Rational points on a modular curve (awaiting review)

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.