Finding rational points on bielliptic genus 2 curves.

*(English)*Zbl 1029.11024Summary: The authors discuss a technique for trying to find all rational points on curves of the form \(Y^2= f_3X^6+ f_2X^4+ f_1X^2+ f_0\), where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty’s theorem may be applied. However, they concentrate on the situation when the rank is at least 2. In this case, they derive an associated family of elliptic curves, defined over a number field \(\mathbb{Q}(\alpha)\). If each of these elliptic curves has rank less than the degree of \(\mathbb{Q}(\alpha): \mathbb{Q}\), then they describe a Chabauty-like technique which may be applied to try to find all the points \((x,y)\) defined over \(\mathbb{Q}(\alpha)\) on the elliptic curves, for which \(x\in \mathbb{Q}\). This in turn allows them to find all \(\mathbb{Q}\)-rational points on the original genus 2 curve. They apply this to give a solution to a problem of Diophantus (where the sextic in \(X\) is irreducible over \(\mathbb{Q}\)), which simplifies the recent solution given in the second author’s Ph.D. thesis, Univ. California, Berkeley (1997). The authors also present two examples where the sextic in \(X\) is reducible over \(\mathbb{Q}\).