Cusps of a modular curve (awaiting review)

The cusps on $X_H$ are the points whose image under the canonical morphism $j\colon X_H\to X(1)\simeq \mathbb P^1$ is $\infty$. It is only the noncuspidal points that parametrize elliptic curves (with level structure).

The cusps of a modular curve $X_H$ correspond to the complement of $Y_H$ in $X_H$, where $Y_H$ is the coarse moduli stack $\mathcal M_H^0$ defined in [MR:0337993, 10.1007/978-3-540-37855-6_4].

The rational cusps (also called $\Q$-cusps) are the cusps fixed by $\Gal_\Q$.