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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 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
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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