Kirchhoff's theorem for Prym varieties

TL;DR

This paper extends Kirchhoff's matrix tree theorem to tropical Prym varieties associated with double covers of metric graphs, providing explicit formulas and studying the properties of the tropical Abel-Prym map.

Contribution

It introduces a Kirchhoff-type formula for the volume of tropical Prym varieties and analyzes the harmonicity and degree of the tropical Abel-Prym map.

Findings

The degree of the tropical Abel-Prym map is 2^{g-1}.

A new proof of the analogous finite graph result using the Ihara zeta function.

The algebraic Abel-Prym map also has degree 2^{g-1}.

Abstract

We prove an analogue of Kirchhoff's matrix tree theorem for computing the volume of the tropical Prym variety for double covers of metric graphs. We interpret the formula in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel-Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is $2^{g-1}$. Along the way, we use the Ihara zeta function to provide a new proof of the analogous result for finite graphs. As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel-Prym map is $2^{g-1}$ as well.