In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form where
is an integer and f is a polynomial of degree
with coefficients in a field
; more precisely, it is the smooth projective curve whose function field defined by this equation.
is an elliptic curve, the case
is a hyperelliptic curve, and the case
Some authors impose additional restrictions, for example, that the integer
should not be divisible by the characteristic of
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.
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.
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
has an affine model given by an equation for some polynomial
has a point defined over
In particular, function field extension
be a superelliptic curve defined over an algebraically closed field
denote the set of roots of
{\displaystyle B={\begin{cases}B'&{\text{ if }}m{\text{ divides }}\deg(f),\\B'\cup \{\infty \}&{\text{ otherwise.
is the set of branch points of the covering map
For an affine branch point
ramification points
For the point at infinity, define integer
Then analogously to the other ramification points,
In particular, the curve is unramified over infinity if and only if its degree
defined as above is connected precisely when
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