Harmonic Morphisms of Arithmetical Structures on Graphs

TL;DR

This work generalizes the graph Jacobian framework to arithmetical structures on graphs, showing that a harmonic morphism $\phi: \Gamma_2 \to \Gamma_1$ pulls back arithmetical structures and induces surjective pushforwards and injective pullbacks on arithmetical critical groups. It introduces a Riemann–Hurwitz formula for arithmetical graphs, relating arithmetical genus across a morphism via degree and ramification data. The results yield divisibility relations among critical groups and genus inequalities, with concrete examples on cycles, wheels, complete graphs, and stars that illustrate the theory. Methodologically, the work relies on the generalized Laplacian $L(\Gamma;S)$ and key matrix identities such as $L_2 \Phi = D_\mu \Phi L_1$ and degree-preserving pullbacks/pushforwards, linking divisors, Jacobians, and ramification data in a unified framework.

Abstract

Let $φ\colon Γ_2 \rightarrow Γ_1$ be a harmonic morphism of connected graphs. We show that an arithmetical structure on $Γ_1$ can be pulled back via $φ$ to an arithmetical structure on $Γ_2$. We then show that some results of Baker and Norine on the critical groups for the usual Laplacian extend to arithmetical critical groups, which are abelian groups determined by the generalized Laplacian associated to these arithmetical structures. In particular, we show that the morphism $φ$ induces a surjective group homomorphism from the arithmetical critical group of $Γ_2$ to that of $Γ_1$ and an injective group homomorphism from the arithmetical critical group of $Γ_1$ to that of $Γ_2$. Finally, we prove a Riemann-Hurwitz formula for arithmetical structures.

Figures (2)

• Figure 1: A harmonic morphism $\phi$ from $\Gamma_2=W_5$ to $\Gamma_2=C_3$.
• Figure 2: A graph with only cyclic critical groups.

Theorems & Definitions (36)

• Example 2.1
• Proposition 2.2
• Example 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• proof
• Example 2.7
• Theorem 3.1
• proof