### On N'eron models, divisors and modular curves, by Bas Edixhoven

Let $p$ be a prime number such that the modular curve $X_0(p)$ has genus
at least two. We show that the only points of the reduction mod $p$ of
$X_0(p)$ with image in the reduction mod $p$ of $J_0(p)$ in the cuspidal
group are the two cusps. This answers a question of Robert Coleman. For the
proof we give a description of the special fibre of the N\'eron model of the
jacobian of a semi-stable curve in terms of divisors. We also study to what
extent the morphism from a semistable curve with given base point to the
N\'eron model of its jacobian is a closed immmersion. Implicitly, logarithmic
structures intervene, and a well-known modular form of weight $p+1$ on
supersingular elliptic curves plays an important role.

Bas Edixhoven <edix@univ-rennes1.fr>