Superelliptic curve

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In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form

ym=f(x),

where m2 is an integer and f is a polynomial of degree d3 with coefficients in a field k; more precisely, it is the smooth projective curve whose function field defined by this equation. The case m=2 and d=3 is an elliptic curve, the case m=2 and d5 is a hyperelliptic curve, and the case m=3 and d4 is an example of a trigonal curve.

Some authors impose additional restrictions, for example, that the integer m should not be divisible by the characteristic of k, that the polynomial f should be square free, that the integers m and d should be coprime, or some combination of these.[1]

The Diophantine problem of finding integer points on a superelliptic curve can be solved by a method similar to one used for the resolution of hyperelliptic equations: a Siegel identity is used to reduce to a Thue equation.

Definition

More generally, a superelliptic curve is a cyclic branched covering

C1

of the projective line of degree m2 coprime to the characteristic of the field of definition. The degree m of the covering map is also referred to as the degree of the curve. By cyclic covering we mean that the Galois group of the covering (i.e., the corresponding function field extension) is cyclic.

The fundamental theorem of Kummer theory implies [citation needed] that a superelliptic curve of degree m defined over a field k has an affine model given by an equation

ym=f(x)

for some polynomial fk[x] of degree m with each root having order <m, provided that C has a point defined over k, that is, if the set C(k) of k-rational points of C is not empty. For example, this is always the case when k is algebraically closed. In particular, function field extension k(C)/k(x) is a Kummer extension.

Ramification

Let C:ym=f(x) be a superelliptic curve defined over an algebraically closed field k, and Bk denote the set of roots of f in k. Define set B={B if m divides deg(f),B{} otherwise. Then B1(k) is the set of branch points of the covering map C1 given by x.

For an affine branch point αB, let rα denote the order of α as a root of f. As before, we assume that 1rα<m. Then eα=m(m,rα) is the ramification index e(Pα,i) at each of the (m,rα) ramification points Pα,i of the curve lying over α𝔸1(k)1(k) (that is actually true for any αk).

For the point at infinity, define integer 0r<m as follows. If s=min{tmtdeg(f)}, then r=msdeg(f). Note that (m,r)=(m,deg(f)). Then analogously to the other ramification points, e=m(m,r) is the ramification index e(P,i) at the (m,r) points P,i that lie over . In particular, the curve is unramified over infinity if and only if its degree m divides deg(f).

Curve C defined as above is connected precisely when m and rα are relatively prime (not necessarily pairwise), which is assumed to be the case.

Genus

By the Riemann-Hurwitz formula, the genus of a superelliptic curve is given by

g=12(m(|B|2)αB(m,rα))+1.

See also

References

  1. Galbraith, S.D.; Paulhus, S.M.; Smart, N.P. (2002). "Arithmetic on superelliptic curves". Mathematics of Computation 71: 394–405. doi:10.1090/S0025-5718-00-01297-7.