Intersection matrices for the minimal regular model of $X_0(N)$ and applications to the Arakelov canonical sheaf

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