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 $m\ge 2$ and $d=\mathrm{deg}f(x)\ge m$, 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 $X^{\hat{I}{´}_n}$, where $\hat{I}{´}_n\to 1/m^2$ 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.

30 pages, accepted for publication in Journal of Number Theory