# Kirchhoff’s theorem for Prym varieties

Yoav Len  Mathematical Institute, University of St Andrews, St Andrews KY16
9SS, UK<EMAIL_ADDRESS>and  Dmitry Zakharov  Department of
Mathematics, Central Michigan University, Mount Pleasant, MI 48859, USA
<EMAIL_ADDRESS>
###### 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.

###### 2010 Mathematics Subject Classification:

14T05; 14H40

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Kirchhoff’s theorem for the Prym group and the Prym variety
  4. 4 The local structure of the Abel–Prym map
  5. 5 Harmonicity of the Abel–Prym map
  6. A The algebraic Abel–Prym map (by Sebastian Casalaina-Martin)

##  1\. Introduction

Kirchhoff’s celebrated matrix tree theorem states that the number of spanning
trees of a connected finite graph $G$, also known as the _complexity_ of $G$,
is equal to the absolute value of the determinant of the reduced Laplacian
matrix of $G$. From a tropical viewpoint, this number is also equal to the
order of the Jacobian group $\operatorname{Jac}(G)$ of $G$.

In [ABKS14], Kirchhoff’s theorem was generalized to metric graphs and given a
geometric interpretation. The Jacobian variety $\operatorname{Jac}(\Gamma)$ of
a metric graph $\Gamma$ of genus $g$ is a real torus of dimension $g$, and its
volume can be computed as a weighted sum over all spanning trees of $\Gamma$.
Given a set $F\subset E(\Gamma)$ of $g$ edges of $\Gamma$ (with respect to a
choice of model), denote by $w(F)$ the product of the lengths of the edges in
$F$. Then (see Theorem 1.5 in [ABKS14])

$\operatorname{Vol}^{2}(\operatorname{Jac}(\Gamma))=\sum_{F}w(F),$ (1)

where the sum is taken over those subsets $F$ such that $\Gamma\backslash F$
is a spanning tree of $\Gamma$.

The weighted matrix-tree theorem can be proved by a direct application of the
Cauchy–Binet formula (see Remark 5.7 in [ABKS14]), but the authors give a
geometric proof in terms of a canonical representability result for tropical
divisor classes, that we briefly recall. Let
$\Phi:\operatorname{Sym}^{g}(\Gamma)\to\operatorname{Pic}^{g}(\Gamma)$ be the
tropical Abel–Jacobi map, sending an effective degree $g$ divisor $D$ to its
linear equivalence class. A divisor $D=P_{1}+\cdots+P_{g}$ is a called a
_break divisor_ if each $P_{i}$ is supported on an edge $e_{i}$ in such a way
that $\\{e_{1},\ldots,e_{g}\\}$ is the complement of a spanning tree of
$\Gamma$. By a result of Mikhalkin and Zharkov [MZ08], the map $\Phi$ has a
canonical continuous section, whose image is the set of break divisors in
$\operatorname{Sym}^{g}(\Gamma)$. Hence $\operatorname{Pic}^{g}(\Gamma)$ (and,
by translation, $\operatorname{Jac}(\Gamma)$) has a canonical cellular
decomposition coming from the cells of $\operatorname{Sym}^{g}(\Gamma)$
parametrized by the spanning trees of $\Gamma$. Computing the volume of
$\operatorname{Jac}(\Gamma)$ in terms of this decomposition gives Equation
(1), where the terms in the right hand side correspond to the volumes of the
individual cells. We note that the results of [ABKS14] can be reinterpreted as
saying that the Abel–Jacobi map $\Phi$ is a _harmonic morphism of polyhedral
spaces of degree one_ (see Remark 2.5).

The purpose of this paper is to prove analogous results for the tropical Prym
variety associated to a free double cover of metric graphs. Given an étale
double cover $f:\widetilde{C}\to C$ of smooth algebraic curves of genera
$2g-1$ and $g$ respectively, the kernel of the norm map
$\operatorname{Nm}:\operatorname{Jac}(\widetilde{C})\to\operatorname{Jac}(C)$
has two connected components, and the even component is an abelian variety of
dimension $g-1$, known as the _Prym variety_
$\operatorname{Prym}(\widetilde{C}/C)$ of the double cover. Prym varieties
have been extensively studied following Mumford’s seminal paper [Mum74], as
they are one of only few instances of abelian varieties that can be described
explicitly. Furthermore, they play a key role in rationality questions for
threefolds [CG72] and in constructing compact hyper-Kähler manifolds [LSV17].

The notion of an étale cover of algebraic curves has two natural analogues in
tropical geometry. One can consider _free covers_
$\pi:\widetilde{\Gamma}\to\Gamma$, which are covering spaces in the
topological sense: the map $\pi$ is a local homeomorphism at each point, and
an isometry if the graphs are metric. It is often necessary to consider the
more general _unramified covers_ , which are finite harmonic morphisms of
metric graphs satisfying a numerical Riemann–Hurwitz condition. This notion
does not have an analogue for finite graphs. The tropicalization of an étale
cover of algebraic curves is an unramified cover of metric graphs, but not
necessarily free.

The tropical Prym variety $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$
associated to an unramified double cover $\pi:\widetilde{\Gamma}\to\Gamma$ of
metric graphs is defined in analogy with its algebraic counterpart [JL18,
Definition 6.2]. Specifically,
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ is the connected component of
the identity of the kernel of the tropical norm map
$\operatorname{Nm}:\operatorname{Jac}(\widetilde{\Gamma})\to\operatorname{Jac}(\Gamma)$
(note that in the tropical case, the kernel has two connected components if
$\pi$ is free, and one if $\pi$ is unramified but not free). As shown in
[LU19, Theorem B], this construction commutes with tropicalization. Namely, if
$\pi$ is the tropicalization of an étale double cover $f:\widetilde{C}\to C$
of algebraic curves, then the tropical abelian variety
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ is the skeleton of the
Berkovich analytification of $\operatorname{Prym}(\widetilde{C}/C)$, and the
corresponding Abel–Prym maps commute (the corresponding result for Jacobians
was proved in [BR15]). This observation has recently led to new results
concerning the dimensions of Brill–Noether loci in Prym varieties [CLRW20,
Corollary B].

In the current paper, we consider only free double covers of finite and metric
graphs. We first compute the order of the _Prym group_
$\operatorname{Prym}(\widetilde{G}/G)$ of a free double cover
$p:\widetilde{G}\to G$ of a finite graph $G$ of genus $g$. The finite group
$\operatorname{Prym}(\widetilde{G}/G)$ is a canonically defined index two
subgroup of the kernel of the norm map
$\operatorname{Nm}:\operatorname{Jac}(\widetilde{G})\to\operatorname{Jac}(G)$.
In the spirit of Kirchhoff’s formula, the order of
$\operatorname{Prym}(\widetilde{G}/G)$ is a weighted sum over certain
$(g-1)$-element subsets of $E(G)$: given a subset $F\subset E(G)$ of $g-1$
edges of $G$, we say that $F$ is an _odd genus one decomposition of rank $r$_
if $G\backslash F$ consists of $r$ connected components of genus one, each
having connected preimage in $\widetilde{G}$.

######  Kirchhoff–Prym formula (Proposition 3.2).

The order of the Prym group $\operatorname{Prym}(\widetilde{G}/G)$ of a free
double cover $p:\widetilde{G}\to G$ of finite graphs is equal to

$|\operatorname{Prym}(\widetilde{G}/G)|=\frac{1}{2}|\operatorname{Ker}\operatorname{Nm}|=\sum_{r=1}^{g}4^{r-1}C_{r},$

where $C_{r}$ is the number of odd genus one decompositions of $G$ of rank
$r$.

This formula has already been obtained by Zaslavsky in the seminal paper
[Zas82] as the determinant of the _signed Laplacian matrix_ of the graph $G$
(see Theorem 8A.4 in _loc. cit._), and was later explicitly interpreted as the
order of the kernel of the norm map by Reiner and Tseng (see Proposition 9.9
in [RT14]). We give an alternative proof, by comparing the Ihara zeta
functions $\zeta(s,\widetilde{G})$ and $\zeta(s,G)$ of the graphs
$\widetilde{G}$ and $G$. By the work of Stark and Terras [ST96, ST00], the
quotient $\zeta(s,\widetilde{G})/\zeta(s,G)$ for a free double cover
$p:\widetilde{G}\to G$ is the $L$-function of the cover evaluated at the
nontrivial representation of the Galois group $\mathbb{Z}/2\mathbb{Z}$, and we
use the $L$-function to compute the order of the Prym group. To the best of
our knowledge, this is the first application of the Ihara zeta function to
tropical geometry.

We then derive a weighted version of the Kirchhoff–Prym formula for the volume
of the Prym variety of a free double cover of metric graphs, in the same way
that Equation (1) generalizes Kirchhoff’s theorem.

######  Theorem A (Theorem 3.4).

The volume of the tropical Prym variety
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ of a free double cover
$\pi:\widetilde{\Gamma}\to\Gamma$ of metric graphs is given by

$\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\sum_{F\subset
E(\Gamma)}4^{r(F)-1}w(F),$

where the sum is taken over all odd genus one decompositions $F$ of $\Gamma$,
and where $w(F)$ is the product of the lengths of the edges in $F$.

In the second part of our paper, we derive a geometric interpretation for the
volume formula for the tropical Prym variety, in the spirit of [ABKS14]. Let
$\pi:\widetilde{\Gamma}\to\Gamma$ be a free double cover of metric graphs, and
let $\iota:\widetilde{\Gamma}\to\widetilde{\Gamma}$ be the associated
involution. Consider the _Abel–Prym map_ $\Psi$ associated to $\pi$

$\Psi:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma),\quad\Psi(D)=D-\iota(D),$

where $\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$ denotes the
component of $\operatorname{Ker}\operatorname{Nm}$ of the same parity as
$g-1$.

Our principal result states that $\Psi$ is a _harmonic morphism of polyhedral
spaces of degree $2^{g-1}$_ (as in Definition 2.12). The space
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ has a natural polyhedral
decomposition, with the top-dimensional cells $C(\widetilde{F})$ indexed by
multisets $\widetilde{F}\subset E(\widetilde{\Gamma})$ of $g-1$ edges of
$\widetilde{\Gamma}$. We define the _degree_ of a top-dimensional cell to be
$\deg_{\Psi}(\widetilde{F})=2^{r(\widetilde{F})-1}$ if $p(\widetilde{F})$
consists of distinct edges and is an odd genus one decomposition of rank
$r(\widetilde{F})$, and zero otherwise. Then the Abel–Prym map $\Psi$
contracts the cell $C(\widetilde{F})$ if and only if
$\deg_{\Psi}(\widetilde{F})=0$. Furthermore $\Psi$ is harmonic with respect to
the degree, meaning that it satisfies a balancing condition around every
codimension one cell of $\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$. This
implies that we can extend the degree function to all of
$\operatorname{Sym}^{g-1}(\widetilde{\Gamma})$ in such a way that the sum of
the degrees in each fiber of $\Psi$ is a finite constant, called the global
degree of $\Psi$. To compute the global degree, we first observe that the
harmonicity of the Abel–Prym map allows us to express the volume of
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ in terms of its degree.
Comparing the result with Theorem A, we find that the global degree is in fact
$2^{g-1}$. The factors $4^{r(F)-1}$ in the weighted Kirchhoff–Prym formula
represent squares of the local degrees of $\Psi$.

Summarizing, we obtain a semi-canonical representability result for tropical
Prym divisors:

######  Theorem B (Theorem 5.1).

The Abel–Prym map
$\Psi:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$
associated to a free double cover $\pi:\widetilde{\Gamma}\to\Gamma$ of metric
graphs is a harmonic morphism of polyhedral spaces of degree $2^{g-1}$. In
particular, there is a degree map
$\deg_{\Psi}:\operatorname{Sym}^{g-1}(\widetilde{\Gamma})\to\mathbb{Z}_{\geq
0}$ such that any element of
$\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$ has exactly $2^{g-1}$
representatives of the form $\widetilde{D}-\iota(\widetilde{D})$ counted with
multiplicity $\deg_{\Psi}(\widetilde{D})$, where $\widetilde{D}$ is an
effective divisor of degree $g-1$.

We note that a divisor in
$\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$ may have infinitely
many representatives of the form $\widetilde{D}-\iota(\widetilde{D})$ with
$\deg_{\Psi}(\widetilde{D})=0$, but a generic divisor in
$\operatorname{Prym}^{[g-1]}(\widetilde{\Gamma}/\Gamma)$ only has
representatives $\widetilde{D}-\iota(\widetilde{D})$ with
$\deg_{\Psi}(\widetilde{D})>0$, and hence finitely many in total.

The canonical representability result of [ABKS14] also holds in the integral
setting, after fixing a generic element
$\lambda\in\operatorname{Jac}(\Gamma)$: given a model $G$ of $\Gamma$, any
class in $\operatorname{Pic}^{g}(G)$ is represented by a unique break divisor
$D\in\operatorname{Sym}^{g}(G)$. Shifting the break divisor by $\lambda$, we
obtain a divisor supported on the complement of a spanning tree of $G$. See
[ABKS14, Remark 4.26] and [BBY19, Example 1.3.4] for more detail. An analogous
correspondence result does not hold for Prym groups. In fact, the discrete
Abel–Prym map
$\operatorname{Sym}^{g-1}(\widetilde{G})\to\operatorname{Prym}^{[g-1]}(\widetilde{G}/G)$
associated to a free double cover $p:\widetilde{G}\to G$ of finite graphs is
not even surjective in general (see Example 2.9).

We believe that suitable generalizations of Theorems A and B hold for
unramified double covers of metric graphs, which is the more general framework
considered in [JL18] and [LU19]. To derive and prove them using the methods of
our paper, it would first be necessary to develop a theory of $L$-functions of
unramified Galois covers of graphs, extending the theory for free covers
developed in [ST96] and [ST00]. Such a theory should be a part of a more
general theory of Ihara zeta functions of graphs of groups. This first step in
this direction is the paper [Zak20] by the second author. It would also be
interesting to determine whether the Prym construction generalizes to other
tropical abelian covers (see [LUZ19]).

###  1.1. The algebraic Abel–Prym map and its tropicalization

Let $C$ be a smooth algebraic curve of genus $g$, and let
$\Phi^{d}:\operatorname{Sym}^{d}(C)\to\operatorname{Pic}^{d}(C)$ be the degree
$d$ Abel–Jacobi map. It is a classical result that $\Phi^{d}$ has degree 1
when $d\leq g$, and is birational when $d=g$ [ACGH85, Chapter 1.3]. The degree
$d$ Abel–Prym map
$\Psi^{d}:\operatorname{Sym}^{d}(\widetilde{C})\to\operatorname{Prym}^{[d]}(\widetilde{C}/C)$
corresponding to an unramified double cover $\pi:\widetilde{C}\to C$ of smooth
algebraic curves is defined by
$\Psi^{d}(\widetilde{D})=\widetilde{D}-\iota(\widetilde{D})$. Unlike the
Abel–Jacobi map, the degree of $\Psi^{d}$ depends non-trivially on the
Brill–Noether type of $C$. For example, if $d=1$ then the degree is equal to
$2$ if $C$ is hyperelliptic and $1$ otherwise. However, the degree of the
Abel–Prym map when $d=g-1$ is always $2^{g-1}$. We are very grateful to
Sebastian Casalaina-Martin for a proof of this result (and a number of others)
about the Abel–Prym map, which we have included as an Appendix to this paper.

Given that the algebraic Abel–Prym map $\Psi^{g-1}$ has degree $2^{g-1}$, it
is tempting to derive Theorem B from the corresponding algebraic statement by
a tropicalization argument (the same argument would also give an alternative
proof of one of the principal results of [ABKS14], namely the existence of a
canonical section of the Abel–Jacobi map). It is well known that the
tropicalization of a degree $d$ map of algebraic curves is a harmonic morphism
of metric graphs of the same degree $d$. However, we are unaware of a suitable
generalization of this result to higher dimension, and the derivation of such
a result is beyond the scope of this paper.

Motivated by this similarity, and by the results of the Appendix, we propose
the following conjecture:

######  Conjecture 1.1.

Let $f:\widetilde{C}\to C$ be an étale double cover of algebraic curves
tropicalizing to a free double cover $\pi:\widetilde{\Gamma}\to\Gamma$ of
metric graphs. Then the degrees of the algebraic and tropical Abel–Prym maps
$\Psi^{d}$ for $d\leq g-2$ associated to $f$ and $\pi$ coincide. In
particular, the degree of $\Psi^{d}$ is bounded by $2^{d}$.

We stress that the tropical and algebraic results presented in this paper are
derived via entirely different techniques, and are independent of each other.

###  1.2. Degenerations of abelian varieties

Polyhedral decompositions of real tori, such as the ones described above,
suggest an interesting connection with degenerations of abelian varieties and
compactifications of their moduli spaces.

The Jacobian of a nodal curve is a semi-abelian variety that is not proper in
general. There are numerous compactifications constructed by various authors
that depend on a choice of degree and an ample line bundle (e.g. [Est01,
Sim94]). In degree $g$, these constructions coincide [Cap94], and the strata
in the compactification are in bijection with certain orientations on the dual
graph of the curve [Chr18, Theorem 3.2.8]. In fact, the same strata are in an
order reversing bijection with the cells in the ABKS decomposition of the
tropical Jacobian [Cap18, Theorem 4.3.4]. More generally, each Simpson and
Esteves compactified Jacobian of $C$ can be constructed from a polyhedral
decomposition of the tropical Jacobian of the dual graph of $C$ [CPS19,
Theorem 1.1]. An analogous statement in degree $g$ holds uniformly over the
moduli space of curves [AAPT19].

The situation is more subtle for Prym varieties. Given an admissible double
cover $\widetilde{C}\to C$ of nodal curves, the identity component of the
kernel of the norm map is, again, a non-proper semi-abelian variety. There are
various approaches for compactifying the Prym variety (e.g. [ABH02, CMGHL17]).
However, unlike the case of Jacobians, the Prym–Torelli map
$\mathcal{R}_{g}\to A_{g-1}$ from the moduli space of étale double covers to
the moduli space of abelian varieties does not extend to the boundary for any
reasonable toroidal compactification of $A_{g-1}$ [Vol02, FS86].

We therefore ask the following.

######  Question 1.2.

Given an admissible double cover $\widetilde{C}\to C$ with tropicalization
$\widetilde{\Gamma}\to\Gamma$, do the cells of the semi-canonical
decomposition of the tropical Prym variety
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ described in Theorem B
correspond to the boundary strata of an appropriate compactification of the
Prym variety $\operatorname{Prym}(\widetilde{C}/C)$?

A positive answer would suggest a path to a natural compactification of the
moduli space of abelian varieties such that the map $\mathcal{R}_{g}\to
A_{g-1}$ extends to the boundary.

### Acknowledgments

We would like to thank Matthew Baker, Samuel Grushevsky, Sam Payne, and Dhruv
Ranganathan for useful discussions, and David Jensen, Martin Ulirsch, and Chi-
Ho Yuen for comments on an older version of the paper. We are very thankful to
Victor Reiner for pointing out the history of the Kirchhoff–Prym formula in
the context of critical groups and signed graphs. We are deeply grateful to
Sebastian Casalaina-Martin for a comprehensive Appendix dedicated to the
algebraic Abel–Prym map.

##  2\. Preliminaries

In this section, we review the necessary material about graphs, metric graphs,
tropical ppavs, Jacobians, Prym varieties, and polyhedral spaces. The only new
material is found in Section 2.5, where we define the Prym group of a free
double cover of graphs. Throughout this paper, we consider both finite and
metric graphs, which we distinguish by using Latin and Greek letters,
respectively. Graphs are allowed to have loops and multi-edges but not legs,
and we do not consider the more general setting of graphs with vertex weights.
All graphs are assumed to be connected unless stated otherwise.

