Bounds for canonical Green's functions at cusps

TL;DR

This paper establishes bounds for the canonical Green's function at cusps of cofinite Fuchsian groups, linking it to scattering constants and zeta functions, and applies these bounds to asymptotic analysis for modular curves.

Contribution

It provides new bounds for the canonical Green's function at cusps of cofinite Fuchsian groups, connecting it with scattering constants, Kronecker limit functions, and the Selberg zeta function, with applications to modular curves.

Findings

Bounded the Green's function using scattering constants and zeta functions.

Derived asymptotic formulas for Green's functions associated with $ ext{Gamma}_0(N)$.

Connected Green's function bounds to Arakelov invariants of modular curves.

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 $N$.