Rational points (reviewed)

For a curve $X$ of genus $g\ge 2$ over $\Q$ (or any number fields) 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 $X(\Q)$, for any given curve $X$, but in specific cases, including when the rank of the Jacobian of $X$ is less than $g$, this can often be accomplished.

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 $y^2+h(x)y=f(x)$ that listed as the minimal equation for the curve.