Fields generated by points on superelliptic curves

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.