Let \(\Gamma \) denote a cofinite Fuchsian subgroup. In the context of Arakelov theory, the canonical Green’s function associated with \(\Gamma \) plays a crucial role in establishing asymptotic behavior for Arakelov invariants of the modular curve related to a congruence subgroup of level N, where N is a positive integer. More precisely, the canonical Green’s functions evaluated at certain cusps contribute to the analytic component of the asymptotic formula for the self-intersection of the relative dualizing sheaf. This article presents a proof demonstrating that the canonical Green’s function of a cofinite Fuchsian subgroup, evaluated at cusps, is bounded by the scattering constants, Kronecker’s limit functions, and the Selberg zeta function associated with the group \(\Gamma \) . As an application, we establish an asymptotic expression for the canonical Green’s function linked to \(\Gamma _0(N)\) , where N is any positive integer.
