On Néron models, divisors and modular curves

Details On Néron models, divisors and modular curves On Néron models, divisors and modular curves

1998-06-15

arxiv.org/abs/math/9806173v1

Mathematics

Number Theory

Scientific paper

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.

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