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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 $\Gal_\Q$.

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  • Review status: beta
  • Last edited by Ciaran Schembri on 2022-11-05 14:13:14
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