Explicit bounds on canonical Green functions of modular curves

TL;DR

This paper establishes explicit bounds on the canonical Green functions for modular curves, which are Riemann surfaces formed from quotients of the upper half-plane by Fuchsian groups, aiding in quantitative analysis of these functions.

Contribution

It provides the first explicit bounds on canonical Green functions for a broad class of modular curves, enhancing understanding of their analytic properties.

Findings

Derived explicit bounds for Green functions on modular curves.

Applied bounds to analyze geometric and spectral properties.

Facilitated quantitative studies of modular curves' Green functions.

Abstract

We prove explicit bounds on canonical Green functions of Riemann surfaces obtained as compactifications of quotients of the upper half-plane by Fuchsian groups.