###  2.1. Graphs and free double covers

We denote the vertex and edge sets of a finite graph $G$ by respectively
$V(G)$ and $E(G)$, and its _genus_ by $g(G)=|E(G)|-|V(G)|+1$. An _orientation_
of a graph $G$ is a choice of direction for each edge, allowing us to define
source and _target_ maps $s,t:E(G)\to V(G)$. For a vertex $v\in V(G)$, the
_tangent space_ $T_{v}G$ is the set of edges emanating from $v$, and the
_valency_ is $\operatorname{val}(v)=\\#T_{v}G$ (where each loop at $v$ counts
twice towards the valency). A _metric graph_ $\Gamma$ is the compact metric
space obtained from a finite graph $G$ by assigning positive lengths
$\ell:E(G)\to\mathbb{R}_{>0}$ to its edges, and identifying each edge $e\in
E(G)$ with a closed interval of length $\ell(e)$. The pair $(G,\ell)$ is
called a model of $\Gamma$, and we define $g(\Gamma)=g(G)$. A metric graph has
infinitely many models, obtained by arbitrarily subdividing edges, but the
genus $g(\Gamma)$ does not depend on the choice of model.

The only maps of finite graphs that we consider in our paper are _free double
covers_ $p:\widetilde{G}\to G$. Such a map consists of a pair of surjective
2-to-1 maps $p:V(\widetilde{G})\to V(G)$ and $p:E(\widetilde{G})\to E(G)$ that
preserve adjacency, and such that the map is an isomorphism in the
neighborhood of every vertex of $\widetilde{G}$. Specifically, for any pair of
vertices $\widetilde{v}$ and $v$ with $p(\widetilde{v})=v$, and for each edge
$e\in E(G)$ attached to $v$, there is a unique edge $\widetilde{e}\in E(G)$
attached to $\widetilde{v}$ that maps to $e$. We say that $p:\widetilde{G}\to
G$ is _oriented_ if $\widetilde{G}$ and $G$ are oriented graphs, and if the
map $p$ preserves the orientation. There is a naturally defined _involution_
$\iota:\widetilde{G}\to\widetilde{G}$ on the source graph that exchanges the
two sheets of the cover. It is easy to see that if $G$ has genus $g$, then any
connected double cover $\widetilde{G}$ of $G$ has genus $2g-1$.

######  Remark 2.1.

If $p:\widetilde{G}\to G$ is a free double cover and $e\in E(G)$ is a loop at
$v$, then the preimage of $e$ is either a pair of loops, one at each of the
two vertices in $p^{-1}(v)$, or a pair of edges connecting the two vertices in
$p^{-1}(v)$ (oriented in the opposite directions if $e$ is oriented).

A free double cover of metric graphs $\pi:\widetilde{\Gamma}\to\Gamma$ is a
free double cover $p:\widetilde{G}\to G$ of appropriate models
$(\widetilde{G},\ell)$ and $(G,\ell)$ of respectively $\widetilde{\Gamma}$ and
$\Gamma$ that preserves edge length, so that
$\ell(p(\widetilde{e}))=\ell(\widetilde{e})$ for all $\widetilde{e}\in
E(\widetilde{\Gamma})$. A free double cover is the same as a finite harmonic
morphism of global degree two and local degree one everywhere, and we do not
consider the more general case of unramified harmonic morphisms of degree two
studied in [JL18] and [LU19]. From a topological viewpoint, free double covers
are the same as normal covering spaces with Galois group
$\mathbb{Z}/2\mathbb{Z}$.

We consistently use the following construction, due to [Wal76], to describe a
double cover $p:\widetilde{G}\to G$ of a graph $G$ of genus $g$.

######  Construction A.

Let $G$ be a graph of genus $g$. Fix a spanning tree $T\subset G$ and a subset
$S\subset\\{e_{0},\ldots,e_{g-1}\\}$ of the edges in the complement of $T$.
Let $\widetilde{T}^{+}$ and $\widetilde{T}^{-}$ be two copies of $T$, and for
a vertex $v\in V(T)=V(G)$ denote $\widetilde{v}^{\pm}$ the corresponding
vertices in $\widetilde{T}^{\pm}$. We define the graph $\widetilde{G}$ as

$\widetilde{G}=\widetilde{T}^{+}\cup\widetilde{T}^{-}\cup\\{\widetilde{e}_{0}^{\pm},\ldots,\widetilde{e}_{g-1}^{\pm}\\}.$

The map $p:\widetilde{G}\to G$ sends $\widetilde{T}^{\pm}$ isomorphically to
$T$ and $\widetilde{e}_{i}^{\pm}$ to $e_{i}$. For $e_{i}\in S$, each of the
two edges $\widetilde{e}^{\pm}_{i}$ above it has one vertex on
$\widetilde{T}^{+}$ and one on $\widetilde{T}^{-}$, while for $e_{i}\notin S$
both vertices of $\widetilde{e}^{\pm}_{i}$ lie on the tree
$\widetilde{T}^{\pm}$. It is clear that if $G$ is connected, then
$\widetilde{G}$ is connected if and only if $S$ is nonempty. In the latter
case, we may and will assume that $e_{0}\in S$, and then
$\widetilde{T}=\widetilde{T}^{+}\cup\widetilde{T}^{-}\cup\\{\widetilde{e}_{0}^{+}\\}$
is a spanning tree for $\widetilde{G}$. We furthermore always assume that the
starting and ending vertices of $\widetilde{e}^{+}_{0}$ lie respectively on
$\widetilde{T}^{+}$ and $\widetilde{T}^{-}$, and conversely for
$\widetilde{e}^{-}_{0}$:

$s(\widetilde{e}^{\pm}_{0})=\widetilde{s(e_{0})}^{\pm},\quad
t(\widetilde{e}^{\pm}_{0})=\widetilde{t(e_{0})}^{\mp}.$

We do not make the same assumptions about the lifts of the remaining edges
$e_{i}\in S$.

The set of connected free double covers of $G$ is thus identified with the set
of nonempty subsets of $\\{e_{0},\ldots,e_{g-1}\\}$. Alternatively, the
fundamental cycle construction defines a basis for $H_{1}(G,\mathbb{Z})$
corresponding to the edges $e_{i}$ (the $i$-th basis element is the unique
cycle supported on $T\cup\\{e_{i}\\}$ and containing $+e_{i}$). The set of
nonempty subsets of $\\{e_{0},\ldots,e_{g-1}\\}$ is then identified with the
set of nonzero elements of
$\operatorname{Hom}(H_{1}(G,\mathbb{Z}),\mathbb{Z}/2\mathbb{Z})=H^{1}(G,\mathbb{Z}/2\mathbb{Z})$
(a subset is identified with its indicator function), and the latter is
canonically identified with the set of connected free double covers of $G$ by
covering space theory.

######  Remark 2.2.

Let $p:\widetilde{G}\to G$ be a free double cover corresponding to a tree
$T\subset G$ and a subset $S\subset E(G)\backslash E(T)$, and let
$G^{\prime}\subset G$ be a subgraph. Then the preimage $p^{-1}(G^{\prime})$ is
connected (equivalently, the restricted cover
$p|_{p^{-1}(G^{\prime})}:p^{-1}(G^{\prime})\to G^{\prime}$ is a nontrivial
free double cover) if and only if there is a cycle on $G^{\prime}$ that
contains an odd number of edges from $S$.

###  2.2. Chip-firing and linear equivalence

We now briefly recall the basic notions of divisor theory for finite and
metric graphs (see [BN07, Section 1] and [LPP12, Section 2] respectively for
details).

Let $G$ be a finite graph. The _divisor group_ $\operatorname{Div}(G)$ of $G$
is the free abelian group on $V(G)$, and the _degree_ of a divisor is the sum
of its coefficients:

$\operatorname{Div}(G)=\left\\{\sum
a_{v}v:a_{v}\in\mathbb{Z}\right\\},\quad\deg\sum a_{v}v=\sum a_{v}.$

A divisor $D=\sum a_{v}v$ is called _effective_ if all $a_{v}\geq 0$, and we
denote the set of divisors of degree $d$ by $\operatorname{Div}^{d}(G)$.

Let $n=|V(G)|$ be the number of vertices, and let $Q$ and $A$ be the $n\times
n$ _valency_ and _adjacency_ matrices:

$Q_{uv}=\delta_{uv}\operatorname{val}(u),\quad A_{uv}=|\\{\text{edges between
$u$ and $v$}\\}|.$ (2)

The _Laplacian_ $L=Q-A$ of $G$ is a symmetric degenerate matrix whose rows and
columns sum to zero. Given a vertex $v$, the divisor obtained via _chip-firing
from_ $v$ is

$D_{v}=-\sum_{u\in V(G)}L_{uv}u.$

Such a divisor has degree zero, hence the set of _principal divisors_
$\operatorname{Prin}(G)$, which are defined as the image of the chip-firing
map

$\mathbb{Z}^{V(G)}\to\mathbb{Z}^{V(G)}=\operatorname{Div}(G),\quad a\mapsto-
La,$

lies inside $\operatorname{Div}^{0}(G)$. The _Picard group_ and _Jacobian_ of
$G$ are defined as

$\operatorname{Pic}(G)=\operatorname{Div}(G)/\operatorname{Prin}(G),\quad\operatorname{Jac}(G)=\operatorname{Div}^{0}(G)/\operatorname{Prin}(G).$

Since any principal divisor has degree zero, the degree function descends to
$\operatorname{Pic}(G)$, and we denote $\operatorname{Pic}^{k}(G)$ the set of
equivalence classes of degree $k$ divisors, so that
$\operatorname{Jac}(G)=\operatorname{Pic}^{0}(G)$. The group
$\operatorname{Pic}(G)$ is infinite, but $\operatorname{Jac}(G)$ is a finite
group whose order is equal to the absolute value of any cofactor of the
Laplacian $L$. _Kirchhoff’s matrix tree theorem_ states that
$|\operatorname{Jac}(G)|$ is equal to the number of spanning trees of $G$ (see
[BS13, Theorem 6.2]).

The Picard variety of a metric graph $\Gamma$ of genus $g$ is defined as
follows (see [BF11]). A _divisor_ on a metric graph $\Gamma$ is a finite
linear combination of the form

$D=a_{1}p_{1}+a_{2}p_{2}+\cdots+a_{k}p_{k},$

where $a_{i}\in\mathbb{Z}$ and $p_{i}$ can be any point of $\Gamma$, and $\deg
D=a_{1}+\cdots+a_{k}$. We denote by $\operatorname{Div}(\Gamma)$ the divisor
group and by $\operatorname{Div}^{k}(\Gamma)$ the set of divisors of degree
$k$. A _rational function_ $M$ on $\Gamma$ is a piecewise-linear real-valued
function with integer slopes. The principal divisor $\operatorname{div}(M)$
associated to $M$ is the degree zero divisor whose value at each point
$p\in\Gamma$ is the sum of the incoming slopes of $M$ at $p$. It is clear that
$\operatorname{div}(M+N)=\operatorname{div}(M)+\operatorname{div}(N)$ and
$\operatorname{div}(-M)=-\operatorname{div}(M)$, so the principal divisors
$\operatorname{Prin}(\Gamma)$ form a subgroup of
$\operatorname{Div}^{0}(\Gamma)$, and the degree function descends to the
quotient:

$\operatorname{Pic}(\Gamma)=\operatorname{Div}(\Gamma)/\operatorname{Prin}(\Gamma),\quad\operatorname{Pic}^{k}(\Gamma)=\\{[D]\in\operatorname{Pic}(\Gamma):\deg
D=k\\}.$

The _Picard variety_ $\operatorname{Pic}^{0}(\Gamma)$ is a real torus of
dimension $g$ and is isomorphic to the Jacobian variety of $\Gamma$, which we
review in the next section, while each $\operatorname{Pic}^{k}(\Gamma)$ is a
torsor over $\operatorname{Pic}^{0}(\Gamma)$.

###  2.3. Tropical abelian varieties

The Jacobian variety of a metric graph $\Gamma$ is a _tropical principally
polarized abelian variety_ (tropical ppav for short). We review the theory of
tropical ppavs, following are [FRSS18] and [LU19], though we have found it
convenient to slightly modify the main definitions (see Remark 2.3). In brief,
a tropical ppav is a real torus $\Sigma$ whose universal cover is equipped
with a distinguished lattice (used to define integral local coordinates on
$\Sigma$, and in general distinct from the lattice defining the torus itself),
and an inner product.

Let $\Lambda$ and $\Lambda^{\prime}$ be finitely generated free abelian groups
of the same rank, and let
$[\cdot,\cdot]:\Lambda^{\prime}\times\Lambda\to\mathbb{R}$ be a nondegenerate
pairing. The triple $(\Lambda,\Lambda^{\prime},[\cdot,\cdot])$ defines a _real
torus with integral structure_
$\Sigma=\operatorname{Hom}(\Lambda,\mathbb{R})/\Lambda^{\prime}$, where the
"integral structure" refers to the lattice
$\operatorname{Hom}(\Lambda,\mathbb{Z})\subset\operatorname{Hom}(\Lambda,\mathbb{R})$,
and where $\Lambda^{\prime}$ is embedded in
$\operatorname{Hom}(\Lambda,\mathbb{R})$ via the assignment
$\lambda^{\prime}\mapsto[\lambda^{\prime},\cdot]$. The transposed data
$(\Lambda^{\prime},\Lambda,[\cdot,\cdot]^{t})$ define the _dual torus_
$\Sigma^{\prime}=\operatorname{Hom}(\Lambda^{\prime},\mathbb{R})/\Lambda$.

Let $\Sigma_{1}=(\Lambda_{1},\Lambda^{\prime}_{1},[\cdot,\cdot]_{1})$ and
$\Sigma_{2}=(\Lambda_{2},\Lambda^{\prime}_{2},[\cdot,\cdot]_{2})$ be two real
tori with integral structure, and let
$f_{*}:\Lambda_{1}^{\prime}\to\Lambda_{2}^{\prime}$ and
$f^{*}:\Lambda_{2}\to\Lambda_{1}$ be a pair of maps satisfying

$[\lambda^{\prime}_{1},f^{*}(\lambda_{2})]_{1}=[f_{*}(\lambda^{\prime}_{1}),\lambda_{2}]_{2}$
(3)

for all $\lambda^{\prime}_{1}\in\Lambda^{\prime}_{1}$ and
$\lambda_{2}\in\Lambda_{2}$. The map $f^{*}$ defines a dual map
$\overline{f}:\operatorname{Hom}(\Lambda_{1},\mathbb{R})\to\operatorname{Hom}(\Lambda_{2},\mathbb{R})$,
and condition (3) implies that
$\overline{f}(\Lambda^{\prime}_{1})\subset\Lambda^{\prime}_{2}$ (in fact,
$\overline{f}|_{\Lambda^{\prime}_{1}}=f_{*}$). Hence the pair $(f_{*},f^{*})$
defines a _homomorphism_ $f:\Sigma_{1}\to\Sigma_{2}$ of real tori with
integral structures. The transposed pair $(f^{*},f_{*})$ defines the _dual
homomorphism_ $f^{\prime}:\Sigma^{\prime}_{2}\to\Sigma^{\prime}_{1}$.

Let $f=(f_{*},f^{*}):\Sigma_{1}\to\Sigma_{2}$ be a homomorphism of real tori
with integral structures
$\Sigma_{i}=(\Lambda_{i},\Lambda^{\prime}_{i},[\cdot,\cdot]_{i})$. We can
naturally associate two real tori to $f$: the connected component of the
identity of the kernel of $f$, denoted $(\operatorname{Ker}f)_{0}$, and the
cokernel $\operatorname{Coker}f$. It is easy to see that
$(\operatorname{Ker}f)_{0}$ and $\operatorname{Coker}f$ also have integral
structures, and the natural maps $i:(\operatorname{Ker}f)_{0}\to\Sigma_{1}$
and $p:\Sigma_{2}\to\operatorname{Coker}f$ are homomorphisms of real tori with
integral structure.

Indeed, let $K=(\operatorname{Coker}f^{*})^{tf}$ be the quotient of
$\operatorname{Coker}f^{*}$ by its torsion subgroup (equivalently, the
quotient of $\Lambda_{1}$ by the saturation of $\operatorname{Im}f^{*}$), and
let $K^{\prime}=\operatorname{Ker}f_{*}$. Then
$\operatorname{Hom}(K,\mathbb{R})$ is naturally identified with the kernel of
the map
$\operatorname{Hom}(\Lambda_{1},\mathbb{R})\to\operatorname{Hom}(\Lambda_{2},\mathbb{R})$
dual to $f^{*}$, and therefore
$(\operatorname{Ker}f)_{0}=(K,K^{\prime},[\cdot,\cdot]_{K})$, where
$[\cdot,\cdot]_{K}:K^{\prime}\times K\to\mathbb{R}$ is the pairing induced by
$[\cdot,\cdot]_{1}$. We note that this pairing is well-defined: given
$\lambda^{\prime}_{1}\in K^{\prime}$ and $\lambda_{2}\in\Lambda_{2}$, Equation
(3) implies that

$[\lambda^{\prime}_{1},f^{*}(\lambda_{2})]_{1}=[f_{*}(\lambda^{\prime}_{1}),\lambda_{2}]_{2}=[0,\lambda_{2}]_{2}=0.$

Therefore, for $\lambda^{\prime}\in K^{\prime}$ and $\lambda\in K$, the
pairing $[\lambda^{\prime},\lambda]_{K}=[\lambda^{\prime},\lambda]_{1}$ does
not depend on a choice of representative for $\lambda\in K$. The natural maps
$i^{*}:\Lambda_{1}\to K$ and $i_{*}:K^{\prime}\to\Lambda_{1}^{\prime}$ define
$(\operatorname{Ker}f)_{0}$ as an integral subtorus of $\Sigma_{1}$.
Similarly, $\operatorname{Coker}f=(C,C^{\prime},[\cdot,\cdot]_{C})$, where
$C=\operatorname{Ker}f^{*}$, $C^{\prime}=(\operatorname{Coker}f_{*})^{tf}$,
the pairing $[\cdot,\cdot]_{C}$ is induced by $[\cdot,\cdot]_{2}$, and $p$ is
given by the natural maps $p_{*}:\Lambda^{\prime}_{2}\to C^{\prime}$ and
$p^{*}:C\to\Lambda_{2}$. We note that a morphism $f$ of real tori with
integral structure has finite kernel if and only if $K$ and $K^{\prime}$ are
trivial, in other words if $f_{*}$ is injective (equivalently, if
$\operatorname{Im}f^{*}$ has finite index in $\Lambda_{1}$).

Let $\Sigma=(\Lambda,\Lambda^{\prime},[\cdot,\cdot])$ be a real torus with
integral structure. A _polarization_ on $\Sigma$ is a map
$\xi:\Lambda^{\prime}\to\Lambda$ (necessarily injective) with the property
that the induced bilinear form

$(\cdot,\cdot):\Lambda^{\prime}\times\Lambda^{\prime}\to\mathbb{R},\quad(\lambda^{\prime},\mu^{\prime})=[\lambda^{\prime},\xi(\mu^{\prime})]$

is symmetric and positive definite. Given a polarization $\xi$ on $\Sigma$,
the pair $(\xi,\xi)$ defines a homomorphism $\eta:\Sigma\to\Sigma^{\prime}$ to
the dual, whose finite kernel is identified with
$\Lambda/\operatorname{Im}\xi$. The pair $(\Sigma,\xi)$ is called a _tropical
polarized abelian variety_. The map $\eta$ is an isomorphism if and only if
$\xi$ is an isomorphism, in which case we say that the polarization $\xi$ is
_principal_.

