Bounds on canonical Green's functions

Abstract

A fundamental object in the theory of arithmetic surfaces is the Green's function associated to the canonical metric. Previous expressions for the canonical Green's function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green's function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green's function through covers and for families of modular curves.

A relation of two metrics on a noncompact hyperbolic Riemann orbisurface

“…The above relation, which relates the two natural metrics defined on a Riemann orbisurface has been proved for compact hyperbolic Riemann surfaces, as a relation of differential forms by Jorgenson and Kramer [6]. The same authors have also extended the key identity to noncompact hyperbolic Riemann surfaces of finite volume in [5].…”

“…(9), it follows that the residual hyperbolic metric is well defined on X . Furthermore, from Proposition 3.3 in [6], for z ∈ X \E, we have…”

“…The key identity has been the most significant technical result of [6], which transforms a problem in Arakelov theory into that of hyperbolic geometry. The key identity has enabled J. Jorgenson and J. Kramer to derive optimal bounds for the canonical Green's function defined on a compact hyperbolic Riemann surface X in terms of invariants coming from the hyperbolic geometry of X .…”

“…The same authors have also extended the key identity to noncompact hyperbolic Riemann surfaces of finite volume in [5]. In this paper, the authors use different methods from [6], and study the behavior of the key identity over a family of degenerating compact hyperbolic Riemann surfaces. Our main theorem can be seen as an extension of their result to elliptic fixed points and cusps at the level of currents acting on the space of singular function C , (X ).…”

“…Our main theorem can be seen as an extension of their result to elliptic fixed points and cusps at the level of currents acting on the space of singular function C , (X ). Our methods are different from the ones employed in [5], and are organized around the original line of proof in [6].…”