Computation of Admissible Arakelov-Green Functions on Metrized Graphs

TL;DR

This paper provides an explicit formula and a computational algorithm for admissible Arakelov-Green functions on metrized graphs, which are key objects in nonarchimedean arithmetic geometry, extending previous formulas.

Contribution

It introduces a new explicit formula for admissible Arakelov-Green functions on metrized graphs and implements an algorithm in SageMath for their computation.

Findings

Derived a new explicit formula for admissible Arakelov-Green functions.

Developed and implemented a SageMath algorithm for practical computation.

Demonstrated the algorithm with computational examples.

Abstract

Metrized graphs are nonarchimedean analogues of Riemann surfaces, and Arakelov-Green functions on these graphs are of fundamental importance for some aspects of arithmetic geometry. In the present paper, we give an explicit formula for an admissible Arakelov-Green function on a metrized graph, extending Cinkir's formula for the canonical Arakelov-Green function. Based on our formula, we present and implement an algorithm in the computer algebra system SageMath for explicitly computing such functions. We illustrate our algorithm with computational examples.