Let $\Sigma=(\Lambda,\Lambda^{\prime},[\cdot,\cdot])$ be a $g$-dimensional
tropical polarized abelian variety. The associated bilinear form
$(\cdot,\cdot)$ on $\Lambda^{\prime}$ extends to an inner product on the
universal cover $V=\operatorname{Hom}(\Lambda,\mathbb{R})$, which we also
denote $(\cdot,\cdot)$, and hence to a translation-invariant Riemannian metric
on $\Sigma$. Let $C\subset\Sigma$ be a parallelotope framed by vectors
$v_{1},\ldots,v_{g}\in V$, then the volume of $C$ is equal to the square root
$\sqrt{\det(v_{i},v_{j})}$ of the Gramian determinant of the $v_{i}$. In
particular, if $\lambda^{\prime}_{1},\ldots,\lambda^{\prime}_{g}$ is a basis
of $\Lambda^{\prime}$, then

$\operatorname{Vol}^{2}(\Sigma)=\det(\lambda^{\prime}_{i},\lambda^{\prime}_{j}).$

Finally, let $f:\Sigma_{1}\to\Sigma_{2}$ be a homomorphism of real tori with
integral structures given by $f^{*}:\Lambda_{2}\to\Lambda_{1}$ and
$f_{*}:\Lambda^{\prime}_{1}\to\Lambda^{\prime}_{2}$, and assume that $f$ has
finite kernel (equivalently, $f_{*}$ is injective). Given a polarization
$\xi_{2}:\Lambda^{\prime}_{2}\to\Lambda_{2}$ on $\Sigma_{2}$ with associated
bilinear form $(\cdot,\cdot)_{2}$, we define the _induced polarization_
$\xi_{1}:\Lambda^{\prime}_{1}\to\Lambda_{1}$ by
$\xi_{1}=f^{*}\circ\xi_{2}\circ f_{*}$. This is indeed a polarization, because
by (3) the associated bilinear form $(\cdot,\cdot)_{1}$ on
$\Lambda^{\prime}_{1}$ is given by

$(\lambda^{\prime}_{1},\mu^{\prime}_{1})_{1}=[\lambda^{\prime}_{1},\xi_{1}(\mu^{\prime}_{1})]_{1}=[\lambda^{\prime}_{1},f^{*}(\xi_{2}(f_{*}(\mu^{\prime}_{1})))]_{2}=[f_{*}(\lambda^{\prime}_{1}),\xi_{2}(f_{*}(\mu^{\prime}_{1}))]_{2}=(f_{*}(\lambda^{\prime}_{1}),f_{*}(\mu^{\prime}_{1}))_{2},$

so it is symmetric and positive definite because $f_{*}$ is injective. Hence,
in particular, an integral subtorus $i:\Pi\to\Sigma$ of a tropical polarized
abelian variety $(\Sigma,\xi)$ has an induced polarization, which we denote
$i^{*}\xi$. We note that the polarization induced by a principal polarization
is not necessarily itself principal.

######  Remark 2.3.

In [LU19], a real torus with integral structure is defined as a torus
$\Sigma=N_{\mathbb{R}}/\Lambda$ with a distinguished lattice $N\subset N_{R}$
in the universal cover, and a morphism $f:\Sigma_{1}\to\Sigma_{2}$ as a map
$\overline{f}:N_{1,\mathbb{R}}\to N_{2,\mathbb{R}}$ satisfying
$\overline{f}(\Lambda_{1})\subset\Lambda_{2}$ and induced by a
$\mathbb{Z}$-linear map $N_{1}\to N_{2}$. It is easy to see that this
definition is equivalent to ours.

###  2.4. The Jacobian of a metric graph

We now construct the Jacobian variety $\operatorname{Jac}(\Gamma)$ of a metric
graph $\Gamma$ of genus $g$ as a tropical ppav, following [BF11] and [LU19].
We first pick an oriented model $G$ of $\Gamma$ and consider the corresponding
simplicial homology groups. Let $A$ be either $\mathbb{Z}$ or $\mathbb{R}$,
and let $C_{0}(G,A)=A^{V(G)}$ and $C_{1}(G,A)=A^{E(G)}$ be respectively the
_simplicial $0$-chain and $1$-chain groups_ of $G$ with coefficients in $A$.
The source and target maps $s,t:E(G)\to V(G)$ induce a boundary map

$d_{A}:C_{1}(G,A)\to C_{0}(G,A),\quad\sum_{e\in E(G)}a_{e}e\mapsto\sum_{e\in
E(G)}a_{e}[t(e)-s(e)],$

and the _first simplicial homology group_ of $G$ with coefficients in $A$ is
$H_{1}(G,A)=\operatorname{Ker}d_{A}$. We also consider the group of _$A$
-valued harmonic $1$-forms_ $\Omega(G,A)$ on $G$, which is a subgroup of the
free $A$-module with basis $\\{de:e\in E(G)\\}$:

$\Omega(G,A)=\left\\{\sum_{e\in
E(G)}\omega_{e}de:\sum_{e:t(e)=v}\omega_{e}=\sum_{e:s(e)=v}\omega_{e}\mbox{
for all }v\in V(G)\right\\}.$

We note that mathematically $H_{1}(G,A)$ and $\Omega(G,A)$ are the same
object, but it is convenient to distinguish them, both for historical purposes
and for clarity of exposition.

We now define an _integration pairing_

$[\cdot,\cdot]:C_{1}(G,A)\times\Omega(G,A)\to\mathbb{R}$

by

$[\gamma,\omega]=\int_{\gamma}\omega=\sum_{e\in
E(G)}\gamma_{e}\omega_{e}\ell(e),\quad\gamma=\sum_{e\in
E(G)}\gamma_{e}e,\quad\omega=\sum_{e\in E(G)}\omega_{e}de.$

By Lemma 2.1 in [BF11], the integration pairing restricts to a perfect pairing
on $H_{1}(G,A)\times\Omega(G,A)$.

Let $G^{\prime}$ be the model of $\Gamma$ obtained by subdividing the edge
$e\in E(G)$ into two edges $e_{1}$ and $e_{2}$, oriented in the same way as
$e$, with $\ell(e_{1})+\ell(e_{2})=\ell(e)$. The natural embedding
$C_{1}(G,A)\to C_{1}(G^{\prime},A)$ sending $e$ to $e_{1}+e_{2}$ restricts to
an isomorphism $H_{1}(G,A)\to H_{1}(G^{\prime},A)$. Similarly, the groups
$\Omega(G,A)$ and $\Omega(G^{\prime},A)$ are naturally isomorphic, and these
isomorphisms preserve the integration pairing. Hence we can define
$\Omega(\Gamma,A)=\Omega(G,A)$ and $H_{1}(\Gamma,A)=H_{1}(G,A)$ for any model
$G$, and by a $1$-chain, or _path_ , on $\Gamma$ we mean a $1$-chain on any
model of $\Gamma$.

We now let $\Lambda=\Omega(\Gamma,\mathbb{Z})$ and
$\Lambda^{\prime}=H_{1}(\Gamma,\mathbb{Z})$, let
$[\cdot,\cdot]:\Lambda^{\prime}\times\Lambda\to\mathbb{R}$ be the integration
pairing, and let $\xi:H_{1}(\Gamma,\mathbb{Z})\to\Omega(\Gamma,\mathbb{Z})$ be
the natural isomorphism sending the $1$-cycle $\sum a_{e}e$ to the $1$-form
$\sum a_{e}de$. We denote
$\Omega^{*}(\Gamma)=\operatorname{Hom}(\Omega(\Gamma,\mathbb{Z}),\mathbb{R})$,
and by the universal coefficient theorem the group
$\operatorname{Hom}(H_{1}(\Gamma,\mathbb{Z}),\mathbb{R})$ is canonically
isomorphic to $H^{1}(\Gamma,\mathbb{R})$. The _Jacobian variety_ and the
_Albanese variety_ of $\Gamma$ are the dual tropical ppavs

$\operatorname{Jac}(\Gamma)=\Omega(\Gamma)^{*}/H_{1}(\Gamma,\mathbb{Z}),\quad\operatorname{Alb}(\Gamma)=H^{1}(\Gamma,\mathbb{R})/\Omega(\Gamma,\mathbb{Z}).$

The group $H_{1}(\Gamma,\mathbb{Z})$ carries an intersection form

$(\cdot,\cdot)=[\cdot,\xi(\cdot)]:H_{1}(\Gamma,\mathbb{Z})\times
H_{1}(\Gamma,\mathbb{Z})\to\mathbb{R},\quad\left(\sum_{e\in
E(G)}\gamma_{e}e,\sum_{e\in E(G)}\delta_{e}e\right)=\sum_{e\in
E(G)}\gamma_{e}\delta_{e}\ell(e)$ (4)

that induces an inner product on $\Omega^{*}(\Gamma)$.

Fix a point $q\in\Gamma$, and for any $p\in\Gamma$ choose a path
$\gamma(q,p)\in C_{1}(\Gamma,\mathbb{Z})$ from $q$ to $p$. Integrating along
$\gamma(q,p)$ defines an element of $\Omega(\Gamma)^{*}$, and choosing a
different path $\gamma^{\prime}(q,p)$ defines the same element modulo
$H_{1}(\Gamma,\mathbb{Z})\subset\Omega(\Gamma)^{*}$. Hence we have a well-
defined _Abel–Jacobi map_ $\Phi_{q}:\Gamma\to\operatorname{Jac}(\Gamma)$ with
base point $q$:

$\Phi_{q}:\Gamma\to\operatorname{Jac}(\Gamma),\quad
p\mapsto\left(\omega\mapsto\int_{\gamma(q,p)}\omega\right).$ (5)

The map $\Phi_{q}$ extends by linearity to $\operatorname{Div}(\Gamma)$, and
its restriction to $\operatorname{Div}^{0}(\Gamma)$ does not depend on the
choice of base point $q$. The tropical analogue of the Abel–Jacobi theorem
(see [MZ08], Theorem 6.3) states that $\Phi_{q}$ descends to a canonical
isomorphism $\operatorname{Pic}^{0}(\Gamma)\simeq\operatorname{Jac}(\Gamma)$.
Since any $\operatorname{Pic}^{k}(\Gamma)$ is a torsor over
$\operatorname{Pic}^{0}(\Gamma)$, we can define
$\operatorname{Vol}(\operatorname{Pic}^{k}(\Gamma))=\operatorname{Vol}(\operatorname{Jac}(\Gamma))$.

Finally, we recall the principal results [ABKS14], which concern the tropical
Jacobi inversion problem. Consider the degree $g$ Abel–Jacobi map

$\Phi:\operatorname{Sym}^{g}(\Gamma)\to\operatorname{Pic}^{g}(\Gamma),\quad\Phi(p_{1},\ldots,p_{g})=p_{1}+\cdots+p_{g}.$

A choice of model $G$ for $\Gamma$ defines a cellular decomposition

$\operatorname{Sym}^{g}(\Gamma)=\bigcup_{F\in\operatorname{Sym}^{g}(E(G))}C(F),$

where for a multiset
$F=\\{e_{1},\ldots,e_{g}\\}\in\operatorname{Sym}^{g}(E(G))$ of $g$ edges of
$G$ the cell $C(F)$ consists of divisors supported on $F$:

$C(F)=\\{p_{1}+\cdots+p_{g}:p_{i}\in e_{i}\\}.$

We say that $F$ is a _break set_ if all $e_{i}$ are distinct and $G\backslash
F$ is a tree, and the set of _break divisors_ is the union of the cells $C(F)$
over all break sets $F$.

The map $\Phi$ is affine linear on each cell $C(F)$, and has maximal rank
precisely when $F$ is a break set. Specifically, the following is true:

  1. (1)

If $F=\\{e_{1},\ldots,e_{g}\\}$ is a break set, then the restriction of $\Phi$
to $C(F)$ is injective, and

$\operatorname{Vol}(\Phi(C(F)))=\frac{w(F)}{\operatorname{Vol}(\operatorname{Jac}(\Gamma))},\quad
w(F)=\operatorname{Vol}(C(F))=\ell(e_{1})\cdots\ell(e_{g}).$ (6)

  2. (2)

If $F$ is not a break set, then the restriction of $\Phi$ to $C(F)$ does not
have maximal rank, and $\operatorname{Vol}(\Phi(C(F)))=0$.

Furthermore, the map $\Phi$ has a unique continuous section whose image is the
set of break divisors. Hence the images of the break cells $C(F)$ cover
$\operatorname{Pic}^{g}(\Gamma)$ with no overlaps in the interior of cells,
and summing together their volumes gives
$\operatorname{Vol}(\operatorname{Jac}(\Gamma))=\operatorname{Vol}(\operatorname{Pic}^{g}(\Gamma))$:

######  Theorem 2.4 (Theorem 1.5 of [ABKS14]).

The volume of the Jacobian variety of a metric graph $\Gamma$ of genus $g$ is
given by

$\operatorname{Vol}^{2}(\operatorname{Jac}(\Gamma))=\sum_{F\subset
E(\Gamma)}w(F),$ (7)

where the sum is taken over $g$-element subsets $F\subset E(\Gamma)$ such that
$\Gamma\backslash F$ is a tree.

######  Remark 2.5.

This result can be interpreted as saying that $\Phi$ is a _harmonic morphism
of polyhedral spaces of degree 1_ , where we define the local degree of $\Phi$
on a cell $C(F)$ to be $1$ if $F$ is a break set and $0$ otherwise. Indeed,
the harmonicity condition ensures that such a map has a unique continuous
section, since each cell of $\operatorname{Pic}^{g}(\Gamma)$ has a unique
preimage in $\operatorname{Sym}^{g}(\Gamma)$ and these preimages fit together
along codimension one cells. Formula (6) then implies that the map $\Phi$ has
a common volume dilation factor
$1/\operatorname{Vol}(\operatorname{Jac}(\Gamma))$ on all non-contracted
cells.

######  Remark 2.6.

We also note that, from the point of view of the Riemannian geometry of
$\operatorname{Jac}(\Gamma)$, the edge lengths on $\Gamma$ are measured in
units of $[\mbox{length}]^{2}$, not $[\mbox{length}]$. This is already clear
from Formula (4) for the intersection form. Hence, for example, if $\Gamma$ is
a circle of length $L$ (in other words consists of a single loop of length $L$
attached to a vertex), then $\Gamma$ is canonically isomorphic to
$\operatorname{Pic}^{1}(\Gamma)$, but the volume of
$\operatorname{Jac}(\Gamma)$ is $\sqrt{L}$, rather than $L$.

###  2.5. Prym groups

We now discuss the Prym group of a free double cover of finite graphs. Unlike
the case of metric graphs (which we treat in Section 2.6), finite groups don’t
have a distinguished connected component of the identity. We therefore require
a notion of parity on elements of the kernel of the norm map.

Let $p:\widetilde{G}\to G$ be a free double cover of graphs. The induced maps
$\operatorname{Nm}:\operatorname{Div}(\widetilde{G})\to\operatorname{Div}(G)$
and
$\iota:\operatorname{Div}(\widetilde{G})\to\operatorname{Div}(\widetilde{G})$
given by

$\operatorname{Nm}\left(\sum a_{v}v\right)=\sum a_{v}p(v),\quad\iota\left(\sum
a_{v}v\right)=\sum a_{v}\iota(v)$

preserve degree and linear equivalence, and descend to give a surjective map
$\operatorname{Nm}:\operatorname{Jac}(\widetilde{G})\to\operatorname{Jac}(G)$
and an isomorphism
$\iota:\operatorname{Jac}(\widetilde{G})\to\operatorname{Jac}(\widetilde{G})$.

A divisor in the kernel
$D\in\operatorname{Ker}\operatorname{Nm}\subset\operatorname{Div}(\widetilde{G})$
has degree zero and can be uniquely represented as $D=E-\iota(E)$, where $E$
is an effective divisor and the supports of $E$ and $\iota(E)$ are disjoint.
We define the _parity_ of $D$ as

$\epsilon(D)=\deg E\bmod 2.$

It turns out that parity respects addition and linear equivalence, and hence
gives a surjective homomorphism from
$\operatorname{Ker}\operatorname{Nm}\subset\operatorname{Jac}(\widetilde{G})$
to $\mathbb{Z}/2\mathbb{Z}$:

######  Proposition 2.7.

Let
$D_{1},D_{2}\in\operatorname{Ker}\operatorname{Nm}\subset\operatorname{Div}^{0}(\widetilde{G})$.

  1. (1)

$\epsilon(D_{1}+D_{2})=\epsilon(D_{1})+\epsilon(D_{2})$.

  2. (2)

If $D_{1}\simeq D_{2}$ then $\epsilon(D_{1})=\epsilon(D_{2})$.

###### Proof.

Suppose that $p:\widetilde{G}\to G$ is defined by a spanning tree $T\subset G$
and a nonempty subset $S\subset E(G)\backslash E(T)$, as in Construction A.
Every divisor
$D\in\operatorname{Ker}\operatorname{Nm}\subset\operatorname{Div}(\widetilde{G})$
is of the form

$D=\sum_{v\in
V(G)}(a_{\widetilde{v}^{+}}\widetilde{v}^{+}+a_{\widetilde{v}^{-}}\widetilde{v}^{-}),$

where $a_{\widetilde{v}^{+}}+a_{\widetilde{v}^{-}}=0$ for each $v\in V(G)$. It
follows that if $D=E-\iota(E)$ then $\deg E=\sum|a_{\widetilde{v}^{+}}|$,
hence

$\epsilon(D)=\sum|a_{\widetilde{v}^{+}}|\bmod 2=\sum
a_{\widetilde{v}^{+}}\bmod 2,$

which is clearly preserved by addition.

To complete the proof, we need to show that any principal divisor in
$\operatorname{Ker}\operatorname{Nm}\subset\operatorname{Div}(\widetilde{G})$
is even. Consider an arbitrary principal divisor

$D=\sum_{v\in
V(G)}(c_{\widetilde{v}^{+}}D_{\widetilde{v}^{+}}+c_{\widetilde{v}^{-}}D_{\widetilde{v}^{-}})$

on $\widetilde{G}$. Its norm is
$\operatorname{Nm}(D)=\sum(c_{\widetilde{v}^{+}}+c_{\widetilde{v}^{-}})D_{v}\in\operatorname{Div}(G)$,
which is the trivial divisor if and only if
$c_{\widetilde{v}^{+}}+c_{\widetilde{v}^{-}}=c$ for a fixed $c\in\mathbb{Z}$
and for all $v\in V(G)$. Therefore, if $\operatorname{Nm}(D)=0$ in
$\operatorname{Div}(G)$, then setting
$a_{v}=c_{\widetilde{v}^{+}}-c=-c_{\widetilde{v}^{-}}$ we see that

$D=cD^{+}+\sum_{v\in V(G)}a_{v}(D_{\widetilde{v}^{+}}-D_{\widetilde{v}^{-}}),$

where $D^{+}$ the principal divisor obtained by firing each vertex
$\widetilde{v}^{+}$ of the top sheet once, and $a_{v}\in\mathbb{Z}$. We now
show that each summand above is even, so $D$ is even as well by the first part
of the proof.

First, we consider divisors of the form
$D_{\widetilde{v}^{+}}-D_{\widetilde{v}^{-}}$ for $v\in V(G)$. Suppose that
the double cover $p$ is described by Construction A. For any vertex $u\in
V(G)$, denote $a_{u}$ and $b_{u}$ the number of edges between $u$ and $v$ in
$E(G)\backslash S$ and $S$ respectively. Then

$(D_{\widetilde{v}^{+}}-D_{\widetilde{v}^{-}})(\widetilde{u}^{\pm})=\pm(a_{u}-b_{u}),$

and

$(D_{\widetilde{v}^{+}}-D_{\widetilde{v}^{-}})(\widetilde{v}^{\pm})=-\sum_{u\neq
v}\mp{}(a_{u}+b_{u}).$

It follows that the contribution from each vertex $u$ to the positive part of
$D_{\widetilde{v}^{+}}-D_{\widetilde{v}^{-}}$ is
$|a_{u}-b_{u}|+a_{u}+b_{u}=\max(2a_{u},2b_{u})$, which is even.

