Critical and ramification points of the modular parametrization of an elliptic curve

Delaunay, Christophe

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 109-124.

Soit $E$ une courbe elliptique définie sur $\mathbb{Q}$ de conducteur $N$ et soit $\varphi$ son revêtement modulaire :

$$\varphi :X_0\left(N\right)\to E\left(ℂ\right).$$

Dans cet article, nous nous intéressons aux points critiques et aux points de ramification de $\varphi$. En particulier, nous expliquons comment donner une étude plus ou moins expérimentale de ces points.

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with conductor $N$ and denote by $\varphi$ the modular parametrization:

$$\varphi :X_0\left(N\right)\to E\left(ℂ\right).$$

In this paper, we are concerned with the critical and ramification points of $\varphi$. In particular, we explain how we can obtain a more or less experimental study of these points.

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