An outline of the Flynn-Chabauty method for curves of genus 2 with an application to the curve C : y2 = x5 − 2x

An outline of the Flynn-Chabauty method for curves of genus 2 with an application to the curve : C : y² = x⁵ − 2x 

We will outline a method, which we call the Flynn-Chabauty method, for determining the set of rational points on a genus 2 curve, of rank 0 or 1, defined over Q. The method is highly developed, and there is an extensive literature on the subject. Since there is no point in re-writing theorems and proofs that have already been presented superbly, our goals are rather modest. This thesis provides, firstly, a summary of some background theory in the arithmetic of genus 2 curves and their Jacobians. Secondly, it provides a summary of the main concepts and results that are necessary to understand and apply the Flynn-Chabauty method, and an outline of the method itself. Third, we provide some MAPLE code to facilitate the application of the Flynn-Chabauty method to specific curves. Fourth, we apply the Flynn-Chabauty method to the curve C : y² = x⁵ − 2x and obtain a complete description of its set of rational points. We hope that this thesis will enable a person with little number-theoretical background to use the Flynn-Chabauty method. We hope that this thesis will also be useful as a guide for the reader who wishes to gain a deeper understanding of the Flynn-Chabauty method.