Fields generated by points on superelliptic curves

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, discriminant bounded by X, and Galois closure Sₙ. For C a fixed curve given by an affine equation yᵐ=f (x) where m 2 and deg f (x) = d m, we find that for all degrees n divisible by (m, d) and sufficiently large, the number of such fields is asymptotically bounded below by X^cₙ, where cₙ 1/m² as n. This bound is determined explicitly by parameterizing x and y by rational functions, counting specializations, and accounting for multiplicity. We then give geometric heuristics suggesting that for n not divisible by (m, d), degree n points may be less abundant than those for which n is divisible by (m, d). Namely, we discuss the obvious geometric sources from which we expect to find points on C and discuss the relationship between these sources and our parametrization. When one a priori has a point on C of degree not divisible by (m, d), we argue that a similar counting argument applies. As a proof of concept we show in the case that C has a rational point that our methods can be extended to bound the number of fields generated by a degree n point of C, regardless of divisibility of n by (m, d).