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

${\displaystyle y^{p}-y=f(x)}$

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

${\displaystyle y^{2}+h(x)y=f(x)}$

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

${\displaystyle y^{p}-y=f(x),}$

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

Ramification

Let ${\displaystyle C:y^{p}-y=f(x)}$ be an Artin–Schreier curve. Rational function $f$ over an algebraically closed field $k$ has partial fraction decomposition

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

for some finite set $B'$ of elements of $k$ and corresponding non-constant polynomials $f_\alpha$ defined over $k$, and (possibly constant) polynomial ${\displaystyle f_{\infty }}$. 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 ${\displaystyle f_{\infty }}$ or it is zero. If that is the case, we define

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

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 ${\displaystyle y^{p}-y=f(x)}$, 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

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

Genus

Since, $p$ does not divide ${\displaystyle \deg(f_{\alpha })}$, ramification indices ${\displaystyle e(P_{\alpha })}$ 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

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

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

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

Generalizations

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

${\displaystyle g(y^{p})=f(x)}$

for some separable polynomial ${\displaystyle g\in k[x]}$ and rational function ${\displaystyle f\in k(x)\backslash g(k(x))}$. Mapping ${\displaystyle (x,y)\mapsto x}$ 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 ${\displaystyle g(y^{p})=a_{m}y^{p^{m}}+a_{m-1}y^{p^{m-1}}+\cdots +a_{1}y^{p}+a_{0}}$, a change of variables can be found so that ${\displaystyle a_{m}=a_{1}=1}$ 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

${\displaystyle C\to C_{m-1}\to \cdots \to C_{0}=\mathbb {P} ^{1},}$

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.