Curves with sharp Chabauty-Coleman bound

TL;DR

This paper constructs specific algebraic curves where Coleman's effective Chabauty bound is exact, enabling precise determination of rational points under certain rank conditions, with explicit examples across various genera.

Contribution

It provides explicit constructions of curves with sharp Chabauty-Coleman bounds and demonstrates their application to determine rational points in new cases.

Findings

Explicit examples of genus two and rank one curves with sharp bounds

Construction of a genus five curve with rational points determined by descent and Coleman's theorem

Validation of Coleman's bound effectiveness in new curve cases

Abstract

We construct curves of each genus $g\geq 2$ for which Coleman's effective Chabauty bound is sharp and Coleman's theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman's bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman's theorem.