Artin–Schreier curve

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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.

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.

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 $a_{0}=0${{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}} }}.

See also

• Artin–Schreier theory
• Hyperelliptic curve
• Superelliptic curve