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

2022/05/23 by Miguel Grados, Grados, Miguel, Anna-Maria von Pippich +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.2205.11437

arxiv created 2022/05/23 · arxiv updated 2022/05/24

Abstract

Let N≥ 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)=ω²/φ(N), with ω² 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_Γlog(N) + o(g_Γlog(N)), as the level N tends to infinity.

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