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

$$y^p - y = f(x)$$

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 covering

$$C \to \mathbb{P}^1$$

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 $$\mathbb{Z}/p\mathbb{Z}$$. 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 model

$$y^p - y = f(x),$$

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 decomposition

$$f(x) = f_\infty(x) + \sum_{\alpha \in B'} f_\alpha\left(\frac{1}{x-\alpha}\right)$$

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 define

$$B = \begin{cases} B' &\text{ if } f_\infty = 0, \\ B'\cup\{\infty\} &\text{ otherwise.}\end{cases}$$

Then the set $$B \subset \mathbb{P}^1(k)$$ is precisely the set of branch points of the covering $$C \to \mathbb{P}^1$$.

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 degree of the cover is a prime number, over each branching point \alpha \in B lies a single ramification point $$P_\alpha$$ with corresponding ramification index equal to

$$e(P_\alpha) = (p - 1)\big(\deg(f_\alpha) + 1\big) + 1.$$

Genus

Since, p does not divide $$\deg(f_\alpha)$$, ramification indices $$e(P_\alpha)$$ are not divisible by p either. Therefore, Riemann-Roch theorem may be used to compute that genus of an Artin–Schreier curve is given by

$$g = \frac{p-1}{2} \left( \sum_{\alpha\in B} \big(\deg(f_\alpha) + 1\big) - 2 \right).$$

For example, for a hyperelliptic curve defined over a field of characteristic p = 2 by equation $$y^2 - y = f(x)$$with f decomposing as above, we have

$$g = \sum_{\alpha\in B} \frac{\deg(f_\alpha) + 1}{2} - 1.$$

Generalizations

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

$$g(y^p) = f(x)$$

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 $$\mathbb{P}^1$$. Separability of defining polynomial g ensures separability of the corresponding function field extension k(C)/k(x). If $$g(y^p) = a_{m} y^{p^m} + a_{m - 1} y^{p^{m-1}} + \cdots + a_{1} y^p + a_0$$, 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 curves

$$C \to C_{m-1} \to \cdots \to C_0 = \mathbb{P}^1,$$

each of degree p, starting with the projective line.

Artin–Schreier theory
Hyperelliptic curve
Superelliptic curve

References

Koblitz, Neal (1989). "Hyperelliptic cryptosystems". Journal of Cryptology 1: 139–150. doi:10.1007/BF02252872.

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.

Mathematics Encyclopedia