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Jun 7, 2021, 9:09:22 PM6/7/21

to nt...@googlegroups.com, Lea Beneish, Chris Keyes

Hi everyone,

For our last seminar of the year, we have Lea Beneish from McGill and Chris Keyes from Emory who will be speaking on "Fields generated by points on superelliptic curves". As usual, come prepared to ask a lot of questions!

Zoom link. Meeting ID: 995 4255 2693 Passcode: C401, Tuesday 10am - 10:55am

Gather.town for lunch, Tuesday 12:30 - 1:30pm

Gather.town for lunch, Tuesday 12:30 - 1:30pm

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 $n$, 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 $gcd(m, d)$.

See you tomorrow!

Isabel

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