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.

Hasse invariant and group cohomology.

Edixhoven, Bas; Khare, Chandrashekhar — 2003

Documenta Mathematica

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

Conrad, Brian; Edixhoven, Bas; Stein, William — 2003

Documenta Mathematica

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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 $X_{0} (N)$

Hasse invariant and group cohomology.

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