Bounds for canonical Green's functions at cusps

Priyanka Majumder; Anna-Maria von Pippich

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arXiv · 2210.04452

Bounds for canonical Green's functions at cusps

Priyanka Majumder·Anna-Maria von Pippich

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Abstract

Let $\Gamma$ be a cofinite Fuchsian subgroup. The canonical Green's function associated with $\Gamma$ arises in Arakelov theory when establishing asymptotics for Arakelov invariants of the modular curve associated with some congruence subgroup of level $N$ with a positive integer $N$. More precisely, in the known cases, canonical Green's functions at certain cusps contribute to the analytic part of the asymptotics for the self-intersection of the relative dualizing sheaf. In this article, we prove canonical Green's function of a cofinite Fuchsian subgroup at cusps bounded by the scattering constants, the Kronecker limit functions, and the Selberg zeta function of the group $\Gamma$. Then as an application, we prove an asymptotic expression of the canonical Green's function associated with $\Gamma_0(N)$, for any positive integer $N$.

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Priyanka Majumder, Anna-Maria von Pippich. 2022-10-10. Bounds for canonical Green's functions at cusps. https://arxiv.org/abs/2210.04452

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