In mathematics, a superelliptic curve is a plane curve with an equation of the form
where the exponent m is fixed and f is a polynomial of degree d. The case m = 2 and d = 3 is an elliptic curve, the case m = 2 and d > 4 is a hyperelliptic curve, and the case m = 3 and d > 3 is an example of a trigonal curve.
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.
More generally, a superelliptic curve is a cyclic branched covering
of the projective line of degree coprime to the characteristic of the field of definition. The degree 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.
for some polynomial of degree with each root having order , provided that has a point defined over , that is, if the set of -rational points of is not empty. For example, this is always the case when is algebraically closed. In particular, function field extension is a Kummer extension.
Let be a superelliptic curve defined over an algebraically closed field , and denote the set of roots of in . Define set
Then is the set of branch points of the covering map given by .
For an affine branch point , let denote the order of as a root of . As before, we assume that . Then
is the ramification index at each of the ramification points of the curve lying over (that is actually true for any ).
For the point at infinity, define integer as follows. If
then . Note that . Then analogously to the other ramification points,
is the ramification index at the points that lie over . In particular, the curve is unramified over infinity if and only if its degree divides .
Curve defined as above is connected precisely when and are relatively prime (not necessarily pairwise), which is assumed to be the case.
By the Riemann-Hurwitz formula, the genus of a superelliptic curve is given by
- Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. 201. Springer-Verlag. p. 361. ISBN 0-387-98981-1. Zbl 0948.11023.
- Koo, Ja Kyung (1991). "On holomorphic differentials of some algebraic function field of one variable over ". Bull. Austral. Math. Soc. 43 (3): 399–405. doi:10.1017/S0004972700029245.
- Lang, Serge (1978). Elliptic Curves: Diophantine Analysis. Grundlehren der mathematischen Wissenschaften. 231. Springer-Verlag. ISBN 0-387-08489-4.
- Shorey, T.N.; Tijdeman, R. (1986). Exponential Diophantine equations. Cambridge Tracts in Mathematics. 87. Cambridge University Press. ISBN 0-521-26826-5. Zbl 0606.10011.
- Smart, N. P. (1998). The Algorithmic Resolution of Diophantine Equations. London Mathematical Society Student Texts. 41. Cambridge University Press. ISBN 0-521-64633-2.