130.85.58.237 at 18:42, 6 December 2013

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In mathematics, a '''superelliptic curve''' is a [[plane curve]] with an equation of the form
:<math>y^m = f(x),</math>
where the exponent <math>m</math> is fixed and <math>f</math> is a [[polynomial]]. The case <math>m = 2</math> is a ''[[hyperelliptic curve]]'': the case <math>m = 3</math> is 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]].

== Definition ==
More generally, a ''superelliptic curve'' is a cyclic [[branched covering]]
:<math>C \to \mathbb{P}^1</math>
of the projective line of degree <math>m \geq 2</math> coprime to the characteristic of the field of definition. The degree <math>m</math> 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 of an algebraic variety|function field]] extension) is [[Cyclic group|cyclic]].

The fundamental theorem of [[Kummer theory]] implies {{Citation needed|date=February 2014}} that a superelliptic curve of degree <math>m</math> defined over a field <math>k</math> has an affine model given by an equation
:<math>y^m = f(x)</math>
for some polynomial <math>f \in k[x]</math> of degree <math>m</math> without repeated roots, provided that <math>C</math> has a point defined over <math>k</math>, that is, if the set <math>C(k)</math> of <math>k</math>-rational points of <math>C</math> is not empty. For example, this is always the case when <math>k</math> is [[Algebraically closed field|algebraically closed]]. In particular, function field extension <math>k(C)/k(x)</math> is a [[Kummer extension]].

==Ramification==
Let <math>C: y^m = f(x)</math> be a superelliptic curve defined over an algebraically closed field <math>k</math>, and <math>B' \subset k</math> denote the set of roots of <math>f</math> in <math>k</math>. Define set
:<math>B = \begin{cases} B' &\text{ if }m\text{ divides }\deg(f), \\ B'\cup\{\infty\} &\text{ otherwise.}\end{cases}</math>
Then <math>B \subset \mathbb{P}^1(k)</math> is the set of branch points of the covering map <math>C \to \mathbb{P}^1</math> given by <math>x</math>.

For an affine branch point <math>\alpha \in B</math>, let <math>r_\alpha</math> denote the order of <math>\alpha</math> as a root of <math>f</math>. Then
:<math>e_\alpha = \frac{m}{(m, r_\alpha)}</math>
is the ramification index <math>e(P_{\alpha, i})</math> at each of the <math>(m, r_\alpha)</math> ramification points <math>P_{\alpha, i}</math> of the curve lying over <math>\alpha \in \mathbb{A}^1(k) \subset \mathbb{P}^1(k)</math> (that is actually true for any <math>\alpha \in k</math>).

For the point at infinity,
:<math>e_\infty = \frac{m}{(m, \deg(f))}</math>
is the ramification index <math>e(P_{\infty, i})</math> at the <math>(m, \deg(f))</math> points <math>P_{\infty, i}</math> that lie over <math>\infty</math>. In particular, the curve is unramified over infinity if and only if its degree <math>m</math> and the degree of the defining polynomial <math>f</math> are relatively prime.

==See also==
* [[Hyperelliptic curve]]
* [[Artin-Schreier curve]]
* [[Kummer theory]]
* [[Superellipse]]

==References==
* {{cite book | first1=Marc | last1=Hindry | first2=Joseph H. | last2=Silverman | authorlink2=Joseph H. Silverman | title=Diophantine Geometry: An Introduction | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | volume=201 | year=2000 | isbn=0-387-98981-1 | zbl=0948.11023 | page=361 }}
* {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Elliptic Curves: Diophantine Analysis | volume=231 | series=Grundlehren der mathematischen Wissenschaften | publisher=[[Springer-Verlag]] | year=1978 | isbn=0-387-08489-4 }}
* {{cite book | last1=Shorey | first1=T.N. | last2=Tijdeman | first2=R. | author2-link=Robert Tijdeman | title=Exponential Diophantine equations | series=Cambridge Tracts in Mathematics | volume=87 | publisher=[[Cambridge University Press]] | year=1986 | isbn=0-521-26826-5 | zbl=0606.10011 }}
* {{cite book | title=The Algorithmic Resolution of Diophantine Equations | volume=41 | series=London Mathematical Society Student Texts | first=N. P. | last=Smart | authorlink=Nigel Smart (cryptographer) | publisher=[[Cambridge University Press]] | year=1998 | isbn=0-521-64633-2 }}

[[Category:Algebraic curves]]

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130.85.58.237 at 18:42, 6 December 2013