Explicit Computation of Certain Arakelov-Green Functions

TL;DR

This paper provides a combinatorial interpretation and explicit formulas for Arakelov-Green functions on metrized graphs, enabling their computation through an algorithm based on electric circuit theory.

Contribution

It introduces a new combinatorial approach and explicit formulas for Arakelov-Green functions, facilitating their computation on metrized graphs.

Findings

Formulas show Arakelov-Green functions are piecewise linear or quadratic.

Derived an algorithm for explicit computation of these functions.

Clarified the connection between Arakelov-Green functions and electric circuit theory.

Abstract

Arakelov-Green functions defined on metrized graphs have important role in relating arithmetical problems on algebraic curves into graph theoretical problems. In this paper, we clarify the combinatorial interpretation of certain Arakelov-Green functions by using electric circuit theory. The formulas we gave clearly show that such functions are piece-wisely defined, and each piece is a linear or quadratic function on each pair of edges of metrized graphs. These formulas lead to an algorithm for explicit computation of Arakelov-Green functions.