Monday, October 11, 2021 12:30 PM
Lea Beneish (Berkeley)

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 $n$, discriminant bounded by $X$, and Galois closure $S_n$. For $C$ a fixed curve given by an affine equation $y^m = f(x)$ where $m \geq 2$ and $deg f(x) = d \geq m$, 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 $X^{c_n}$ , where $c_n \to$1/m^2$as$n \to \infty$. 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$\gcd(m, d)$, degree$n$points may be less abundant than those for which$n$is divisible by$\gcd(m, d)$. 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$\gcd(m, d)$, 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$\gcd(m, d)\$. This talk is based on joint work with Christopher Keyes.