On Néron models, divisors and modular curves

1998/06/15 by Bas Edixhoven, Edixhoven, Bas
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.math/9806173

arxiv created 1998/06/15 · arxiv updated 2009/12/01

Abstract

Let p be a prime number such that the modular curve X₀(p) has genus at least two. We show that the only points of the reduction mod p of X₀(p) with image in the reduction mod p of J₀(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 Néron 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 Néron 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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