As for $D^{+}$, a direct calculation shows that

$D^{+}=\sum_{v\in V(G)}D_{\widetilde{v}^{+}}=\sum_{e\in
S}\left(\widetilde{s(e)}^{-}+\widetilde{t(e)}^{-}-\widetilde{s(e)}^{+}-\widetilde{t(e)}^{+}\right),$

hence $D^{+}$ is even, and the proof is complete.

∎

######  Definition 2.8.

The Prym group
$\operatorname{Prym}(\widetilde{G}/G)\subset\operatorname{Jac}(\widetilde{G})$
of a free double cover $p:\widetilde{G}\to G$ is the subgroup of even divisors
in $\operatorname{Ker}\operatorname{Nm}$.

It is clear that the order of the Prym group is equal to

$|\operatorname{Prym}(\widetilde{G}/G)|=\frac{1}{2}|\operatorname{Ker}\operatorname{Nm}|=\frac{|\operatorname{Jac}(\widetilde{G})|}{2|\operatorname{Jac}(G)|},$

and one of the principal results of our paper is a combinatorial formula (14)
for $|\operatorname{Prym}(\widetilde{G}/G)|$. For now, we illustrate with an
example.

$v_{1}$$v_{2}$$v_{3}$$v_{4}$$e_{1}$$e_{3}$$e_{4}$$e_{5}$$e_{2}$$\widetilde{v}_{1}^{-}$$\widetilde{v}_{2}^{-}$$\widetilde{v}_{3}^{-}$$\widetilde{v}_{4}^{-}$$\widetilde{v}_{1}^{+}$$\widetilde{v}_{2}^{+}$$\widetilde{v}_{3}^{+}$$\widetilde{v}_{4}^{+}$
Figure 1. An example of a free double cover

######  Example 2.9.

Consider the free double cover $p:\widetilde{G}\to G$ shown in Fig. 1. In
terms of Construction A, we can describe it by choosing $T\subset G$ to be the
tree containing $e_{3}$, $e_{4}$, and $e_{5}$, and setting
$S=\\{e_{1},e_{2}\\}$. Using Kirchhoff’s theorem, we find that
$|\operatorname{Jac}(G^{\prime})|=64$ and $|\operatorname{Jac}(G)|=4$,
therefore $\operatorname{Ker}\operatorname{Nm}$ and
$\operatorname{Prym}(\widetilde{G}/G)$ have orders $16$ and $8$, respectively.
The group $\operatorname{Ker}\operatorname{Nm}$ is spanned by the divisors
$D_{i}=\widetilde{v}_{i}^{+}-\widetilde{v}_{i}^{-}$, where $i=1,2,3,4$, and an
exhaustive calculation using Dhar’s burning algorithm gives a complete set of
relations on the $D_{i}$:

$2D_{1}=0,\quad 8D_{2}=0,\quad D_{4}=D_{1}+4D_{2},\quad D_{3}=3D_{2}.$

It follows that
$\operatorname{Ker}\operatorname{Nm}\simeq\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/8\mathbb{Z}$
with generators $D_{1}$ and $D_{2}$, and hence
$\operatorname{Prym}(\widetilde{G}/G)\simeq\mathbb{Z}/8\mathbb{Z}$ with
generator $D_{1}+D_{2}$.

We note that the Abel–Prym map
$\widetilde{G}=\operatorname{Sym}^{1}(\widetilde{G})\to\operatorname{Prym}^{1}(\widetilde{G}/G)$
sending $\widetilde{v}^{\pm}_{i}$ to $\pm D_{i}$ is not surjective: both sets
have eight elements, but the images of $\widetilde{v}^{\pm}_{1}$ are equal, as
well as those of $\widetilde{v}^{\pm}_{4}$.

###  2.6. Prym varieties

Finally, we recall the definition of the Prym variety of a free double cover
$\pi:\widetilde{\Gamma}\to\Gamma$ of metric graphs (see [JL18] and [LU19]). As
in the finite case, the cover $\pi$ induces a surjective _norm_ map

$\operatorname{Nm}:\operatorname{Pic}^{0}(\widetilde{\Gamma})\to\operatorname{Pic}^{0}(\Gamma),\quad\operatorname{Nm}\left(\sum
a_{i}\widetilde{p}_{i}\right)=\sum a_{i}\pi(\widetilde{p}_{i}),$

and a corresponding involution
$\iota:\operatorname{Pic}^{0}(\widetilde{\Gamma})\to\operatorname{Pic}^{0}(\widetilde{\Gamma})$.

The kernel $\operatorname{Ker}\operatorname{Nm}$ consists of divisors having a
representative of the form $E-\iota(E)$ for some effective divisor $E$ on
$\widetilde{\Gamma}$. Indeed, suppose that $\widetilde{D}$ is a divisor on
$\widetilde{\Gamma}$ such that $\operatorname{Nm}(\widetilde{D})\simeq 0$.
Then $\operatorname{Nm}(\widetilde{D})+\operatorname{div}{f}=0$ for some
piecewise linear function $f$. Defining $\tilde{f}(x)=f(\pi(x))$, we see that
$\widetilde{D}$ is equivalent to a divisor whose pushforward is the zero
divisor on the nose. Furthermore, the parity of $E$ is well-defined, and
$\operatorname{Ker}\operatorname{Nm}$ has two connected components
corresponding to the parity of $E$ [JL18, Proposition 6.1] (note that, in the
more general case when $\pi$ is a dilated unramified double cover,
$\operatorname{Ker}\operatorname{Nm}$ has only one connected component).

######  Definition 2.10.

The _Prym variety_
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)\subset\operatorname{Pic}^{0}(\widetilde{\Gamma})$
of the free double cover $\pi:\widetilde{\Gamma}\to\Gamma$ of metric graphs is
the connected component of the identity of
$\operatorname{Ker}\operatorname{Nm}$.

The Prym variety $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ has the
structure of a tropical ppav, which we now describe. Denote
$\widetilde{\Lambda}=\Omega(\widetilde{\Gamma},\mathbb{Z})$,
$\widetilde{\Lambda}^{\prime}=H_{1}(\widetilde{\Gamma},\mathbb{Z})$,
$\Lambda=\Omega(\Gamma,\mathbb{Z})$ and
$\Lambda^{\prime}=H_{1}(\Gamma,\mathbb{Z})$. Choose an oriented model
$p:\widetilde{G}\to G$ for $\pi$, and consider the pushforward and pullback
maps $\pi_{*}:H_{1}(\widetilde{\Gamma},\mathbb{Z})\to
H_{1}(\Gamma,\mathbb{Z})$ and
$\pi^{*}:\Omega(\Gamma,\mathbb{Z})\to\Omega(\widetilde{\Gamma},\mathbb{Z})$
defined by

$\pi_{*}\left[\sum_{\widetilde{e}\in
E(\widetilde{G})}a_{\widetilde{e}}\widetilde{e}\right]=\sum_{\widetilde{e}\in
E(\widetilde{G})}a_{\widetilde{e}}\pi(\widetilde{e}),\quad\pi^{*}\left[\sum_{e\in
E(G)}a_{e}de\right]=\sum_{e\in
E(G)}a_{e}(d\widetilde{e}^{+}+d\widetilde{e}^{-}).$

It is easy to verify that the maps $\pi_{*}$ and $\pi^{*}$ satisfy Equation
(3) with respect to the integration pairings on $\widetilde{\Gamma}$ and
$\Gamma$. Therefore the pair $(\pi_{*},\pi^{*})$ defines a homomorphism
$\operatorname{Nm}:\operatorname{Jac}(\widetilde{\Gamma})\to\operatorname{Jac}(\Gamma)$
of real tori with integral structure (by Proposition 2.2.3 in [LU19], this
homomorphism is identified with the norm homomorphism
$\operatorname{Nm}:\operatorname{Pic}^{0}(\widetilde{\Gamma})\to\operatorname{Pic}^{0}(\Gamma)$
under the Abel–Jacobi isomorphism, justifying the notation). Hence
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ is in fact the real torus
with integral structure
$(\operatorname{Ker}\operatorname{Nm})_{0}=(K,K^{\prime},[\cdot,\cdot]_{K})$,
where $K=(\operatorname{Coker}\pi^{*})^{tf}$,
$K^{\prime}=\operatorname{Ker}\pi_{*}$, and $[\cdot,\cdot]_{K}$ is the pairing
induced by the integration pairing on $\widetilde{\Gamma}$. Alternatively, we
can describe $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ as the quotient

$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)=\frac{\operatorname{Ker}\overline{\pi}:\Omega^{*}(\widetilde{\Gamma})\to\Omega^{*}(\Gamma)}{\operatorname{Ker}\pi_{*}:H_{1}(\widetilde{\Gamma},\mathbb{Z})\to
H^{1}(\Gamma,\mathbb{Z})},$

where $\overline{\pi}$ is the map dual to $\pi^{*}$.

The polarization
$\widetilde{\xi}:H_{1}(\widetilde{\Gamma},\mathbb{Z})\to\Omega(\widetilde{\Gamma},\mathbb{Z})$
on $\operatorname{Jac}(\widetilde{\Gamma})$ induces a polarization
$i^{*}\widetilde{\xi}:K^{\prime}\to K$ on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, and Theorem 2.2.7 in [LU19]
states that there exists a _principal_ polarization $\psi:K^{\prime}\to K$ on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ such that
$i^{*}\widetilde{\xi}=2\psi$. Hence
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ is a tropical ppav. We note
that the inner product $(\cdot,\cdot)_{P}$ on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ induced by the principal
polarization $\psi$ is half of the restriction of the inner product
$(\cdot,\cdot)_{\widetilde{\Gamma}}$ from
$\operatorname{Jac}(\widetilde{\Gamma})$. In other words for
$\gamma,\delta\in\operatorname{Ker}\pi_{*}$ we have

$(\gamma,\delta)_{P}=[\gamma,\psi(\delta)]=\frac{1}{2}[\gamma,\widetilde{\xi}(\delta)]=\frac{1}{2}(\gamma,\delta)_{\widetilde{\Gamma}}=\frac{1}{2}\sum_{\widetilde{e}\in
E(\widetilde{\Gamma})}\gamma_{\widetilde{e}}\delta_{\widetilde{e}}\ell(\widetilde{e}),\quad\gamma=\sum_{\widetilde{e}\in
E(\widetilde{\Gamma})}\gamma_{\widetilde{e}}\widetilde{e},\quad\delta=\sum_{\widetilde{e}\in
E(\widetilde{\Gamma})}\delta_{\widetilde{e}}\widetilde{e},$ (8)

and similarly for the induced product on $\operatorname{Ker}\overline{\pi}$.
When discussing the metric properties of
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, such as its volume, we
always employ the inner product $(\cdot,\cdot)_{P}$ induced by the principal
polarization.

We use a set of explicit coordinates on the torus
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, or more accurately on its
universal cover $\operatorname{Ker}\overline{\pi}$. Choose a basis

$\widetilde{\gamma}_{j}=\sum_{\widetilde{e}\in
E(\widetilde{G})}\widetilde{\gamma}_{j,\widetilde{e}}\widetilde{e},\quad
j=1,\ldots,g-1$

for $\operatorname{Ker}\pi_{*}:H_{1}(\widetilde{\Gamma},\mathbb{Z})\to
H_{1}(\Gamma,\mathbb{Z})$. The principal polarization
$\psi=\frac{1}{2}\widetilde{\xi}$ gives a corresponding basis of the second
lattice $(\operatorname{Coker}\pi^{*})^{tf}$:

$\omega_{j}=\psi(\widetilde{\gamma}_{j})=\frac{1}{2}\sum_{\widetilde{e}\in
E(\widetilde{G})}\widetilde{\gamma}_{j,\widetilde{e}}d\widetilde{e},\quad
j=1,\ldots,g-1.$

Let $\omega^{*}_{j}$ denote the basis of
$\operatorname{Ker}\overline{\pi}:\Omega^{*}(\widetilde{\Gamma})\to\Omega^{*}(\Gamma)$
dual to the $\omega_{j}$, so that $\omega^{*}_{j}(\omega_{k})=\delta_{jk}$,
then elements of $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ can be given
(locally uniquely) as linear combinations of the $\omega^{*}_{j}$.

We compute for future reference the volume of the unit cube
$C(\omega^{*}_{1},\ldots,\omega^{*}_{g-1})$ in the coordinate system defined
by the $\omega^{*}_{j}$. We know that
$\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\sqrt{\det
G}$, where $G_{ij}=(\widetilde{\gamma}_{i},\widetilde{\gamma}_{j})_{P}$ is the
Gramian matrix of the basis $\widetilde{\gamma}_{j}$. The
$\widetilde{\gamma}_{j}$, viewed as elements of
$\operatorname{Ker}\overline{\pi}$, are themselves a basis, so we can write
$\omega^{*}_{i}=\sum_{j}A_{ij}\widetilde{\gamma}_{j}$ for some matrix
$A_{ij}$. Pairing with $\omega_{j}$ and using that
$[\widetilde{\gamma}_{i},\omega_{j}]=G_{ij}$, we see that $A$ is in fact the
inverse matrix of $G$. Hence we see that

$\operatorname{Vol}(C(\omega^{*}_{1},\ldots,\omega^{*}_{g-1}))=\det(\omega^{*}_{i},\omega^{*}_{j})=\det
G^{-1}\det(\widetilde{\gamma}_{i},\widetilde{\gamma}_{j})\det
G^{-1}=\frac{1}{\det
G}=\frac{1}{\operatorname{Vol}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))}.$
(9)

In particular, this volume does not depend on the choice of basis
$\widetilde{\gamma}_{j}$.

######  Remark 2.11.

The definition of the Prym group for a free double cover of finite graphs is
consistent with the definition for metric graphs in the following sense. Let
$p:\widetilde{G}\to G$ be a free double cover of finite graphs, and let
$\pi:\widetilde{\Gamma}\to\Gamma$ be the corresponding double cover of metric
graphs, where $\widetilde{\Gamma}$ and $\Gamma$ are obtained from respectively
$\widetilde{G}$ and $G$ by setting all edge lengths to $1$. Then
$\operatorname{Jac}(\widetilde{G})$ is naturally a subgroup of
$\operatorname{Jac}(\widetilde{\Gamma})$, consisting of divisors supported at
the vertices, and
$\operatorname{Prym}(\widetilde{G}/G)=\operatorname{Jac}(\widetilde{G})\cap\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$.

###  2.7. Polyhedral spaces and harmonic morphisms

The spaces $\operatorname{Sym}^{d}(\Gamma)$, $\operatorname{Jac}(\Gamma)$, and
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ are examples of _rational
polyhedral spaces_ , which are topological spaces locally modeled on rational
polyhedral sets in $\mathbb{R}^{n}$. A rational polyhedral space comes
equipped with a structure sheaf, pulled back from the sheaf of affine
$\mathbb{Z}$-linear functions on the embedded polyhedra. We shall not require
the general theory of rational polyhedral spaces, in particular we shall use
only the polyhedral decomposition and not the sheaf of affine functions. See
[MZ14], [GS19] for details.

A rational polyhedral space $P$ is a finite union of polyhedra, which we call
_cells_. We only consider compact polyhedral spaces. The intersection of any
two cells is either empty or a face of each. A polyhedral space $P$ is
_equidimensional of dimension $n$_ if each maximal cell of $P$ (with respect
to inclusion) has dimension $n$, and is _connected through codimension one_ if
the complement in $P$ of all cells of codimension two is connected. A map
$f:P\to Q$ of polyhedral spaces is locally given by affine $\mathbb{Z}$-linear
transformations, and is required to map each cell of $P$ surjectively onto a
cell of $Q$. We say that $f$ _contracts_ a cell $C$ of $P$ if
$\dim(f(C))<\dim(C)$.

We use an ad hoc definition of harmonic morphisms of polyhedral spaces,
modelled on the corresponding definition for metric graphs.

######  Definition 2.12.

[cf. Definition 2.5 in [LR18]] Let $f:P\to Q$ be a map of equidimensional
polyhedral spaces of the same dimension, and let $\deg_{f}$ be a non-negative
integer-valued function defined on the top-dimensional cells of $P$. Let $C$
be a codimension 1 cell of $P$ mapping surjectively onto a codimension one
cell $D$ of $Q$. We say that $f$ is _harmonic_ at $C$ (with respect to the
degree function $\deg_{f}$) if the following condition holds: for any
codimension zero cell $N$ of $Q$ adjacent to $D$, the sum

$\deg_{f}(C)=\sum_{M\subset f^{-1}(N),\,M\supset C}\deg_{f}(M)$ (10)

of the degrees $\deg_{f}M$ over all codimension zero cells $M$ of $P$ adjacent
to $C$ and mapping to $N$ is the same, in other words does not depend on the
choice of $N$. We say that $f$ is _harmonic_ if $f$ is harmonic at every
codimension one cell of $P$, and in addition if $f(C)=0$ on a codimension zero
cell $C$ if and only if $f$ contracts $C$.

Given a harmonic morphism $f:P\to Q$, Equation (10) extends the degree
function $\deg_{f}$ to codimension one cells of $P$. If $Q$ is connected
through codimension one, we can similarly define the degree on cells of any
codimension, and hence on all of $P$ (note, however, that for a cell $C$ of
positive codimension, $\deg_{f}(C)=0$ does not imply that $C$ is contracted).
The function $\deg_{f}$ is locally constant in fibers: given $p\in P$ and an
open neighborhood $V\ni f(p)$, there exists an open neighborhood $U\ni p$ such
that $f(U)\subset V$, and such that for any $q\in V$ the sum of the degrees
over all points of $f^{-1}(q)\cap U$ is the same (in particular, this sum is
finite). It follows that a harmonic morphism to a target connected through
codimension one is surjective, and has a well-defined _global degree_ , which
is the sum of the degrees over all points of any fiber.

The structure of a rational polyhedral space on a tropical abelian variety,
such as $\operatorname{Jac}(\Gamma)$ and
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, is induced by the integral
structure: an affine linear function is $\mathbb{Z}$-linear if it has integer
slopes with respect to the embedded lattice. The correct definition of a
rational polyhedral structure on $\operatorname{Sym}^{d}(\Gamma)$, however,
requires some care. The space $\operatorname{Sym}^{d}(\Gamma)$ has a natural
cellular decomposition, with top-dimensional cells $C(F)$ indexed by
$d$-tuples $F\subset E(G)$ of edges of a suitably chosen model $G$ of $\Gamma$
(see Eq. (21)). If the $d$-tuple $F=\\{e_{1},\ldots,e_{d}\\}$ contains neither
loops nor repeated edges, then $C(F)$ is the parallelotope obtained by taking
the Cartesian product of the $e_{i}$ inside $\mathbb{R}^{d}$. If $F$ contains
loop edges, then the corresponding cell will have self-gluings along some of
its boundary cells. A more serious issue arises if $F$ has repeated edges; in
this case the cell $C(F)$ is a quotient of a parallelotope by a finite
permutation action, which requires introducing additional cells at the
appropriate diagonals.

A complete description of the polyhedral structure on
$\operatorname{Sym}^{d}(\Gamma)$ is given in the paper [BU18]. However,
neither of the aforementioned issues arise in our paper. First, we can always
choose a loopless model for any tropical curve. Second, and more importantly,
all the calculations in this paper concern only those cells $C(F)$ for which
$F$ has no repeated edges. Specifically, we are interested in the structure of
the Abel–Prym map
$\Psi^{d}:\operatorname{Sym}(\widetilde{\Gamma})\to\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$
associated to a double cover $\widetilde{\Gamma}\to\Gamma$, and one of our
first results (Thm. 4.1 part (1)) states that $\Psi^{d}$ contracts all those
cells $C(F)$ where $F$ has repeated edges. Hence, all top-dimensional cells of
the symmetric product $\operatorname{Sym}^{d}(\Gamma)$ can be assumed to be
parallelotopes.

