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...

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Bibliographic Details
Published in:arXiv.org 2022-05
Main Authors: Grados, Miguel, Anna-Maria von Pippich
Format: Article
Language:English
Subjects:
Intersections
Subgroups
Online Access:Get full text
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Summary: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.
ISSN:2331-8422