Rank-Favorable Bounds for Rational Points on Superelliptic Curves of Small Rank

TL;DR

This paper extends rank-favorable bounds for rational points from hyperelliptic to superelliptic curves, showing fewer points when the Jacobian's rank is small, which supports conjectures about rational points on most curves.

Contribution

It generalizes Stoll's rank-favorable bounds from hyperelliptic to superelliptic curves, highlighting differences relevant to uniformity conjectures.

Findings

Bounded rational points for small rank on superelliptic curves

Extension of Chabauty's method to superelliptic curves

Implications for uniformity conjectures in number theory

Abstract

Let $C$ be a curve of genus at least three defined over a number field, and let $r$ be the rank of the rational points of 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, noting in the process some differences that ought to inform uniformity conjectures for general curves. Our results have stark implications for bounding numbers of rational points, since $r$ is expected to be small for "most" curves.