Artin–Schreier curve

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 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 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.

See also

• Artin–Schreier theory
• Hyperelliptic curve
• Superelliptic curve

References

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.