Intersection matrices for the minimal regular model of X 0 ( N ) ${X}_0(N)$ and applications to the Arakelov canonical sheaf

Abstract

Let $N>1$ be an integer coprime to 6 such that $N\notin \{5,7,13\}$ and let $g=g\left(N\right)$ be the genus of the modular curve $X_0\left(N\right)$. We compute the intersection matrices relative to special fibres of the minimal regular model of $X_0\left(N\right)$. Moreover, we prove that the self-intersection of the Arakelov canonical sheaf of $X_0\left(N\right)$ is asymptotic to $3g\log N$, for $N\to +\infty$.