|Title: Rank-Favorable Bounds for Rational Points on Superelliptic Curves|
|Speaker: Noam Kantor of Emory University|
|Contact: Noam Kantor, email@example.com|
|Date: 2017-04-03 at 3:00PM|
|Venue: MSC: N304|
Let $C$ be a curve of genus at least two, and let $r$ be the rank of the rational points on its Jacobian. Under mild hypotheses on $r$, recent results by Katz, Rabinoff, Zureick-Brown, and Stoll bound the number of rational points on $C$ by a constant that depends only on its genus. Yet one expects an even stronger bound that depends favorably on $r$: when $r$ is small, there should be fewer points on $C$. In a 2013 paper, Stoll established such a ``rank-favorable" bound for hyperelliptic curves using Chabauty's method. In the present work we extend Stoll's results to superelliptic curves. We also discuss a possible strategy for proving a rank-favorable bound for arbitrary curves based on the metrized complexes of Amini and Baker. Our results have stark implications for bounding the number of rational points on a curve, since $r$ is expected to be small for ``most" curves.
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