The rational points on a modular curve $X_H$ may be divided into three categories: cuspidal points, CM points, and non-cuspidal non-CM points.

Genus zero curves with a rational point are isomorphic to $\mathbb P^1$ and therefore have infinitely many rational points. Genus one curves with a rational point whose Jacobian has positive Mordell-Weil rank are isomorphic to an elliptic curve with infinitely many rational points. In all other cases, the set of rational points is finite.

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- Review status: beta
- Last edited by Bjorn Poonen on 2022-03-24 18:36:52

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

- 2022-03-24 18:36:52 by Bjorn Poonen
- 2022-03-20 20:41:39 by Andrew Sutherland
- 2022-03-20 20:40:48 by Andrew Sutherland
- 2022-03-20 20:40:31 by Andrew Sutherland

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