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 list of **known rational 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.

**Knowl status:**

- Review status: reviewed
- Last edited by Kiran S. Kedlaya on 2019-09-05 19:15:51

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**History:**(expand/hide all)

- 2019-09-05 19:15:51 by Kiran S. Kedlaya (Reviewed)
- 2019-04-20 16:11:37 by Jennifer Paulhus (Reviewed)
- 2018-05-24 16:18:36 by John Cremona (Reviewed)

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