Fields generated by points on superelliptic curves

TL;DR

This paper establishes a lower bound on the number of field extensions generated by algebraic points on superelliptic curves over , showing asymptotic growth related to the curve's parameters and degree divisibility.

Contribution

It provides the first asymptotic lower bounds for the number of fields generated by points on superelliptic curves with fixed degree and bounded discriminant, including geometric heuristics.

Findings

Lower bound ^{\u03b4_n} for the number of such fields

Asymptotic ^{1/m^2} growth as degree n increases

Degree n points may be less abundant when n is not divisible by (gcd(m, d))

Abstract

We give an asymptotic lower bound on the number of field extensions generated by algebraic points on superelliptic curves over $\mathbb{Q}$ with fixed degree $n$ and discriminant bounded by $X$. For $C$ a fixed such curve given by an affine equation $y^m = f(x)$ where $m \geq 2$ and $d= \mathrm{deg}\ f (x) \geq m$, we find that for all degrees $n$ divisible by $\gcd(m, d)$ and sufficiently large, the number of such fields is asymptotically bounded below by $X^{\delta_n}$, where $\delta_n \to 1/m^2$ as $n \to \infty$. We then give geometric heuristics suggesting that for n not divisible by $\gcd(m, d)$, degree $n$ points may be less abundant than those for which $n$ is divisible by $\gcd(m,d)$ and provide an example of conditions under which a curve is known to have finitely many points of certain degrees.