EUDML

Let $N \geq 1$ be an integer. Let $X_{0} (N)$ be the modular curve over $Z$ , as constructed by Katz and Mazur. The minimal resolution of $X_{0} (N)$ over $Z [1 / 6]$ is computed. Let $p \geq 5$ be a prime, such that $N = p^{2} M$ , with $M$ prime to $p$ . Let $n = (p^{2} - 1) / 2$ . It is shown that $X_{0} (N)$ has stable reduction at $p$ over $Q [\sqrt[n]{p}]$ , and the fibre at $p$ of the stable model is computed.

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Hasse invariant and group cohomology.

$J_{1} (p)$ has connected fibers.

Néron models and tame ramification

Rational elliptic curves are modular

Rational torsion points on elliptic curves over number fields

The weight in Serre's conjectures on modular forms.

On the prime-to- $p$ part of the groups of connected components of Néron models

Minimal resolution and stable reduction of $X_{0} (N)$