Explicit Computation of Certain Arakelov-Green Functions

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.