# Hyper-elliptic curve

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2010 Mathematics Subject Classification: Primary: 14H45 [MSN][ZBL]

A non-singular projective model of the affine curve \$y^2=f(x)\$, where \$f(x)\$ is a polynomial without multiple roots of odd degree \$n\$ (the case of even degree \$2k\$ may be reduced to that of odd degree \$2k-1\$). The field of functions on a hyper-elliptic curve (a field of hyper-elliptic functions) is a quadratic extension of the field of rational functions; in this sense it is the simplest field of algebraic functions except for the field of rational functions. Hyper-elliptic curves are distinguished by the condition of the existence of a one-dimensional linear series \$g_2'\$ of divisors of degree 2, defining a morphism of order 2 of the hyper-elliptic curve onto the projective straight line. The genus of a hyper-elliptic curve is \$g =(n-1)/2\$, so that, for various odd \$n\$, hyper-elliptic curves are birationally inequivalent.

For \$n=1\$, \$g=0\$ one obtains the projective straight line; for \$n=3\$, \$g=1\$ an elliptic curve is obtained. Traditionally, curves of genus 0 and 1 are not called hyper-elliptic curves. The fractions of regular differential forms generate a subfield of genus 0 on a hyper-elliptic curve of genus \$g>1\$; this property is a complete characterization of hyper-elliptic curves. A further characterization is that hyper-elliptic curves have exactly \$2g+2\$ Weierstrass points.

How to Cite This Entry:
Hyper-elliptic curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyper-elliptic_curve&oldid=34818
This article was adapted from an original article by V.E. Voskresenskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article