##  3\. Kirchhoff’s theorem for the Prym group and the Prym variety

In this section, we give combinatorial formulas for the order (14) of the Prym
group of a free double cover $p:\widetilde{G}\to G$ of finite graphs, and the
volume (17) of the Prym variety of a free double cover
$\pi:\widetilde{\Gamma}\to\Gamma$ of metric graphs.

Formula (14) had already been obtained by Zaslavsky (see Theorem 8A.4 in
[Zas82]). Specifically, a free double cover $p:\widetilde{G}\to G$ induces the
structure of a _signed graph_ on $G$: defining the cover $p$ in terms of
Construction A with respect to a spanning tree $T\subset G$, we attach a
negative sign $-$ to each edge $e\in S\subset E(G)\backslash E(T)$ and a
positive sign $+$ to all other edges. Zaslavsky then defines the _signed
Laplacian matrix_ of $G$ and shows that its determinant is given by (14) (note
that the signed Laplacian is non-singular, unlike the ordinary Laplacian).
Reiner and Tseng specifically interpret the determinant of the signed
Laplacian as the order of the Prym group
$\operatorname{Prym}(\widetilde{G}/G)$ (see Proposition 9.9 in [RT14]).

We give an alternative proof of (14) using the Ihara zeta function. Given a
free double cover $p:\widetilde{G}\to G$, the orders of
$\operatorname{Jac}(\widetilde{G})$ and $\operatorname{Jac}(G)$ can be
computed from the corresponding zeta functions $\zeta(\widetilde{G},s)$ and
$\zeta(G,s)$ using Northshield’s class number formula [Nor98]. Hence the ratio
$|\operatorname{Prym}(\widetilde{G}/G)|=|\operatorname{Jac}(\widetilde{G})|/2|\operatorname{Jac}(G)|$
is given by the ratio of the zeta functions. This is equal to the Artin–Ihara
$L$-function of the cover and can be explicitly computed from the
corresponding determinantal formula, derived by Stark and Terras (see [ST96]
and [ST00]).

The volume formula (17) is new, to the best of our knowledge, and is derived
from (14) by a scaling argument.

###  3.1. The Ihara zeta function and the Artin–Ihara $L$-function

The Ihara zeta function $\zeta(s,G)$ of a finite graph $G$ is the graph-
theoretic analogue of the Dedekind zeta function of a number field and is
defined as an Euler product over certain equivalence classes of closed paths
on $G$. We recall its definition and properties (see [Ter10] for an elementary
treatment).

Let $G$ be a graph with $n=\\#(V(G))$ vertices and $m=\\#(E(G))$ edges. A
_path_ $P$ of _length_ $k=\ell(P)$ is a sequence $P=e_{1}\cdots e_{k}$ of
oriented edges of $G$ such that $t(e_{i})=s(e_{i+1})$ for $i=1,\ldots,k-1$. We
say that a path $P$ is _closed_ if $t(e_{k})=s(e_{1})$ and _reduced_ if
$e_{i+1}\neq\overline{e}_{i}$ for $i=1,\ldots,k-1$ and
$e_{1}\neq\overline{e}_{k}$. We can define positive integer powers of closed
paths by concatenation, and a closed reduced path $P$ is called _primitive_ if
there does not exist a closed path $Q$ such that $P=Q^{k}$ for some $k\geq 2$.
We consider two reduced paths to be _equivalent_ if they differ by a choice of
starting point, i.e. we set $e_{1}\cdots e_{k}\sim e_{j}\cdots e_{k}\cdot
e_{1}\cdots e_{j-1}$ for all $j=1,\ldots k$. A _prime_ $\mathfrak{p}$ of $G$
is an equivalence class of primitive paths, and has a well-defined length
$\ell(\mathfrak{p})$. We note that a primitive path and the same path
traversed in the opposite direction represent distinct primes.

The _Ihara zeta function_ $\zeta(s,G)$ of a graph $G$ is the product

$\zeta(s,G)=\prod_{\mathfrak{p}}(1-s^{\ell(\mathfrak{p})})^{-1}$

over all primes $\mathfrak{p}$ of $G$, where $s$ is a complex variable. This
product is usually infinite, converges for sufficiently small $s$, and extends
to rational function.

The _three-term determinant formula_ , due to Bass [Bas92] (see also [Ter10]),
expresses the reciprocal $\zeta(s,G)^{-1}$ as an explicit polynomial

$\zeta(s,G)^{-1}=(1-s^{2})^{g-1}\det(I_{n}-As+(Q-I_{n})s^{2}),$

where $Q$ and $A$ are the valency and adjacency matrices (see (2)), and
$g=m-n+1$ is the genus of $G$. It is clear from this formula that
$\zeta(s,G)^{-1}$ vanishes at $s=1$ to order at least $g$, because for $s=1$
the matrix inside the determinant is equal to the Laplacian $L$ of $G$ and
$\det L=0$. In fact, the order of vanishing is equal to $g$, and Northshield
[Nor98] shows that the leading Taylor coefficient computes the complexity,
i.e. the order of the Jacobian of $G$:

$\zeta(s,G)^{-1}=(-1)^{g-1}2^{g}(g-1)|\operatorname{Jac}(G)|(s-1)^{g}+O((s-1)^{g+1}).$
(11)

This result may be viewed as a graph-theoretic analogue of the class number
formula.

The analogy with number theory was further reinforced by Stark and Terras, who
developed (see [ST96] and [ST00]) a theory of $L$-functions of Galois covers
of graphs, as follows. Let $p:\widetilde{G}\to G$ be a free Galois cover of
graphs with Galois group $K$ (we do not define these, since we only consider
free double covers, which are Galois covers with $K=\mathbb{Z}/2\mathbb{Z}$),
and fix a representation $\rho$ of $K$. Given a prime $\mathfrak{p}$ of $G$,
choose a representative $P$ with starting vertex $v\in V(G)$, and choose a
vertex $\widetilde{v}\in V(\widetilde{G})$ lying over $v$. The path $P$ lifts
to a unique path $\widetilde{P}$ in $\widetilde{G}$ starting at
$\widetilde{v}$ and mapping to $P$, and the terminal vertex of $\widetilde{P}$
also maps to $v$. The _Frobenius element_ $F(P,\widetilde{G}/G)\in K$ is the
unique element of the Galois group mapping $\widetilde{v}$ to the terminal
vertex of $\widetilde{P}$. The _Artin–Ihara $L$-function_ is now defined as
the product

$L(s,\rho,\widetilde{G}/G)=\prod_{\mathfrak{p}}\det(1-\rho(F(P,\widetilde{G}/G))s^{\ell(\mathfrak{p})})^{-1}$

taken over the primes $\mathfrak{p}$ of $G$, where for each prime
$\mathfrak{p}$ we pick an arbitrary representative $P$ (Frobenius elements
corresponding to different representatives of $\mathfrak{p}$ are conjugate, so
the determinant is well-defined).

Similarly to the zeta function, the product defining the $L$-function
converges to a rational function and is given by a determinant formula. Pick a
spanning tree $T\subset G$ and index its preimages in $\widetilde{G}$, called
the _sheets_ of the cover, by the elements of $K$. Given an edge $e\in E(G)$,
the _Frobenius element_ $F(e)\in K$ is equal to $h^{-1}g$, where $h$ and $g$
are respectively the indices of the sheets of the source and the target of
$e$. Let $d$ be the degree of $\rho$, and define the $nd\times nd$ _Artinized
valency_ and _Artinized adjacency_ matrices as

$Q_{\rho}=Q\otimes I_{d},\quad(A_{\rho})_{uv}=\sum\rho(F(e)),$

where in the right hand side we sum over all edges $e$ between $u$ and $v$.
The three-term determinant formula for the $L$-function states that

$L(s,\rho,\widetilde{G}/G)^{-1}=(1-s^{2})^{(g-1)d}\det(I_{nd}-A_{\rho}s+(Q_{\rho}-I_{nd})s^{2}).$
(12)

Finally, we relate the zeta and $L$-functions associated to a free Galois
cover $p:\widetilde{G}\to G$ with Galois group $K$. First of all, the zeta
functions of $\widetilde{G}$ and $G$ are equal to the $L$-function evaluated
at respectively the right regular and trivial representations $\rho_{K}$ and
$1_{K}$:

$\zeta(s,\widetilde{G})=L(s,\rho_{K},\widetilde{G}/G),\quad\zeta(s,G)=L(s,1_{K},\widetilde{G}/G).$

Furthermore, for a reducible representation $\rho=\rho_{1}\oplus\rho_{2}$ the
$L$-function factors as

$L(s,\rho,\widetilde{G}/G)=L(s,\rho_{1},\widetilde{G}/G)L(s,\rho_{2},\widetilde{G}/G).$

It follows that the zeta function of $\widetilde{G}$ has a factorization

$\zeta(s,\widetilde{G})=\zeta(s,G)\prod_{\rho}L(s,\rho,\widetilde{G}/G)^{d(\rho)},$
(13)

where the product is taken over the distinct nontrivial irreducible
representations of $K$.

###  3.2. The order of the Prym group

We now specialize to the case where $K=\mathbb{Z}/2\mathbb{Z}$ in order to
compute the order of the Prym group of a free double cover $p:\widetilde{G}\to
G$ of finite graphs. By (11), the leading Taylor coefficients of the zeta
functions $\zeta(s,\widetilde{G})^{-1}$ and $\zeta(s,G)^{-1}$ at $s=1$ compute
respectively the orders $|\operatorname{Jac}(\widetilde{G})|$ and
$|\operatorname{Jac}(G)|$. Since $\zeta^{-1}(s,\widetilde{G})$ is the product
of $\zeta^{-1}(s,G)$ with the inverse of the $L$-function evaluated at the
nontrivial representation of $\mathbb{Z}/2\mathbb{Z}$, the leading Taylor
coefficient of the latter computes the order of the Prym.

By the results of [ABKS14], the Jacobian group of a graph $G$ of genus $g$
(and, by extension, the Jacobian variety of a metric graph) admits a non-
canonical combinatorial description in terms of certain $g$-element subsets of
$E(G)$, specifically the complements of spanning trees. We now give an
analogous definition for $(g-1)$-element subsets of $E(G)$, which, as we shall
see, enumerate the elements of $\operatorname{Prym}(\widetilde{G}/G)$, and
control the geometry of the Prym varieties of double covers of metric graphs.

######  Definition 3.1.

Let $G$ be a graph of genus $g$, and let $p:\widetilde{G}\to G$ be a connected
free double cover. A subset $F\subset E(G)$ of $g-1$ edges of $G$ is called a
_genus one decomposition of rank_ $r$ if the graph $G\backslash
F=G_{0}\cup\cdots\cup G_{r-1}$ has $r$ connected components, each of which has
genus one. We say that a genus one decomposition $F$ is _odd_ if the preimage
of each $G_{k}$ in $\widetilde{G}$ is connected.

We note that when removing edges from a graph we never remove vertices, even
isolated ones. A simple counting argument shows that if $F\subset E(G)$ is a
subset such that each connected component of $G\backslash F$ has genus one,
then $F$ consists of $g-1$ edges, and a genus one decomposition cannot have
rank greater than $g$.

A genus one graph has two free double covers: the disconnected trivial cover
and a unique nontrivial connected cover. Hence we can equivalently require
that the restriction of the cover $p$ to each $G_{k}$ is a nontrivial free
double cover. If the cover $p$ is described by Construction A with respect to
a choice of spanning tree $T\subset G$ and a nonempty subset $S\subset
E(G)\backslash E(T)$, then a genus one decomposition $F\subset E(G)$ is odd if
and only if each $G_{k}$ has an odd number of edges from $S$ on its unique
cycle (see Remark 2.2).

######  Theorem 3.2.

Let $G$ be a graph of genus $g$, and let $p:\widetilde{G}\to G$ be the
connected free double cover determined by $T\subset G$ and $S\subset
E(G)\backslash E(T)$. The order of the Prym group
$\operatorname{Prym}(\widetilde{G}/G)$ is equal to

$|\operatorname{Prym}(\widetilde{G}/G)|=\frac{1}{2}|\operatorname{Ker}\operatorname{Nm}|=\sum_{r=1}^{g}4^{r-1}C_{r},$
(14)

where $C_{r}$ is the number of odd genus one decompositions of $G$ of rank
$r$.

###### Proof.

Denote $n=|V(G)|$ and $m=|E(G)|=n+g-1$. According to (13), the zeta function
of $\widetilde{G}$ is the product of the zeta function of $G$ and the
$L$-function of the cover $\widetilde{G}/G$ evaluated at the nontrivial
representation $\rho$ of the Galois group $\mathbb{Z}/2\mathbb{Z}$:

$\zeta(s,\widetilde{G})^{-1}=\zeta(s,G)^{-1}L(s,\widetilde{G}/G,\rho)^{-1}.$

The class number formula (11) gives the leading Taylor coefficients at $s=1$:

$\zeta(s,\widetilde{G})^{-1}=2^{2g-1}(2g-2)|\operatorname{Jac}(\widetilde{G})|(s-1)^{2g-1}+O\left((s-1)^{2g}\right),$
$\zeta(s,G)^{-1}=(-1)^{g-1}2^{g}(g-1)|\operatorname{Jac}(G)|(s-1)^{g}+O\left((s-1)^{g+1}\right).$

The leading coefficient of the $L$-function is found directly from (12) (note
that, unlike in formula (11), the determinant does not vanish at $s=1$):

$L(s,\rho,\widetilde{G}/G)^{-1}=(-1)^{g-1}2^{g-1}\det(Q_{\rho}-A_{\rho})(s-1)^{g-1}+O\left((s-1)^{g}\right).$

Therefore, comparing the expansions of $L(s,\rho,\widetilde{G}/G)^{-1}$ with
$\zeta(s,\widetilde{G})^{-1}/\zeta(s,G)^{-1}$, we see that

$|\operatorname{Prym}(\widetilde{G}/G)|=\frac{|\operatorname{Jac}(\widetilde{G})|}{2|\operatorname{Jac}(G)|}=\frac{1}{4}\det(Q_{\rho}-A_{\rho}).$

We now calculate this $n\times n$ determinant. First of all, $Q_{\rho}=Q$
since $\rho$ is one-dimensional. The Frobenius element $F(e)$ of an edge $e\in
E(G)$ is the nontrivial element of $\mathbb{Z}/2\mathbb{Z}$, and hence
$\rho(F(e))=-1$, if and only if $e\in S$. Putting this together, we see that
the matrix $Q_{\rho}-A_{\rho}$ has the following form:

$(Q_{\rho}-A_{\rho})_{uv}=\left\\{\begin{array}[]{cc}|\\{\text{edges from $u$
to $v$ in $S$}\\}|-|\\{\text{edges from $u$ to $v$ not in $S$}\\}|,&u\neq
v,\\\ 4|\\{\text{loops at $u$ in $S$}\\}|+|\\{\text{non-loops at
$u$}\\}|,&u=v.\end{array}\right.$

The matrix $Q_{\rho}-A_{\rho}$ turns out to be equal to the _signed Laplacian
matrix_ of the graph $G$ (see Proposition 9.5 in [RT14]), and its determinant
is computed using a standard argument involving an appropriate factorization
and the Cauchy–Binet formula. We only give a sketch of these calculations,
since they are not new (see Proposition 9.9 in  _loc. cit._).

Pick an orientation on $G$. We factorize the signed Laplacian as
$Q_{\rho}-A_{\rho}=B_{S}(G){}^{t}B_{S}(G)$, where

$(B_{S}(G))_{ve}=\left\\{\begin{array}[]{cc}1,&\text{$t(e)=v$ and $s(e)\neq
v$, or $s(e)=v$, $t(e)\neq v$, and $e\in S$,}\\\ -1,&\text{$s(e)=v$, $t(e)\neq
v$, and $e\notin S$,}\\\ 2,&\text{$s(e)=t(e)=v$ and $e\in S$,}\\\
0,&\text{otherwise}.\end{array}\right.$

is the $n\times m$ _$S$ -twisted adjacency matrix_ $B_{S}(G)$ of the graph
$G$, whose rows and columns are indexed by respectively $V(G)$ and $E(G)$. By
the Cauchy–Binet formula, we have

$|\operatorname{Prym}(\widetilde{G}/G)|=\frac{1}{4}\det(Q_{\rho}-A_{\rho})=\frac{1}{4}\sum_{F\subset
E(G),\,|F|=g-1}\det B_{S}(G\backslash F)^{2}.$ (15)

Here the sum is taken over all subsets $F$ of $E(G)$ consisting of $m-n=g-1$
elements, and $B_{S}(G\backslash F)$ is the matrix obtained from $B_{S}(G)$ by
deleting the columns corresponding to the edges that are in $F$, or,
equivalently, the $S$-twisted adjacency matrix of the graph $G\backslash F$.

Let $F\subset E(G)$ be such a subset, and let $G\backslash
F=G_{0}\cup\cdots\cup G_{r-1}$ be the decomposition of $G$ into connected
components. The matrix $B_{S}(G\backslash F)$ is block-diagonal, with blocks
$B_{S}(G_{k})$ corresponding to the $G_{k}$. A block-diagonal matrix has
nonzero determinant only if all blocks are square, meaning that $g(G_{k})=1$
for all $k$, in which case

$\det B_{S}(G\backslash F)^{2}=\prod_{k=0}^{r-1}\det B_{S}(G_{k})^{2}.$ (16)

The quantity $\det B_{S}(G_{k})^{2}$ for a genus one graph $G_{k}$ is computed
by induction on the extremal edges (if any), and turns out to be equal to $4$
if the unique cycle of $G_{k}$ has an odd number of edges from $S$, and $0$ if
the number is even. Hence, only odd genus one decompositions contribute to the
sum (15), and the contribution of a decomposition of rank $r$ is equal to
$4^{r}$. This completes the proof.

∎

######  Example 3.3.

Consider the free double cover $p:\widetilde{G}\to G$ shown in Figure 1. Here
$g-1=1$, and it is easy to see that any edge of $G$ is an odd genus one
decomposition. The edges $e_{1}$, $e_{2}$, $e_{4}$, and $e_{5}$ are
decompositions of rank one, while $e_{3}$ is a decomposition of rank two.
Hence by (14)

$|\operatorname{Prym}(\widetilde{G}/G)|=4+1\cdot 4=8,$

which agrees with the calculations in Example 2.9.

###  3.3. The volume of the tropical Prym variety

In this section, we prove a weighted version of Theorem 3.2 that gives the
volume of the Prym variety of a free double cover of metric graphs. Let
$\pi:\widetilde{\Gamma}\to\Gamma$ be such a cover, where $\widetilde{\Gamma}$
and $\Gamma$ have genera $2g-1$ and $g-1$, respectively. Choose a model $G$
for $\Gamma$. Similarly to the discrete case, an _odd genus one decomposition_
$F$ of $\Gamma$ of _rank_ $r(F)$ (with respect to the choice of model $G$) is
a subset $F\subset E(G)$ of (necessarily) $g-1$ edges of $G$ such that
$E(G)\backslash F$ consists of $r(F)$ connected components of genus one, each
having a connected preimage in $\widetilde{\Gamma}$. For such an $F$, we
denote by $w(F)$ the product of the lengths of the edges in $F$.

######  Theorem 3.4.

The volume of the Prym variety of a free double cover
$\pi:\widetilde{\Gamma}\to\Gamma$ of metric graphs is given by

$\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\sum_{F\subset
E(\Gamma)}4^{r(F)-1}w(F),$ (17)

where the sum is taken over all odd genus one decompositions $F$ of $\Gamma$.

