Self-intersection of the relative dualizing sheaf on modular curves X(N)

Let $N\geq 3$ be a composite, odd, and square-free integer and let $\Gamma$ be the principal congruence subgroup of level $N$. Let $X(N)$ be the modular curve of genus $g_{\Gamma}$ associated to $\Gamma$. In this article, we study the Arakelov invariant $e(\Gamma)=\bar{\omega}^2/\varphi(N)$, with $\bar{\omega}^2$ denoting the self-intersection of the relative dualizing sheaf for the minimal regular model of $X(N)$, equipped with the Arakelov metric, and $\varphi(N)$ is the Euler's phi function. Our main result is the asymptotics $e(\Gamma) = 2g_{\Gamma}\log(N) + o(g_{\Gamma}\log(N))$, as the level $N$ tends to infinity.