For a curve $X$ of genus $g\ge 2$ over $\Q$ (or any number field) the set of rational points $X(\Q)$ is finite, by a theorem of Faltings. At present no algorithm is known that explicitly computes a provably complete list of the points in $X(\Q)$, but one can conduct a search among points of bounded height to obtain a partial (and possibly, but not always proved to be, complete) list of known points.
Rational points on hyperelliptic curves are written in projective coordinates with respect to the weighted homogeneous equation $y^2+h(x,z)y=f(x,z)$ of degree $2g+2$ that is a smooth projective model for the curve $X$, where $y$ has weight $g+1$, while $x$ and $z$ both have weight 1. This homogeneous equation is uniquely determined by the affine equation $y^2+h(x)y=f(x)$ listed as the minimal equation for the curve.
- Review status: reviewed
- Last edited by Jennifer Paulhus on 2019-04-20 16:11:37