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 $Γ$ be the principal congruence subgroup of level $N$. Let $X(N)$ be the modular curve of genus $g_Γ$ associated to $Γ$. In this article, we study the Arakelov invariant $e(Γ)=\barω^2/φ(N)$, with $\barω^2$ denoting the self-intersection of the relative dualizing sheaf for the minimal regular model of $X(N)$, equipped with the Arakelov metric, and $φ(N)$ is the Euler's phi function. Our main result is the asymptotics $e(Γ) = 2g_Γ\log(N) + o(g_Γ\log(N))$, as the level $N$ tends to infinity.

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