Superelliptic curve

In mathematics, a superelliptic curve is a plane curve with an equation of the form

y^m = f(x),

where the exponent m is fixed and f is a polynomial of degree d. The case m = 2 and d > 4 is a hyperelliptic curve: the case m = 3 and d = 4 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

C \to \mathbb{P}^1

of the projective line of degree $m \geq 2$ 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 that a superelliptic curve of degree $m$ defined over a field $k$ has an affine model given by an equation

y^m = f(x)

for some polynomial $f \in k[x]$ 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 $C: y^m = f(x)$ be a superelliptic curve defined over an algebraically closed field $k$ , and $B' \subset k$ denote the set of roots of $f$ in $k$ . Define set

B = \begin{cases} B' &\text{ if }m\text{ divides }\deg(f), \\ B'\cup\{\infty\} &\text{ otherwise.}\end{cases}

Then $B \subset \mathbb{P}^1(k)$ is the set of branch points of the covering map $C \to \mathbb{P}^1$ given by $x$ .

For an affine branch point $\alpha \in B$ , let $r_\alpha$ denote the order of $\alpha$ as a root of $f$ . As before, we assume that $1 \leq r_\alpha < m$ . Then

e_\alpha = \frac{m}{(m, r_\alpha)}

is the ramification index $e(P_{\alpha, i})$ at each of the $(m, r_\alpha)$ ramification points $P_{\alpha, i}$ of the curve lying over $\alpha \in \mathbb{A}^1(k) \subset \mathbb{P}^1(k)$ (that is actually true for any $\alpha \in k$ ).

For the point at infinity, define integer $0 \leq r_\infty < m$ as follows. If

s = \min \{t \in \mathbb{Z} | mt \geq \deg(f) \},

then $r_\infty = ms - \deg(f)$ . Note that $(m, r_\infty) = (m, \deg(f))$ . Then analogously to the other ramification points,

e_\infty = \frac{m}{(m, r_\infty)}

is the ramification index $e(P_{\infty, i})$ at the $(m, r_\infty)$ points $P_{\infty, i}$ that lie over $\infty$ . 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 $r_\alpha$ 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 = \frac12 \left( m (|B| - 2) - \sum_{\alpha \in B} (m, r_\alpha)\right) + 1.

References

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 $\mathbb {C}$ ". 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.

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Superelliptic curve

Definition

Ramification

Genus

See also

References