Regular Functions on Formal-Analytic Arithmetic Surfaces

TL;DR

This paper explores the properties of regular functions on certain formal-analytic arithmetic surfaces, linking algebraic and complex-analytic conjectures, and introduces a Green's function pushforward formula related to Arakelov theory.

Contribution

It establishes a connection between a conjecture on the size of the ring of regular functions and a complex-analytic conjecture, and introduces a new Green's function pushforward formula.

Findings

The ring of regular functions can have continuum cardinality under certain conjectures.

A polynomial approximation argument demonstrates the abundance of regular functions.

A new formula for Green's functions relates to Arakelov degree.

Abstract

In this paper, we show that for a broad class of pseudoconvex formal-analytic arithmetic surfaces over $\text{Spec}(\mathbb{Z})$, those which admit a nonconstant monic such regular function, that a conjecture of Bost-Charles that the ring of regular functions has continuum cardinality is implied by a purely complex-analytic conjecture. Under the conjecture, a Fekete-Szego-type approximation argument produces a polynomial "large" relative to the regular function, which in turn yields continuum many distinct regular functions. We also introduce a formula for the pushforward by a holomorphic function of the equilibrium Green's functions for our bordered Riemann surface with boundary, a formula which has constant term related to Arakelov degree.