## Report MI 1999-15

**P.J. Bruin**

*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 *x*^{2}+*y*^{3}=*z*^{8}, 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.

**Document:** 1999-15.dvi (), 1999-15.ps (), 1999-15.ps.gz ()

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

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