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

Abstract

Let $N\ge 3$ 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 $e(\Gamma )={\bar{\omega }}^2/\phi (N)$, 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 $\phi (N)$ 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.