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Fields generated by points on superelliptic curves

Lea Beneish, McGill University and Chris Keyes, Emory University
Tuesday, June 8, 2021 - 10:00am to 10:55am

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 \$\operatorname{deg} f(x) = d \geq m\$, we find that for all degrees \$n\$ divisible by \$\operatorname{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 \$\operatorname{gcd}(m, d)\$, degree \$n\$ points may be less abundant than those for which \$n\$ is divisible by \$\operatorname{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 \$\operatorname{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 \$\operatorname{gcd}(m, d)\$.

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