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 $Q$ with fixed degree n and discriminant bounded by X. For C a fixed such curve given by an affine equation $y^m=f\left(x\right)$ where $m\ge 2$ and $d=\mathrm{deg}f\left(x\right)\ge m$, 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 $X^{{\delta }_n}$, 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.