Elliptic Curves Induced by Diophantine Triples

This chapter introduces elliptic curves induced by Diophantine triples and discusses their properties and applications. One of the main applications is the construction of infinite families of rational Diophantine sextuples. The four known different constructions of such families are presented. Another important application of elliptic curves induced by Diophantine triples is constructing high-rank elliptic curves with certain torsion groups. We present details on the construction of some record curves over \(\mathbb {Q}(t)\) and \(\mathbb {Q}\) . We also discuss some other connections between Diophantine m-tuples and elliptic curves.