Diagonal genus 5 curves, elliptic curves over $\mathbb {Q}(t)$, and rational diophantine quintuples

Abstract

The problem of finding all possible extensions of a given rational diophantine quadruple to a rational diophantine quintuple is equivalent to the determination of the set of rational points on a certain curve of genus 5 that can be written as an intersection of three diagonal quadrics in $\mathbb P^4$. We discuss how one can (try to) determine the set of rational points on such a curve. We apply our approach to the original question in several cases. In particular, we show that Fermat's diophantine quadruple (1,3,8,120) can be extended to a rational diophantine quintuple in only one way, namely by 777480/8288641. We then discuss a method that allows us to find the Mordell-Weil group of an elliptic curve $E$ defined over the rational function field $\mathbb Q(t)$ when $E$ has full $\mathbb Q(t)$-rational 2-torsion. This builds on recent results of Dujella, Gusi\'c and Tadi\'c. We give several concrete examples to which this method can be applied. One of these results implies that there is only one extension of the diophantine quadruple $\bigl(t-1,t+1,4t,4t(4t^2-1)\bigr)$ over $\mathbb Q(t)$.