######  Remark 3.5.

The right hand side of formula (17) is defined with respect to a choice of
model $G$ for $\Gamma$. Let $G^{\prime}$ be the model obtained from $G$ by
subdividing an edge $e\in E(G)$ into edges $e^{\prime}_{1}$ and
$e^{\prime}_{2}$, so that $\ell(e)=\ell(e^{\prime}_{1})+\ell(e^{\prime}_{2})$.
If $e\in F$ for some odd genus one decomposition $F$ of $G$, then
$(F\backslash\\{e\\})\cup\\{e^{\prime}_{1}\\}$ and
$(F\backslash\\{e\\})\cup\\{e^{\prime}_{2}\\}$ are odd genus one
decompositions of $G^{\prime}$ of the same rank as $F$, and vice versa. It
follows that the right hand side is invariant under edge subdivision, and
hence does not depend on the choice of model for $\Gamma$. We also note that
$\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))$ is
computed with respect to the intrinsic principal polarization on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, which is half of the
restriction of the principal polarization on
$\operatorname{Jac}(\widetilde{\Gamma})$.

We first establish the relationship between the volumes of the three tropical
ppavs $\operatorname{Jac}(\widetilde{\Gamma})$, $\operatorname{Jac}(\Gamma)$,
and $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. To compute the last of
the three volumes, we define (building on Construction A) an explicit basis
for the kernel of the pushforward map
$\pi_{*}:H_{1}(\widetilde{\Gamma},/\mathbb{Z})\to H_{1}(\Gamma,\mathbb{Z})$,
which we also use later.

Let $G$ be a graph. Introduce the $\mathbb{Z}$-valued bilinear pairing

$\langle\cdot,\cdot\rangle:C_{1}(G,\mathbb{Z})\times
C_{1}(G,\mathbb{Z})\to\mathbb{Z},\quad\left\langle\sum_{e\in
E(G)}a_{e}e,\sum_{e\in E(G)}b_{e}e\right\rangle=\sum_{e\in E(G)}a_{e}b_{e}.$
(18)

We note that this pairing does not take edge lengths into account, and is not
to be confused with the integration pairing $(\cdot,\cdot)$ on a metric graph.

######  Construction B.

Let $\pi:\widetilde{\Gamma}\to\Gamma$ be a connected free double cover of
metric graphs. Choose an oriented model $p:\widetilde{G}\to G$, and suppose
that the cover $p$ is given by Construction A with respect to a spanning tree
$T\subset G$ and a nonempty subset $S\subset E(G)\backslash
E(T)=\\{e_{0},\ldots,e_{g-1}\\}$ containing $e_{0}$. In this construction, we
define an explicit basis of the kernel of the pushforward map
$p_{*}:H_{1}(\widetilde{G},\mathbb{Z})\to H_{1}(G,\mathbb{Z})$, as well as
bases for $H_{1}(\widetilde{G},\mathbb{Z})$ and $H_{1}(G,\mathbb{Z})$. We use
these bases to compute Gramian determinants, hence we view them to be
unordered sets.

We first construct a basis for $H_{1}(G,\mathbb{Z})$. Let $\gamma_{i}\in
H_{1}(G,\mathbb{Z})$ for $i=0,\ldots,g-1$ denote the unique cycle of
$T\cup\\{e_{i}\\}$ such that $\langle\gamma_{i},e_{i}\rangle=1$. It is a
standard fact that

$\mathcal{B}=\\{\gamma_{0},\ldots,\gamma_{g-1}\\}$

is a basis of $H_{1}(G,\mathbb{Z})$, and furthermore any $\gamma\in
H_{1}(G,\mathbb{Z})$ can be explicitly decomposed in terms of $\mathcal{B}$ as
follows:

$\gamma=\langle\gamma,e_{1}\rangle\gamma_{1}+\cdots+\langle\gamma,e_{g}\rangle\gamma_{g}.$

Similarly, let $\widetilde{\gamma}_{0}\in H_{1}(\widetilde{G},\mathbb{Z})$ and
$\widetilde{\gamma}_{i}^{\pm}\in H_{1}(\widetilde{G},\mathbb{Z})$ for
$i=1,\ldots,g-1$ denote the unique cycle of respectively
$\widetilde{T}\cup\\{\widetilde{e}^{-}_{0}\\}$ and
$\widetilde{T}\cup\\{\widetilde{e}_{i}^{\pm}\\}$ such that respectively
$\langle\widetilde{\gamma}_{0},\widetilde{e}^{-}_{0}\rangle=1$ and
$\langle\widetilde{\gamma}^{\pm}_{i},\widetilde{e}^{\pm}_{i}\rangle=1$ for
$i=1,\ldots,g-1$. Then

$\widetilde{\mathcal{B}}=\\{\widetilde{\gamma}_{0},\widetilde{\gamma}_{1}^{\pm},\ldots,\widetilde{\gamma}_{g-1}^{\pm}\\}$

is a basis of $H_{1}(\widetilde{G},\mathbb{Z})$, and we similarly have

$\widetilde{\gamma}=\langle\widetilde{\gamma},\widetilde{e}^{-}_{0}\rangle\widetilde{\gamma}_{0}+\langle\widetilde{\gamma},\widetilde{e}^{+}_{1}\rangle\widetilde{\gamma}^{+}_{1}+\cdots+\langle\widetilde{\gamma},\widetilde{e}^{+}_{g-1}\rangle\widetilde{\gamma}^{+}_{g-1}+\langle\widetilde{\gamma},\widetilde{e}^{-}_{1}\rangle\widetilde{\gamma}^{-}_{1}+\cdots+\langle\widetilde{\gamma},\widetilde{e}^{-}_{g-1}\rangle\widetilde{\gamma}^{-}_{g-1}$

for any $\widetilde{\gamma}\in H_{1}(\widetilde{G},\mathbb{Z})$.

We now compute the action of the pushforward map
$p_{*}:H_{1}(\widetilde{G},\mathbb{Z})\to H_{1}(G,\mathbb{Z})$ and the
involution map $\iota_{*}:H_{1}(\widetilde{G},\mathbb{Z})\to
H_{1}(\widetilde{G},\mathbb{Z})$ on the basis $\widetilde{\mathcal{B}}$. The
cycle $\widetilde{\gamma}_{0}$ starts at the vertex
$s(\widetilde{e}_{0}^{-})=\widetilde{s(e_{0})}^{-}$ on the lower sheet
$\widetilde{T}^{-}$, then proceeds via $+\widetilde{e}_{0}^{-}$ to the vertex
$t(\widetilde{e}_{0}^{-})=\widetilde{t(e_{0})}^{+}$ on the upper sheet
$\widetilde{T}^{+}$, then to $\widetilde{s(e_{0})}^{+}$ via a unique path in
$\widetilde{T}^{+}$, then back to
$t(\widetilde{e}_{0}^{+})=\widetilde{t(e_{0})}^{-}$ on $T^{-}$ via
$+\widetilde{e}_{0}^{+}$, and then back to $\widetilde{s(e_{0})}^{-}$ via a
unique path in $\widetilde{T}^{-}$. In other words,

$\widetilde{\gamma}_{0}=\widetilde{e}_{0}^{+}+\widetilde{e}_{0}^{-}+\mbox{edges
of
}\widetilde{T}^{\pm},\quad\iota_{*}(\widetilde{\gamma}_{0})=\widetilde{e}_{0}^{+}+\widetilde{e}_{0}^{-}+\mbox{edges
of }\widetilde{T}^{\pm},\quad p_{*}(\widetilde{\gamma}_{0})=2e_{0}+\mbox{edges
of }T,$

therefore computing the intersection numbers with $\widetilde{\mathcal{B}}$
and $\mathcal{B}$ we see that

$\iota_{*}(\widetilde{\gamma}_{0})=\widetilde{\gamma}_{0},\quad
p_{*}(\widetilde{\gamma}_{0})=2\gamma_{0}.$

Now consider the cycle $\widetilde{\gamma}_{i}^{+}$ for $e_{i}\in
S\backslash\\{e_{0}\\}$. We introduce the index

$\sigma_{i}=\left\\{\begin{array}[]{cc}+1,&s(\widetilde{e}_{i}^{+})=s(e_{i})^{+},\\\
-1,&s(\widetilde{e}_{i}^{+})=s(e_{i})^{-}.\end{array}\right.$

If $\sigma_{i}=1$, then the cycle $\widetilde{\gamma}_{i}^{+}$ starts at
$s(\widetilde{e}_{i}^{+})=\widetilde{s(e_{i})}^{+}$ on $\widetilde{T}^{+}$,
then moves to $t(\widetilde{e}_{i}^{+})=\widetilde{t(e_{i})}^{-}$ on
$\widetilde{T}^{-}$ via $\widetilde{e}_{i}^{+}$, and then back to
$\widetilde{s(e_{i})}^{+}$ on $\widetilde{T}^{+}$ via a unique path in
$\widetilde{T}$. This path crosses from $\widetilde{T}^{-}$ to
$\widetilde{T}^{+}$, and hence must contain the edge $-\widetilde{e}_{0}^{+}$.
If $\sigma_{i}=-1$, then $\widetilde{\gamma}_{i}^{+}$ crosses from
$\widetilde{T}^{+}$ to $\widetilde{T}^{-}$, and hence contains
$\widetilde{e}_{0}^{+}$. Similarly, we calculate that the cycle
$\widetilde{\gamma}_{i}^{-}$ contains the edge
$\sigma_{i}\widetilde{e}_{0}^{+}$. In other words, for $e_{i}\in
S\backslash\\{e_{0}\\}$ we have

$\widetilde{\gamma}_{i}^{\pm}=\widetilde{e}_{i}^{\pm}\mp\sigma_{i}\widetilde{e}_{0}^{+}+\mbox{edges
of
}\widetilde{T}^{\pm},\quad\iota_{*}(\widetilde{\gamma}_{i}^{\pm})=\widetilde{e}_{i}^{\mp}\mp\sigma_{i}\widetilde{e}_{0}^{-}+\mbox{edges
of }\widetilde{T}^{\pm},\quad
p_{*}(\widetilde{\gamma}^{\pm}_{i})=e_{i}\mp\sigma_{i}e_{0}+\mbox{edges of
}T,$

and hence computing the intersection numbers we see that

$\iota_{*}(\widetilde{\gamma}_{i}^{\pm})=\widetilde{\gamma}_{i}^{\mp}\mp\sigma_{i}\widetilde{\gamma}_{0},\quad
p_{*}(\widetilde{\gamma}^{\pm}_{i})=\gamma_{i}\mp\sigma_{i}\gamma_{0},\quad
e_{i}\in S\backslash\\{e_{0}\\}.$

Finally, for $e_{i}\notin S$ the cycle $\gamma_{i}^{\pm}$ is contained in
$\widetilde{T}^{\pm}\cup\\{\widetilde{e}_{i}^{\pm}\\}$ and hence does not
contain the edge $\widetilde{e}_{0}^{+}$. It follows that

$\iota_{*}(\widetilde{\gamma}_{i}^{\pm})=\widetilde{e}_{i}^{\mp}+\mbox{edges
of }\widetilde{T}^{\pm},\quad
p_{*}(\widetilde{\gamma}_{i}^{\pm})=e_{i}+\mbox{edges of }T,\quad e_{i}\notin
S,$

and therefore

$\iota_{*}(\widetilde{\gamma}_{i}^{\pm})=\widetilde{\gamma}_{i}^{\mp},\quad
p_{*}(\widetilde{\gamma}_{i}^{\pm})=\gamma_{i},\quad e_{i}\notin S.$

It is now clear that

$\widetilde{\mathcal{B}}^{\prime}_{2}=\\{\widetilde{\gamma}_{i}^{+}-\iota_{*}(\widetilde{\gamma}_{i}^{+})\\}_{i=1}^{g-1}=\\{\widetilde{\gamma}_{i}^{+}-\widetilde{\gamma}_{i}^{-}+\sigma_{i}\widetilde{\gamma}_{0}\\}_{e_{i}\in
S\backslash\\{e_{0}\\}}\cup\\{\widetilde{\gamma}_{i}^{+}-\widetilde{\gamma}_{i}^{-}\\}_{e_{i}\notin
S}$ (19)

is a basis for $\operatorname{Ker}p_{*}$.

In Example 5.5, we explicitly construct this basis for a double cover
$\pi:\widetilde{\Gamma}\to\Gamma$ with $g=3$.

We now establish the relationship between the volumes of our three tropical
ppavs.

######  Proposition 3.6.

Let $\pi:\widetilde{\Gamma}\to\Gamma$ be a free double cover of metric graphs.
Then the volumes of $\operatorname{Jac}(\widetilde{\Gamma})$,
$\operatorname{Jac}(\Gamma)$, and
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$ are related as

$\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\frac{\operatorname{Vol}^{2}(\operatorname{Jac}(\widetilde{\Gamma}))}{2\operatorname{Vol}^{2}(\operatorname{Jac}(\Gamma))},$

where the volume of each tropical ppav is calculated using its intrinsic
principal polarization.

###### Proof.

We first introduce the following alternative basis $\mathcal{B}^{\prime}$ for
$H_{1}(G,\mathbb{Z})$:

$\mathcal{B}^{\prime}=\\{\gamma_{0}\\}\cup\\{\gamma_{i}-\sigma_{i}\gamma_{0}\\}_{e_{i}\in
S\backslash\\{e_{0}\\}}\cup\\{\gamma_{i}\\}_{e_{i}\notin S}.$ (20)

We now compute the pullback $\widetilde{\mathcal{B}}^{\prime}_{1}$ of
$\mathcal{B}^{\prime}$ to $H_{1}(\widetilde{G},\mathbb{Z})$ via the map

$p^{*}:H_{1}(G,\mathbb{Z})\to H_{1}(\widetilde{G},\mathbb{Z}),\quad\sum_{e\in
E(G)}a_{e}e\mapsto\sum_{e\in E(G)}a_{e}(\widetilde{e}^{+}+\widetilde{e}^{-}).$

Since $\gamma_{i}$ consists of $+e_{i}$ and edges of $T$, we have

$p^{*}(\gamma_{i})=\widetilde{e}^{+}_{i}+\widetilde{e}^{-}_{i}+\mbox{edges of
}T^{\pm}$

for $i=0,\ldots,g-1$. Computing intersection numbers as before, we see that

$p^{*}(\gamma_{0})=\widetilde{\gamma}_{0},\quad
p^{*}(\gamma_{i})=\widetilde{\gamma}^{+}_{i}+\widetilde{\gamma}^{-}_{i},\quad
i=1,\ldots,g-1.$

Hence

$\widetilde{\mathcal{B}}^{\prime}_{1}=p^{*}(\mathcal{B}^{\prime})=\\{\widetilde{\gamma}_{0}\\}\cup\\{\widetilde{\gamma}_{i}^{+}+\widetilde{\gamma}_{i}^{-}-\sigma_{i}\widetilde{\gamma}_{0}\\}_{e_{i}\in
S\backslash\\{e_{0}\\}}\cup\\{\widetilde{\gamma}_{i}^{+}+\widetilde{\gamma}_{i}^{-}\\}_{e_{i}\notin
S}.$

Let $(\cdot,\cdot)_{\widetilde{G}}$ and $(\cdot,\cdot)_{G}$ denote the
intersection pairings (4) on $H_{1}(\widetilde{G},\mathbb{Z})$ and
$H_{1}(G,\mathbb{Z})$, respectively, and let
$(\cdot,\cdot)_{P}=\frac{1}{2}(\cdot,\cdot)_{\widetilde{G}}$ denote the
intersection pairing on $\operatorname{Ker}p_{*}$ corresponding to the
principal polarization on $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. We
add the corresponding subscripts to each Gramian determinant, in order to keep
track of the inner product that is used to compute it. Thus the volumes of
$\operatorname{Jac}(G)$ and $\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$
are given by

$\operatorname{Vol}^{2}(\operatorname{Jac}(\Gamma))=\operatorname{Gram}_{G}(\mathcal{B}^{\prime}),\quad\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\operatorname{Gram}_{P}(\widetilde{\mathcal{B}}^{\prime}_{2}).$

We now consider the set
$\widetilde{\mathcal{B}}^{\prime}=\widetilde{\mathcal{B}}^{\prime}_{1}\cup\widetilde{\mathcal{B}}^{\prime}_{2}$,
which is a basis for the vector space $H_{1}(\widetilde{\Gamma},\mathbb{Q})$.
By appropriately ordering the basis elements, the change-of-basis matrix from
$\widetilde{\mathcal{B}}$ to $\widetilde{\mathcal{B}}^{\prime}$ becomes block-
triangular with a $1$ in the top left corner and a block
$\begin{pmatrix}1&1\\\ -1&1\end{pmatrix}$ for each edge
$e_{i},i=1,2\ldots,g-1$. Its determinant is therefore $\pm 2^{g-1}$, and it
follows that

$\operatorname{Vol}^{2}(\operatorname{Jac}(\widetilde{\Gamma}))=\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}})=2^{2-2g}\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}).$

We now compute $\operatorname{Gram}(\widetilde{\mathcal{B}}^{\prime})$ using
its block structure. First, we note that
$\iota_{*}(\widetilde{\gamma}^{\prime}_{1})=\widetilde{\gamma}^{\prime}_{1}$
for all
$\widetilde{\gamma}^{\prime}_{1}\in\widetilde{\mathcal{B}}^{\prime}_{1}$ and
$\iota_{*}(\widetilde{\gamma}^{\prime}_{2})=-\widetilde{\gamma}^{\prime}_{2}$
for all
$\widetilde{\gamma}^{\prime}_{2}\in\widetilde{\mathcal{B}}^{\prime}_{2}$.
Since $\iota_{*}$ preserves the pairing $(\cdot,\cdot)_{\widetilde{G}}$, it
follows that
$(\widetilde{\gamma}^{\prime}_{1},\widetilde{\gamma}^{\prime}_{2})_{\widetilde{G}}=0$
for all
$\widetilde{\gamma}^{\prime}_{1}\in\widetilde{\mathcal{B}}^{\prime}_{1}$ and
all $\widetilde{\gamma}^{\prime}_{2}\in\widetilde{\mathcal{B}}^{\prime}_{2}$,
therefore

$\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime})=\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_{1})\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_{2}).$

Since $(\cdot,\cdot)_{P}=\frac{1}{2}(\cdot,\cdot)_{\widetilde{G}}$, it is
clear that

$\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_{2})=2^{g-1}\operatorname{Gram}_{P}(\widetilde{\mathcal{B}}^{\prime}_{2}).$

Finally, for any $\gamma_{1},\gamma_{2}\in H_{1}(G,\mathbb{Z})$ we have
$(p^{*}(\gamma_{1}),p^{*}(\gamma_{2}))_{\widetilde{G}}=2(\gamma_{1},\gamma_{2})_{G}$,
and therefore

$\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_{1})=2^{g}\operatorname{Gram}_{G}(\mathcal{B}^{\prime}),$

because $\widetilde{\mathcal{B}}^{\prime}_{1}$ is the pullback of
$\mathcal{B}^{\prime}$. Putting all this together, we have

$\frac{\operatorname{Vol}^{2}(\operatorname{Jac}(\widetilde{\Gamma}))}{2\operatorname{Vol}^{2}(\operatorname{Jac}(\Gamma))}=\frac{2^{2-2g}\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime})}{2\operatorname{Gram}_{G}(\mathcal{B}^{\prime})}=\frac{2^{1-2g}\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_{1})\operatorname{Gram}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_{2})}{\operatorname{Gram}_{G}(\mathcal{B}^{\prime})}=\operatorname{Gram}_{P}(\widetilde{\mathcal{B}}^{\prime}_{2}),$

which is equal to
$\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))$, as
required.

∎

The proof of Theorem 3.4 now follows from Theorem 3.2 and Equation (7) by an
elementary scaling argument.

