Chabauty methods using covers on curves of genus 2
Abstract. In this article, we consider the problem of determining the set of rational points on curves of genus 2. Given a curve C of genus 2 over a number field, we demonstrate a finite set of degree 16 unramified covers such that the rational points of C are covered by the rational points of the covers. We show that these covers map to elliptic curves, making Chabauty methods practically feasible on them. This gives a possibility to determine the rational points on genus 2 curves with Jacobians of higher rank. As an example, we determine the primitive rational solutions to the equation x2+y3=z8, which involves finding the rational points on some genus 2 curves with Jacobians of rank 2.
An archive of scripts for the algebraic number theory package KASH is available in the ASCII-file 1999-14.sh. Instructions on how to unpack it can be found within.
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This paper is part of the thesis Chabauty methods and covering techniques applied to generalised Fermat equations by the same author.
Report printed: August 16, 1999