Finding Rational Points on Bielliptic Genus 2 Curves

We discuss a technique for trying to find all rational points on curves of the form Y 2 = f3X + f2X + f1X + f0, 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, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field Q(α). If each of these elliptic curves has rank less than the degree of Q(α) : Q, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x, y) defined over Q(α) on the elliptic curves, for which x ∈ Q. This in turn allows us to find all Q-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over Q), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over Q.