###### Proof of Theorem 3.4.

The right hand side of (17) is a homogeneous degree $g-1$ polynomial in the
edge lengths of $\Gamma$, and so is the left hand side, being the determinant
of a $(g-1)\times(g-1)$ Gramian matrix. Hence, it is sufficient to prove
Equation (17) in the case when $\Gamma$, and hence $\widetilde{\Gamma}$, have
integer edge lengths. Choose a model $p:\widetilde{G}\to G$ for $\pi$ such
that each edge of $\widetilde{G}$ and $G$ has length one. In this case
$\operatorname{Vol}(F)=1$ for any set of edges, hence by Kirchhoff’s theorem
and (7) we have

$\operatorname{Vol}^{2}(\operatorname{Jac}(\widetilde{\Gamma}))=|\operatorname{Jac}(\widetilde{G})|,\quad\operatorname{Vol}^{2}(\operatorname{Jac}(\Gamma))=|\operatorname{Jac}(G)|.$

It follows by Proposition 3.6 that

$\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}/\Gamma))=\frac{\operatorname{Vol}^{2}(\operatorname{Jac}(\widetilde{\Gamma}))}{2\operatorname{Vol}^{2}(\operatorname{Jac}(\Gamma))}=\frac{|\operatorname{Jac}(\widetilde{G})|}{2|\operatorname{Jac}(G)|}=|\operatorname{Prym}(\widetilde{G}/G)|.$

But $|\operatorname{Prym}(\widetilde{G}/G)|$ can be computed using (14), which
agrees with the right hand side of (17) when all edge lengths are equal to
one. This completes the proof.

∎

######  Example 3.7.

Let $\Gamma$ be the genus two dumbbell graph, with loops $e_{1}$ and $e_{2}$
of lengths $x_{1}$ and $x_{2}$, connected by a bridge $e_{3}$ of length
$x_{3}$. The unique spanning tree of $\Gamma$ consists of the edge $e_{3}$.
The graph $\Gamma$ has two topologically distinct connected free double covers
$\pi_{1}:\widetilde{\Gamma}_{1}\to\Gamma$ and
$\pi_{2}:\widetilde{\Gamma}_{2}\to\Gamma$, corresponding to flipping the edges
$S_{1}=\\{e_{1},e_{2}\\}$ and $S_{2}=\\{e_{1}\\}$ (see Figure 2).

$\Gamma$$e_{1}$$e_{3}$$e_{2}$$\widetilde{\Gamma}_{1}$$\widetilde{e}^{+}_{1}$$\widetilde{e}^{-}_{1}$$\widetilde{e}^{-}_{3}$$\widetilde{e}^{+}_{3}$$\widetilde{e}^{+}_{2}$$\widetilde{e}^{-}_{2}$$\Gamma$$e_{1}$$e_{3}$$e_{2}$$\widetilde{\Gamma}_{2}$$\widetilde{e}^{+}_{1}$$\widetilde{e}^{-}_{1}$$\widetilde{e}^{-}_{3}$$\widetilde{e}^{+}_{3}$$\widetilde{e}^{-}_{2}$$\widetilde{e}^{+}_{2}$
Figure 2. Two free double covers of the dumbbell graph. Flipped edges are
blue.

For the cover $\pi_{1}$, the odd genus one decompositions are $\\{e_{1}\\}$
and $\\{e_{2}\\}$ of rank one, and $\\{e_{3}\\}$ of rank two. For $\pi_{2}$,
the only odd genus one decomposition is $\\{e_{2}\\}$ of rank one. Hence
Theorem 3.4 states that

$\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}_{1}/\Gamma))=x_{1}+x_{2}+4x_{3},\quad\operatorname{Vol}^{2}(\operatorname{Prym}(\widetilde{\Gamma}_{2}/\Gamma))=x_{2}.$

Note that in each case the Prym variety is a circle, and the square of its
volume is its circumference (see Remark 2.6).

##  4\. The local structure of the Abel–Prym map

In the remaining two chapters, we provide a geometrization of the volume
formula (17) for the Prym variety of a free double cover of graphs, in the
spirit of the analogous formula (7) for the volume of the Jacobian variety of
a metric graph derived in [ABKS14].

Let $\pi:\widetilde{\Gamma}\to\Gamma$ be a free double cover of metric graphs
and let $\iota:\widetilde{\Gamma}\to\widetilde{\Gamma}$ be the associated
involution. For any integer $d$, we denote
$\operatorname{Prym}^{[d]}(\widetilde{\Gamma}/\Gamma)$ the connected component
of the kernel of the pushforward map
$\operatorname{Nm}:\operatorname{Jac}(\widetilde{\Gamma})\to\operatorname{Jac}(\Gamma)$
having the same parity as $d$, so that
$\operatorname{Prym}^{[d]}(\widetilde{\Gamma}/\Gamma)=\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$
if $d$ is even, and $\operatorname{Prym}^{[d]}(\widetilde{\Gamma}/\Gamma)$ is
the odd connected component if $d$ is odd. In this section, we study the
_Abel–Prym map_

$\Psi^{d}:\operatorname{Sym}^{d}(\widetilde{\Gamma})\to\operatorname{Prym}^{[d]}(\widetilde{\Gamma}/\Gamma),\quad\Psi^{d}(\widetilde{D})=\widetilde{D}-\iota(\widetilde{D}),$
(21)

for $d\leq g-1$. The space $\operatorname{Sym}^{d}(\widetilde{\Gamma})$ has a
natural cellular structure, with cells enumerated by the edges and vertices of
$\widetilde{\Gamma}$ supporting the divisor. The restriction of $\Psi^{d}$ to
each cell is an affine linear map, and we determine the cells on which
$\Psi^{d}$ has maximal rank.

Choose an oriented model $p:\widetilde{G}\to G$ for $\pi$ such that
$\widetilde{G}$ and $G$ have no loops. Let $0\leq k\leq d$, let
$\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{k}\\}$ be a multiset
of $k$ edges of $\widetilde{G}$, and let $\widetilde{Z}$ be an effective
divisor of degree $d-k$ supported on $V(\widetilde{G})$. Denote by
$C^{k}(\widetilde{F},\widetilde{Z})$ the $k$-dimensional set of effective
divisors on $\widetilde{\Gamma}$ of the form
$\widetilde{D}=\widetilde{P}_{1}+\cdots+\widetilde{P}_{k}+\widetilde{Z}$,
where each $\widetilde{P}_{i}$ lies on $\widetilde{f}_{i}$. Any effective
degree $d$ divisor on $\widetilde{\Gamma}$ can be split up in such a way
(uniquely if each point lies in the interior of an edge), hence we have a
cellular decomposition

$\operatorname{Sym}^{d}(\widetilde{\Gamma})=\bigcup_{k=0}^{d}\bigcup_{\widetilde{F},\widetilde{Z}}C^{k}(\widetilde{F},\widetilde{Z}),$
(22)

where the union is taken over all
$\widetilde{F}\in\operatorname{Sym}^{k}(E(\widetilde{G}))$ and
$\widetilde{Z}\in\operatorname{Sym}^{d-k}(V(\widetilde{G}))$.

The principal result of this section describes the cells
$C^{k}(\widetilde{F},\widetilde{Z})$ that are not contracted by the Abel–Prym
map $\Psi^{d}$. It is clear that the divisor $\widetilde{Z}$ plays no role in
this question, hence we assume that $k=d$, $\widetilde{Z}=0$, and only
consider the top-dimensional cells, which we denote

$C(\widetilde{F})=C^{d}(\widetilde{F},0)=\\{\widetilde{P}_{1}+\cdots+\widetilde{P}_{d}:\widetilde{P}_{i}\in\widetilde{f}_{i}\\}\subset\operatorname{Sym}^{d}(\widetilde{\Gamma}),\quad\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{d}\\}\in\operatorname{Sym}^{d}(E(\widetilde{G})).$

######  Theorem 4.1.

Let $\pi:\widetilde{\Gamma}\to\Gamma$ be a free double cover of metric graphs
with oriented loopless model $p:\widetilde{G}\to G$, and let
$\Psi^{d}:\operatorname{Sym}^{d}(\widetilde{\Gamma})\to\operatorname{Prym}^{[d]}(\widetilde{\Gamma}/\Gamma)$
be the degree $d$ Abel–Prym map, where $1\leq d\leq g-1$. Let
$\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{d}\\}\subset
E(\widetilde{G})$ be a multiset of edges of $\widetilde{G}$, let
$C(\widetilde{F})\subset\operatorname{Sym}^{d}(\widetilde{\Gamma})$ be the
corresponding top-dimensional cell, and denote $F=\\{f_{1},\ldots,f_{d}\\}$,
where $f_{i}=p(\widetilde{f}_{i})$.

  1. (1)

If the edges in $F$ are not distinct (in particular, if the edges in
$\widetilde{F}$ are not distinct), then $\Psi^{d}$ contracts
$C(\widetilde{F})$.

  2. (2)

If the edges in $F$ are distinct, then $\Psi^{d}$ does not contract
$C(\widetilde{F})$ if and only if the preimage under $p$ of each connected
component of $G\backslash F$ is connected.

The proof of the first part of the theorem is quite elementary: for any
$\widetilde{D}\in C(\widetilde{F})$ we construct a nearby divisor
$\widetilde{D}^{\prime}$ such that
$\Psi^{d}(\widetilde{D})=\widetilde{D}-\iota(\widetilde{D})$ is linearly
equivalent to
$\Psi^{d}(\widetilde{D}^{\prime})=\widetilde{D}^{\prime}-\iota(\widetilde{D}^{\prime})$
via an explicit rational function. To prove the second part, we calculate the
matrix of $\Psi^{d}$ with respect to a convenient basis, and compute its rank.
This part, and the necessary constructions, will occupy the greater part of
this section.

###### Proof of Theorem 4.1, part (1).

Let $\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{d}\\}$ be a
multiset such that not all $f_{i}=p(\widetilde{f}_{i})$ are distinct. Without
loss of generality we assume that $f_{1}=f_{2}$, which means that either
$\widetilde{f}_{1}=\widetilde{f}_{2}$ or
$\widetilde{f}_{1}=\iota(\widetilde{f}_{2})$. Let
$\widetilde{D}=\widetilde{P}_{1}+\cdots+\widetilde{P}_{d}$ be a point of
$C(\widetilde{F})$, where each $\widetilde{P}_{i}$ lies in the interior of
$\widetilde{f}_{i}$.

If $\widetilde{f}_{1}=\widetilde{f}_{2}$, we can assume that either
$\widetilde{P}_{1}=\widetilde{P}_{2}$, or that the direction from
$\widetilde{P}_{1}$ to $\widetilde{P}_{2}$ is positive with respect to the
orientation. Denote
$\widetilde{D}^{\prime}=\widetilde{P}^{\prime}_{1}+\widetilde{P}^{\prime}_{2}+\widetilde{P}_{3}+\cdots+\widetilde{P}_{d}$,
where $\widetilde{P}^{\prime}_{1}$ and $\widetilde{P}^{\prime}_{2}$ are
obtained by moving $\widetilde{P}_{1}$ and $\widetilde{P}_{2}$ a small
distance of $\varepsilon>0$ in respectively the negative and the positive
directions along $\widetilde{f}_{1}=\widetilde{f}_{2}$. Then the divisor

$\Psi^{d}(\widetilde{D})-\Psi^{d}(\widetilde{D}^{\prime})=\widetilde{D}-\iota(\widetilde{D})-\widetilde{D}^{\prime}+\iota(\widetilde{D}^{\prime})=\widetilde{P}_{1}+\widetilde{P}_{2}-\widetilde{P}^{\prime}_{1}-\widetilde{P}^{\prime}_{2}-\iota(\widetilde{P}_{1})-\iota(\widetilde{P}_{2})+\iota(\widetilde{P}^{\prime}_{1})+\iota(\widetilde{P}^{\prime}_{2})$

is the principal divisor of a piecewise linear function on
$\widetilde{\Gamma}$. Indeed, consider the function
$M:\widetilde{\Gamma}\to\mathbb{R}$ having the following slopes on the edges
of $\widetilde{\Gamma}$:

  * •

On $\widetilde{f}_{1}=\widetilde{f}_{2}$, $M$ has slope zero to the left of
$\widetilde{P}_{1}^{\prime}$, slope $+1$ on
$[\widetilde{P}^{\prime}_{1},\widetilde{P}_{1}]$, slope zero on
$[\widetilde{P}_{1},\widetilde{P}_{2}]$, slope $-1$ on
$[\widetilde{P}_{2},\widetilde{P}^{\prime}_{2}]$, and slope zero to the right
of $\widetilde{P}^{\prime}_{2}$.

  * •

On $\iota(\widetilde{f}_{1})=\iota(\widetilde{f}_{2})$, $M$ has slope zero to
the left of $\iota(\widetilde{P}_{1})$, slope $-1$ on
$[\iota(\widetilde{P}^{\prime}_{1}),\iota(\widetilde{P}_{1})]$, slope zero on
$[\iota(\widetilde{P}_{1}),\iota(\widetilde{P}_{2})]$, slope $+1$ on
$[\iota(\widetilde{P}_{2}),\iota(\widetilde{P}^{\prime}_{2})]$, and slope zero
to the right of $\iota(\widetilde{P}^{\prime}_{2})$.

  * •

The function $M$ has zero slope on all other edges of $\widetilde{\Gamma}$.

The net changes of $M$ along $\widetilde{f}_{1}=\widetilde{f}_{2}$ and
$\iota(\widetilde{f}_{1})=\iota(\widetilde{f}_{2})$ cancel out, hence the
function $M$ is continuous, and it is clear that
$\Psi^{d}(\widetilde{D})-\Psi^{d}(\widetilde{D}^{\prime})$ is the divisor of
$M$. Therefore, $\Psi^{d}$ is not locally injective at $\widetilde{D}$.

The case $\widetilde{f}_{1}=\iota(\widetilde{f}_{2})$ is similar. We consider
$\widetilde{D}^{\prime}=\widetilde{P}^{\prime}_{1}+\widetilde{P}^{\prime}_{2}+\widetilde{P}_{3}+\cdots+\widetilde{P}_{d}$,
where $\widetilde{P}^{\prime}_{1}$ and $\widetilde{P}^{\prime}_{2}$ are
obtained by moving $\widetilde{P}_{1}$ and $\widetilde{P}_{2}$ a small
distance of $\varepsilon>0$ in the same direction along respectively
$\widetilde{f}_{1}$ and $\widetilde{f}_{2}=\iota(\widetilde{f}_{1})$. It is
easy to check that $\Psi^{d}(\widetilde{D})-\Psi^{d}(\widetilde{D}^{\prime})$
is a principal divisor, hence $\Psi^{d}$ is not locally injective at
$\widetilde{D}$.

∎

To prove part (2) of Theorem 4.1, we give an explicit description of the
Abel–Prym map $\Psi^{d}$ on a cell $C(\widetilde{F})$. We first consider the
case $d=1$. Fix a base point $q\in\widetilde{\Gamma}$, and for each point
$p\in\widetilde{\Gamma}$ fix a path $\gamma(q,p)$ from $q$ to $p$. The
Abel–Prym map $\Psi^{1}$ is a difference of Abel–Jacobi maps (5):

$\Psi^{1}:\widetilde{\Gamma}\to\operatorname{Prym}^{[1]}(\widetilde{\Gamma}/\Gamma)\subset\operatorname{Jac}(\widetilde{\Gamma}),\quad\Psi^{1}(p)(\omega)=\int_{\gamma(q,p)}\omega-\int_{\gamma(q,\iota(p))}\omega.$

It is more convenient to work with the even component
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$. The odd component
$\operatorname{Prym}^{[1]}(\widetilde{\Gamma}/\Gamma)$ is a torsor over
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, and we can pass to the even
component by translating by any element of the odd component, for example the
element $\Psi^{1}(\iota(q))=\int_{\gamma(q,\iota(q))}$. We can further assume
that the path $\gamma(q,\iota(p))$ in the formula above consists of
$\gamma(q,\iota(q))$ followed by the path $\iota_{*}(\gamma(q,p))$. Putting
this together, we obtain the translated Abel–Prym map, which we also denote
$\Psi^{1}$ by abuse of notation:

$\Psi^{1}:\widetilde{\Gamma}\to\operatorname{Prym}(\widetilde{\Gamma}/\Gamma),\quad\Psi^{1}(p)(\omega)=\int_{\gamma(q,p)}\omega-\int_{\iota_{*}(\gamma(q,p))}\omega.$
(23)

Choose a basis $\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{g-1}$ for
$\operatorname{Ker}\pi_{*}:H_{1}(\widetilde{\Gamma},\mathbb{Z})\to
H_{1}(\Gamma,\mathbb{Z})$. As explained in Subsection 2.6, the functionals
$\omega^{*}_{j}\in\Omega^{*}(\widetilde{\Gamma})$ dual to
$\omega_{j}=\psi(\widetilde{\gamma}_{j})$ define a coordinate system on
$\operatorname{Prym}(\widetilde{\Gamma}/\Gamma)$, so we can write

$\Psi^{1}(p)=\int_{\gamma(q,p)}-\int_{\iota_{*}(\gamma(q,p))}=\sum_{j=1}^{g-1}a_{j}(p)\omega_{j}^{*},$

where we find the coefficients $a_{j}(p)$ by pairing with $\omega_{j}$:

$a_{j}(p)=\int_{\gamma(q,p)}\omega_{j}-\int_{\iota_{*}(\gamma(q,p))}\omega_{j}.$

We now assume that $p$ lies on the interior of an edge $\widetilde{f}\in
E(\widetilde{\Gamma})$, which we identify using the orientation with the
segment $(0,\ell(\widetilde{f}))$. Pick
$x_{1},x_{2}\in(0,\ell(\widetilde{f}))$ such that $x_{1}<x_{2}$, then we see
that

$a_{j}(x_{2})-a_{j}(x_{1})=\int_{\gamma(q,x_{2})}\omega_{j}-\int_{\iota_{*}(\gamma(q,x_{2}))}\omega_{j}-\int_{\gamma(q,x_{1})}\omega_{j}+\int_{\iota_{*}(\gamma(q,x_{1}))}\omega_{j}=\int_{\gamma(x_{1},x_{2})}\omega_{j}-\int_{\iota_{*}(\gamma(x_{1},x_{2}))}\omega_{j}.$

We can assume that $\gamma(x_{1},x_{2})$ is the segment
$[x_{1},x_{2}]\subset(0,\ell(\widetilde{f}))$. The integral of $\omega_{j}$
over $\gamma(x_{1},x_{2})$ is equal to the length $x_{2}-x_{1}$ multiplied by
the coefficient with which $\widetilde{f}$ occurs in $\omega_{j}$, which is
$\frac{1}{2}\langle\widetilde{\gamma}_{j},\widetilde{f}\rangle$ (the
$\frac{1}{2}$ coefficient comes from using the principal polarization of the
Prym). Similarly, the integral of $\omega_{j}$ over
$\iota_{*}(\gamma(x_{1},x_{2}))$ is equal to
$\frac{1}{2}(x_{2}-x_{1})\langle\widetilde{\gamma}_{j},\iota(\widetilde{f})\rangle$,
and therefore

$a_{j}(x_{2})-a_{j}(x_{1})=\frac{1}{2}(x_{2}-x_{1})\langle\widetilde{\gamma}_{j},\widetilde{f}-\iota(\widetilde{f})\rangle,$

where $\langle\cdot,\cdot\rangle$ is the edge pairing (18). It follows that,
with respect to the coordinate vectors $\omega^{*}_{j}$ defined by the basis
$\widetilde{\gamma}_{j}$, the restriction of the Abel–Prym map (23) to an edge
$\widetilde{f}=[0,\ell(\widetilde{f})]$ is an affine $\mathbb{Z}$-linear map
of the form

$\Psi^{1}(x)=\frac{1}{2}\sum_{j=1}^{g-1}\langle\widetilde{\gamma}_{j},\widetilde{f}-\iota(\widetilde{f})\rangle\omega^{*}_{j}x+C,$

where $C$ is a constant vector.

