Abstract
In Chapter 8 we have seen that elliptic curves and associated tools allow us to solve many Diophantine problems, essentially those coming from cubic or hyperelliptic quartic equations. The goal of the present chapter is to give some idea of the methods that are used for more general equations. For instance, Diophantus himself poses a problem (Problem 17 of book VI of the Arabic manuscript of Arithmetica [Ses]) that is equivalent to finding a nontrivial rational point on the curve defined by the equation $$ y^2 = x^6 + x^2 + 1. $$ This is a curve of genus 2, whereas elliptic curves are curves of genus 1, and it is the only example of a curve of genus greater than or equal to 2 considered by Diophantus. In this chapter we will be interested in curves of genus greater than or equal to 2.
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