) for some rational function f 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

y ^{2} + h(x) y = f(x) for some polynomials f and h .

Definition

More generally, an Artin-Schreier curve defined over an algebraically closed field of characteristic

p is a branched coveringC \to P ^{1} of the projective line of degree p . Such a cover is necessarily cyclic, that is, the Galois group of the corresponding algebraic function field extension is the cyclic group Z /p Z . In other words, k(C)/k(x) is an Artin–Schreier extension.

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

k has an affine modely ^{p} - y = f(x), for some rational function f \in k(x) that is not equal for z ^{p} - z for any other rational function z . In other words, if we define polynomial g(z) = z ^{p} - z , then we require that f \in k(x) \backslash g(k(x)) .

Ramification

Let

C: y ^{p} - y = f(x) be an Artin–Schreier curve.Rational function f over an algebraically closed field k has partial fraction decompositionf(x) = f _{infty(x)} + \sum _{\alpha} f _{\alpha\left( (1)/(x - \alpha)} \right) for some finite set B' of elements of k and corresponding non-constant polynomials f _{\alpha} defined over k , and (possibly constant) polynomial f _{infty} .After a change of coordinates, f can be chosen so that the above polynomials have degrees coprime to p , and the same either holds for f _{infty} or it is zero. If that is the case, we defineB = \begin{cases} B' & if f _{infty} = 0 , \ B'\cup\{ inf ty\} & otherwise. \end{cases} Then the set B \subset P ^{1(k)} is precisely the set of branch points of the covering C \to P ^{1} .

For example, Artin–Schreier curve

y ^{p} - y = f(x) , where f 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

\alpha \in B lies a single ramification point P _{\alpha} with corresponding different (not to confused with the ramification index) equal toe(P _{\alpha)} = (p - 1 ) ( \deg(f _{\alpha)} + 1 ) + 1 .

Genus

Since

p does not divide \deg(f _{\alpha)} , ramification indices e(P _{\alpha)} are not divisible by p either. Therefore, the Riemann–Roch theorem may be used to compute that the genus of an Artin–Schreier curve is given byg = (p -1)/(2) \left( \sum _{\alpha\in} ( \deg(f _{\alpha)} + 1 ) - 2 \right).

For example, for a hyperelliptic curve defined over a field of characteristic

p = 2 by equation y ^{2} - y = f(x) with f decomposed as above,g = \sum _{\alpha\in} (\deg(f _{\alpha)} +1)/(2) - 1 .

Generalizations

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

k of characteristic p by an equationg(y ^{p)} = f(x) for some separable polynomial g \in k[x] and rational function f \in k(x) \backslash g(k(x)) . Mapping (x, y) \mapsto x yields a covering map from the curve C to the projective line P ^{1} . Separability of defining polynomial g ensures separability of the corresponding function field extension k(C)/k(x) . If g(y ^{p)} = a _{m} y ^{p ^{m}} + a _{m} y ^{p ^{m-1}} + … + a _{1} y ^{p} + a _{0} , a change of variables can be found so that a _{m} = a _{1} = 1 and a _{0} = 0 . 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 curvesC \to C _{m-1} \to … \to C _{0} = P ^{1,} each of degree p , starting with the projective line.

See also

• Artin–Schreier theory
• Hyperelliptic curve
• Superelliptic curve

References

• 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 . 10.1016/j.laa.2013.06.012. 1202.4183.

Notes and References

1. Koblitz . Neal . 1989 . Hyperelliptic cryptosystems . Journal of Cryptology . 1. 3. 139–150 . 10.1007/BF02252872.
2. Sullivan . Francis J. . 1975 . p-Torsion in the class group of curves with too many automorphisms . Archiv der Mathematik . 26. 1 . 253–261 . 10.1007/BF01229737.