This formula readily generalizes to any degree. Let
$\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{d}\\}$ be a set of
distinct edges of $\widetilde{G}$. We identify the cell $C(\widetilde{F})$
with the parallelotope
$[0,\ell(\widetilde{f}_{1})]\times\cdots\times[0,\ell(\widetilde{f}_{d})]$,
where the point $(x_{1},\ldots,x_{d})$ corresponds to the divisor
$\widetilde{D}=\widetilde{P}_{1}+\cdots+\widetilde{P}_{d}$, where
$\widetilde{P}_{i}$ lies on $\widetilde{f}_{i}$ at a distance of $x_{i}$ from
the starting vertex. Translating by any odd Prym divisor and moving to the
even component if $d$ is odd, we see that the restriction of the Abel–Prym map
(21) to the cell $C(\widetilde{F})$ is affine $\mathbb{Z}$-linear:

$\Psi^{d}(x_{1},\ldots,x_{d})=\sum_{i=1}^{d}\sum_{j=1}^{g-1}(\Psi^{d})_{ji}\omega^{*}_{j}x_{i}+C\in\operatorname{Prym}(\widetilde{\Gamma}/\Gamma),$

where $(\Psi^{d})_{ji}$ is the $(g-1)\times d$ matrix

$(\Psi^{d})_{ji}=\frac{1}{2}\left\langle\widetilde{\gamma}_{j},\widetilde{f}_{i}-\iota(\widetilde{f}_{i})\right\rangle,$
(24)

and $C$ is some constant vector.

To prove part (2) of Theorem 4.1, we compute the rank of $(\Psi^{d})_{ji}$
with respect to a carefully chosen basis $\widetilde{\gamma}_{j}$ of
$\operatorname{Ker}\pi_{*}$. Specifically, the "if" and "only if" statements
will require slightly different choices of basis.

###### Proof of Theorem 4.1, part (2).

We consider the decomposition of $G\backslash F$ into connected components,
which we enumerate as follows:

$G\backslash F=G_{0}\cup\cdots\cup G_{r-1}.$

Before proceeding, we derive a relationship between the genera
$g_{k}=|E(G_{k})|-|V(G_{k})|+1$ of the components $G_{k}$:

$g_{0}+\cdots+g_{r-1}=\sum_{k=0}^{r-1}|E(G_{k})|-\sum_{k=0}^{r-1}|V(G_{k})|+r=|E(G)|-d-|V(G)|+r=g+r-d-1.$
(25)

We now consider the two possibilities.

The preimage of one of the connected components is disconnected. Equivalently,
we assume that the restriction of the cover $p$ to one of the connected
components is isomorphic to the trivial free double cover. Let $G_{k}$ be a
connected component of genus $g_{k}$. If $p^{-1}(G_{k})$ is connected, then
Construction B, applied to the cover $p|_{p^{-1}(G_{k})}:p^{-1}(G_{k})\to
G_{k}$, produces $g_{k}-1$ linearly independent cycles $\gamma_{kl}\in
H_{1}(\widetilde{G},\mathbb{Z})$ that are supported on $p^{-1}(G_{k})$ and
that lie in $\operatorname{Ker}p_{*}$. However, if the restriction of $p$ to,
say, $G_{k}$ is trivial, then we can find $g_{k}$ such cycles, by applying
$(\operatorname{Id}-\iota)_{*}$ to the lifts of a linearly independent
collection of cycles on $G_{k}$. In this case, it follows from (25) that there
are at least

$(g_{0}-1)+(g_{1}-1)+\ldots+g_{k}+\ldots+(g_{r-1}-1)=g-d$

linearly independent cycles $\gamma_{kl}\in H_{1}(\widetilde{G},\mathbb{Z})$,
lying in the kernel of $p_{*}$ and supported on

$p^{-1}(G_{0})\cup\ldots\cup
p^{-1}(G_{r-1})=\widetilde{G}\backslash(\widetilde{F}\cup\iota(\widetilde{F})).$

Any such cycle $\gamma_{kl}$ pairs trivially with each $\widetilde{f}_{i}$ and
$\iota(\widetilde{f}_{i})$. Therefore, by completing these cycles to a basis
of $\operatorname{Ker}p_{*}$ (passing to $\mathbb{Q}$-coefficients if
necessary), we see that the matrix (24) of $\Psi^{d}$ with respect to this
basis has at least $g-d$ rows of zeroes, hence has rank less than $d$ and
contracts $C(\widetilde{F})$.

The preimage of each connected component is connected. Equivalently, the
restriction of $p$ to each connected component is a nontrivial free double
cover. This implies that $g_{k}\geq 1$ for each $k$, since any free double
cover of a tree is trivial.

We show that the matrix (24) has rank $d$ with respect to an explicit choice
of basis $\widetilde{\gamma}_{i}$ of $\operatorname{Ker}\pi_{*}$. The
construction of this basis is somewhat involved, and will be used again in the
proof of Theorem 5.1, so we typeset it separately.

######  Construction C.

Let $\pi:\widetilde{\Gamma}\to\Gamma$ be a connected free double cover of
metric graphs of genera $2g-1$ and $g$, respectively, and let
$p:\widetilde{G}\to G$ be an oriented model. Let
$\widetilde{F}=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{d}\\}\subset
E(\widetilde{G})$ be a set of $d$ edges so that the edges
$f_{i}=p(\widetilde{f}_{i})$ are distinct, and denote $F=p(\widetilde{F})$.
Let

$G\backslash F=G_{0}\cup\cdots\cup G_{r-1}$

be the decomposition of $G\backslash F$ into connected components, and further
assume that $p^{-1}(G_{k})$ is connected for each $k$. In this Construction,
we elaborate on Constructions A and B by carefully choosing a spanning tree
$T\subset G$ and a corresponding basis
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{g-1}$ for
$\operatorname{Ker}\pi_{*}:H_{1}(\widetilde{G},\mathbb{Z})\to
H_{1}(G,\mathbb{Z})$, with respect to which the matrix of the Abel–Prym map on
the cell $C(\widetilde{F})$ has a convenient triangular structure. This
construction involves a number of choices and relabelings, so we break it down
into steps. At the same time, we will contract the portions of the graphs
$\widetilde{G}$ and $G$ that are irrelevant to our intersection calculations.

_Contracting the $G_{k}$ and choosing the spanning tree $T\subset G$._ Choose
a spanning tree $T_{k}$ for each connected component $G_{k}$. Denote by
$G^{c}$ the graph obtained from $G$ by contracting each subtree $T_{k}$ to a
separate vertex $v_{k}$. Specifically, the vertex set of $G^{c}$ is
$V(G^{c})=\\{v_{0},\ldots,v_{r-1}\\}$, and the edge set of $G^{c}$ is the set
of edges $G$ that are not in any $T_{k}$. There is a natural contraction map
$(\cdot)^{c}:G\to G^{c}$; this map sends each vertex and edge of $T_{k}$ to
$v_{k}$, and sends each edge of $G$ that is not in any $T_{k}$ to the
corresponding edge of $G^{c}$. The contracted graph $G^{c}$ has the same genus
$g$ as $G$, hence it has $g+r-1$ edges, namely the edges
$F=\\{f_{1},\ldots,f_{d}\\}$ and $g+r-d-1$ loops corresponding to the
uncontracted edges of the $G_{k}$. Choose a spanning tree $T^{c}$ for the
contracted graph $G^{c}$; the $r-1$ edges of $T^{c}$ are a subset of the edges
$\\{f_{1},\ldots,f_{d}\\}$.

Before proceeding, we relabel the edges $\widetilde{f}_{i}$ and
$f_{i}=p(\widetilde{f}_{i})$ so that $E(T^{c})=\\{f_{1},\ldots,f_{r-1}\\}$. We
then choose $v_{0}$ as the root vertex of $T_{c}$, and further relabel and
reorient the edges $f_{1},\ldots,f_{r-1}$ away from $v_{0}$. Specifically, we
require that, along the unique path in $T^{c}$ starting at $v_{0}$ and ending
at any other vertex, the edges are oriented in the direction of the path and
appear in increasing order. Finally, we relabel the vertices
$v_{1},\ldots,v_{r-1}$ so that $t(f_{j})=v_{j}$ for $j=1,\ldots,r-1$; this
implies that $s(f_{j})=v_{\alpha(j)}$ for some index $\alpha(j)<j$.

We now form a spanning tree $T$ for $G$ by joining the subtrees $T_{k}$ with
the edges of $T^{c}$ (viewed as edges of $G$):

$T=T_{0}\cup\cdots\cup T_{r-1}\cup\\{f_{1},\ldots,f_{r-1}\\}.$

We denote the complementary edges of $T$ in $G$ by $E(G)\backslash
E(T)=\\{e_{0},e_{1},\ldots,e_{g-1}\\}$, and we explain below how the $e_{j}$
are chosen.

_Choosing $e_{0}$ and contracting the preimages of the $G_{k}$._ We now
describe the cover $p$ using the spanning tree $T$ and Construction A. The
tree $T$ has two disjoint lifts $\widetilde{T}^{\pm}$ to $\widetilde{G}$, and
we denote $\widetilde{T}^{\pm}_{k}=\widetilde{T}^{\pm}\cap p^{-1}(T_{k})$ the
corresponding lifts of the $T_{k}$. For each $i=1,\ldots,d$, each of the trees
$\widetilde{T}^{\pm}$ contains exactly one of the two edges
$\widetilde{f}_{i}$ and $\iota(\widetilde{f}_{i})$. The cover
$p:f^{-1}(G_{0})\to G_{0}$ is not trivial, so we can pick an edge $e_{0}\in
E(G_{0})\backslash E(T_{0})$ having a lift
$\widetilde{e}_{0}=\widetilde{e}_{0}^{+}$ that connects
$\widetilde{T}_{0}^{+}$ and $\widetilde{T}_{0}^{-}$. Then

$\widetilde{T}=\widetilde{T}^{+}\cup\widetilde{T}^{-}\cup\\{\widetilde{e}_{0}^{+}\\}$

is a spanning tree for $\widetilde{G}$. We note that, by our labeling
convention, for $k=1,\ldots,r-1$ the target vertex $t(\widetilde{f}_{k})$ lies
on either $\widetilde{T}^{+}_{k}$ or $\widetilde{T}^{-}_{k}$, while
$t(\iota(\widetilde{f}_{k}))$ lies on the other subtree.

We now perform a contraction on the graph $\widetilde{G}$ that is consistent
with the contraction $(\cdot)^{c}:G\to G^{c}$ defined above. The tree
$\widetilde{T}^{+}_{0}\cup\widetilde{T}^{-}_{0}\cup\\{\widetilde{e}_{0}^{+}\\}$
is a spanning tree for the preimage $p^{-1}(G_{0})$, while for each
$k=1,\ldots,r-1$ the two disjoint trees $\widetilde{T}^{+}_{k}$ and
$\widetilde{T}^{-}_{k}$ form a spanning forest for $p^{-1}(G_{k})$. Let
$\widetilde{G}^{c}$ denote the graph obtained from $\widetilde{G}$ by
contracting
$\widetilde{T}^{+}_{0}\cup\widetilde{T}^{-}_{0}\cup\\{\widetilde{e}_{0}^{+}\\}$
to a vertex $\widetilde{v}_{0}$, and contracting each
$\widetilde{T}^{\pm}_{k}$ to a separate vertex $\widetilde{v}^{\pm}_{k}$. We
denote the contraction map by $(\cdot)^{c}:\widetilde{G}\to\widetilde{G}^{c}$,
and for a non-contracted edge $\widetilde{e}\in E(\widetilde{G})$ (i.e. for
any edge neither in
$\widetilde{T}^{+}_{0}\cup\widetilde{T}^{-}_{0}\cup\\{\widetilde{e}_{0}^{+}\\}$
nor in any $\widetilde{T}^{\pm}_{k}$) we denote
$(\widetilde{e})^{c}=\widetilde{e}$ by abuse of notation. The double cover
$p:\widetilde{G}\to G$ descends to a map $p:\widetilde{G}^{c}\to G^{c}$ (which
is almost a double cover of graphs, except that $v_{0}$ and $e_{0}$ each have
a single preimage), and the projections commute with the contractions. The
contraction $\widetilde{T}^{c}$ of $\widetilde{T}$ is a spanning tree for
$\widetilde{G}^{c}$, having vertex and edge sets

$V(\widetilde{T}^{c})=V(\widetilde{G}^{c})=\\{\widetilde{v}_{0},\widetilde{v}^{\pm}_{1},\ldots,\widetilde{v}^{\pm}_{r-1}\\},\quad
E(\widetilde{T}^{c})=\\{\widetilde{f}_{1},\ldots,\widetilde{f}_{r-1},\iota(\widetilde{f}_{1}),\ldots,\iota(\widetilde{f}_{r-1})\\}.$

The tree $\widetilde{T}^{c}$ can be viewed as two copies of $T^{c}$ joined at
the common vertex $\widetilde{v}_{0}$.

_Labeling the complementary edges._ Finally, we label the complementary edges

$E(G)\backslash E(T)=E(G_{0})\backslash E(T_{0})\cup\cdots\cup
E(G_{r-1})\backslash
E(T_{r-1})\cup\\{f_{r},\ldots,f_{d}\\}=\\{e_{0},\ldots,e_{g-1}\\}.$

Each of the $e_{j}$ is either an edge of a subgraph $G_{k}$ that does not lie
on the spanning tree $T_{k}$, or is one of the $f_{k}$. Since the restriction
of the double cover $p:\widetilde{G}\to G$ to each of the $G_{k}$ is
nontrivial, for each $k=1,\ldots,r-1$ we can choose an edge in
$E(G_{k})\backslash E(T_{k})$ whose preimages cross $\widetilde{T}^{\pm}_{k}$,
and we label this edge $e_{k}$. We pick the preimage
$\widetilde{e}_{k}=\widetilde{e}_{k}^{+}\in E(\widetilde{G})\backslash
E(\widetilde{T})$ in such a way that the source vertex $s(\widetilde{e}_{k})$
lies on the same subtree $\widetilde{T}^{+}_{k}$ or $\widetilde{T}^{-}_{k}$ as
the target vertex $t(\widetilde{f}_{k})$. Furthermore, for $k=r,\ldots,d$, we
denote $e_{k}=f_{k}$ and
$\widetilde{e}_{k}=\widetilde{e}^{+}_{k}=\widetilde{f}_{k}$. The remaining
edges $e_{k}$ for $k=d+1,\ldots,g-1$ and their preimages
$\widetilde{e}^{\pm}_{k}$ are labeled arbitrarily. We note that in this case,
$e_{k}\in S$ for $k=0,\ldots,r-1$.

To help follow the intersection calculations in the following proofs, we
summarize the structure of the graphs $\widetilde{G}^{c}$ and $G^{c}$ and the
map $p:\widetilde{G}^{c}\to G^{c}$:

  1. (1)

The vertices of the graphs $\widetilde{G}^{c}$ and $G^{c}$ are

$V(\widetilde{G}^{c})=\\{\widetilde{v}_{0},\widetilde{v}^{\pm}_{1},\ldots,\widetilde{v}^{\pm}_{r-1}\\},\quad
V(G^{c})=\\{v_{0},v_{1},\ldots,v_{r-1}\\}.$

The map $p$ sends $\widetilde{v}_{0}$ to $v_{0}$ and $\widetilde{v}^{\pm}_{k}$
to $v_{k}$.

  2. (2)

The edges of the graph $G^{c}$ are as follows:

$E(G^{c})=\\{f_{1},\ldots,f_{r-1},e_{0},\ldots,e_{r-1},e_{r}=f_{r},\ldots,e_{d}=f_{d},e_{d+1},\ldots,e_{g-1}\\}.$

The $r-1$ edges $\\{f_{1},\ldots,f_{r-1}\\}$ are the edges of the spanning
tree $T_{c}$; these are oriented in increasing order away from the root vertex
$v_{0}$, so that $t(f_{j})=v_{j}$ for $j=1,\ldots,r-1$. The remaining $g$
edges $\\{e_{0},\ldots,e_{g-1}\\}$ are split into three groups: for
$j=0,\ldots,r-1$ the edge $e_{j}$ is a loop at $v_{j}$, for $j=r,\ldots,d$ the
edge $e_{j}=f_{j}$ may or may not be a loop, and for $j=d+1,\ldots,g-1$ each
remaining $e_{j}$ is a loop at one of the $v_{k}$.

  3. (3)

The edges of the graph $\widetilde{G}^{c}$ are as follows. For
$j=1,\ldots,r-1$, each $f_{j}$ has two preimages labeled $\widetilde{f}_{j}$
and $\iota(\widetilde{f}_{j})$; these $2r-2$ edges form the spanning tree
$\widetilde{T}^{c}$. Furthermore, for $j=1,\ldots,r-1$ the target vertex
$t(\widetilde{f}_{j})$ is one of the two vertices $\widetilde{v}_{j}^{\pm}$.
The edge $e_{0}$ has a unique preimage $\widetilde{e}^{-}_{0}$, while for
$j=1,\ldots,g-1$ each $e_{j}$ has two preimages
$\widetilde{e}_{j}^{+}=\widetilde{e}_{j}$ and
$\widetilde{e}_{j}^{-}=\iota(\widetilde{e}_{j})$. For $j=1,\ldots,r-1$ the
edges $\widetilde{e}_{j}$ and $\iota(\widetilde{e}_{j})$ (lying over the loop
$e_{j}$) connect $\widetilde{v}_{j}^{+}$ and $\widetilde{v}_{j}^{-}$ in
opposite directions, in such a way that
$s(\widetilde{e}_{j})=t(\widetilde{f}_{j})$. For $j=r,\ldots,d$ we have
$\widetilde{e}_{j}=\widetilde{f}_{j}$, and for each $j=d+1,\ldots,g-1$ the two
edges $\widetilde{e}_{j}$ and $\iota(\widetilde{e}_{j})$ over $e_{j}$ are
either parallel edges between $\widetilde{v}^{+}_{k}$ and
$\widetilde{v}^{-}_{k}$ for some $k$, or form a pair of loops.

We now employ Construction B to produce a basis
$\widetilde{\gamma}_{1},\ldots,\widetilde{\gamma}_{g-1}$ of
$\operatorname{Ker}\pi_{*}:H_{1}(\widetilde{G},\mathbb{Z})\to
H_{1}(G,\mathbb{Z})$, with respect to the chosen spanning tree $T\subset G$.
Let $\widetilde{\gamma}_{0}$ and $\widetilde{\gamma}_{j}^{\pm}$ for
$j=1,\ldots,g-1$ be the unique cycle of respectively
$\widetilde{T}\cup\\{\widetilde{e}^{-}_{0}\\}$ and
$\widetilde{T}\cup\\{\widetilde{e}^{\pm}_{j}\\}$ such that
$\langle\widetilde{\gamma}_{0},\widetilde{e}^{-}_{0}\rangle=1$ and
$\langle\widetilde{\gamma}^{\pm}_{j},\widetilde{e}^{\pm}_{j}\rangle=1$ for
$j=1,\ldots,g-1$. Then the cycles

$\widetilde{\gamma}_{j}=\widetilde{\gamma}_{j}^{+}-\iota_{*}(\widetilde{\gamma}_{j}^{+})=\left\\{\begin{array}[]{cc}\widetilde{\gamma}^{+}_{j}-\widetilde{\gamma}^{-}_{j}+\sigma_{j}\widetilde{\gamma}_{0},&e_{j}\in
S,\\\ \widetilde{\gamma}^{+}_{j}-\widetilde{\gamma}^{-}_{j},&e_{j}\notin
S,\end{array}\right.,\quad j=1,\ldots,g-1$ (26)

form a basis for $\operatorname{Ker}\pi_{*}$.

In Example 5.5, we deploy this construction for a specific cover
$\pi:\widetilde{\Gamma}\to\Gamma$ with $g=3$.