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

TL;DR

This paper investigates the asymptotic behavior of the self-intersection of the relative dualizing sheaf on modular curves of level N, revealing it grows proportionally to the genus times log(N) as N increases.

Contribution

It provides the first asymptotic formula for the Arakelov invariant of modular curves with square-free level N, linking geometric invariants to level growth.

Findings

The invariant e(Γ) asymptotically equals 2g_Γ log(N).

The study establishes a growth rate for the self-intersection of the dualizing sheaf.

Results connect geometric properties of modular curves with their level N.

Abstract

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.