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##### Department of Mathematics, University of California San Diego

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## Local solubility in families of superelliptic curves

##### Abstract:

If we choose at random an integral binary form $f(x, z)$ of fixed degree $d$, what is the probability that the superelliptic curve with equation  $C \colon: y^m = f(x, z)$ has a $p$-adic point, or better, points everywhere locally? In joint work with Lea Beneish, we show that the proportion of forms $f(x, z)$ for which $C$ is everywhere locally soluble is positive, given by a product of local densities. By studying these local densities, we produce bounds which are suitable enough to pass to the large $d$ limit. In the specific case of curves of the form $y^3 = f(x, z)$ for a binary form of degree 6, we determine the probability of everywhere local solubility to be 96.94%, with the exact value given by an explicit infinite product of rational function expressions.

[pre-talk at 1:20PM]

### APM 6402 and Zoom; see https://www.math.ucsd.edu/~nts/

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