2.1 Graphs and free double covers

We denote the vertex and edge sets of a finite graph G respectively by $V(G)$ 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 $T_vG$ is the set of edges emanating from v and the valency is $\operatorname {\mathrm {val}}(v)=\#T_vG$ (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 article 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 neighbourhood 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 $2g-1$ .

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 $(G,\ell )$ respectively of $\widetilde {\Gamma }$ 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 2 and local degree 1 everywhere and we do not consider the more general case of unramified harmonic morphisms of degree 2 studied in [Reference Jensen and LenJL18] and [Reference Len and UlirschLU19]. 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 [Reference WallerWal76], 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

$$ \begin{align*}\widetilde{G}=\widetilde{T}^+\cup \widetilde{T}^- \cup \{\widetilde{e}_0^{{\kern1.5pt}\pm},\ldots,\widetilde{e}_{g-1}^{{\kern1.5pt}\pm}\}. \end{align*} $$

The map $p:\widetilde {G}\to G$ sends $\widetilde {T}^{\pm }$ isomorphically to T and $\widetilde {e}_i^{{\kern1.5pt}\pm }$ to $e_i$ . For $e_i\in S$ , each of the two edges $\widetilde {e}^{{\kern1.5pt}\pm }_i$ above it has one vertex on $\widetilde {T}^+$ and one on $\widetilde {T}^-$ , while for $e_i\notin S$ both vertices of $\widetilde {e}^{{\kern1.5pt}\pm }_i$ 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 $\widetilde {T}=\widetilde {T}^+\cup \widetilde {T}^-\cup \{\widetilde {e}_0^{{\kern1.5pt}+}\}$ is a spanning tree for $\widetilde {G}$ . We furthermore always assume that the starting and ending vertices of $\widetilde {e}^{{\kern1.5pt}+}_0$ lie respectively on $\widetilde {T}^+$ and $\widetilde {T}^-$ and conversely for $\widetilde {e}^{-}_0$ :

$$ \begin{align*}s(\widetilde{e}^{{\kern1.5pt}\pm}_0)=\widetilde{s(e_0)}^{\pm},\quad t(\widetilde{e}^{{\kern1.5pt}\pm}_0)=\widetilde{t(e_0)}^{\mp}. \end{align*} $$

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 ith 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 $\operatorname {\mathrm {Hom}}(H_1(G,\mathbb {Z}),\mathbb {Z}/2\mathbb {Z})=H^1(G,\mathbb {Z}/2\mathbb {Z})$ (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.

2.2 Chip-firing and linear equivalence

We now briefly recall the basic notions of divisor theory for finite and metric graphs (see [Reference Baker and NorineBN07, Section 1] and [Reference Lim, Payne and PotashnikLPP12, Section 2] respectively for details).

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

$$ \begin{align*}\operatorname{\mathrm{Div}}(G)=\left\{\sum a_v v:a_v\in \mathbb{Z}\right\},\quad \deg \sum a_v v=\sum a_v. \end{align*} $$

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

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

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

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

$$ \begin{align*}D_v=-\sum_{u\in V(G)}L_{uv}u. \end{align*} $$

Such a divisor has degree 0; hence, the set of principal divisors $\operatorname {\mathrm {Prin}}(G)$ , which are defined as the image of the chip-firing map

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

lies inside $\operatorname {\mathrm {Div}}^0(G)$ . The Picard group and Jacobian of G are defined as

$$ \begin{align*}\operatorname{\mathrm{Pic}}(G)=\operatorname{\mathrm{Div}}(G)/\operatorname{\mathrm{Prin}}(G),\quad \operatorname{\mathrm{Jac}}(G)=\operatorname{\mathrm{Div}}^0(G)/\operatorname{\mathrm{Prin}}(G). \end{align*} $$

Since any principal divisor has degree 0, the degree function descends to $\operatorname {\mathrm {Pic}}(G)$ and we denote $\operatorname {\mathrm {Pic}}^k(G)$ the set of equivalence classes of degree k divisors, so that $\operatorname {\mathrm {Jac}}(G)=\operatorname {\mathrm {Pic}}^0(G)$ . The group $\operatorname {\mathrm {Pic}}(G)$ is infinite, but $\operatorname {\mathrm {Jac}}(G)$ 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 $|\operatorname {\mathrm {Jac}}(G)|$ is equal to the number of spanning trees of G (see [Reference Baker and ShokriehBS13, Theorem 6.2]).

The Picard variety of a metric graph $\Gamma $ of genus g is defined as follows (see [Reference Baker and FaberBF11]). A divisor on a metric graph $\Gamma $ is a finite linear combination of the form

$$ \begin{align*}D = a_1 p_1 + a_2 p_2 +\cdots + a_k p_k, \end{align*} $$

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 $\operatorname {\mathrm {Div}}(\Gamma )$ the divisor group and by $\operatorname {\mathrm {Div}}^k(\Gamma )$ 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 $\operatorname {\mathrm {div}} (M)$ associated to M is the degree 0 divisor whose value at each point $p\in \Gamma $ is the sum of the incoming slopes of M at p. It is clear that $\operatorname {\mathrm {div}}(M+N)=\operatorname {\mathrm {div}}(M)+\operatorname {\mathrm {div}}(N)$ and $\operatorname {\mathrm {div}}(-M)=-\operatorname {\mathrm {div}}(M)$ , so the principal divisors $\operatorname {\mathrm {Prin}}(\Gamma )$ form a subgroup of $\operatorname {\mathrm {Div}}^0(\Gamma )$ and the degree function descends to the quotient:

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

The Picard variety $\operatorname {\mathrm {Pic}}^0(\Gamma )$ 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 $\operatorname {\mathrm {Pic}}^k(\Gamma )$ is a torsor over $\operatorname {\mathrm {Pic}}^0(\Gamma )$ .

2.3 Tropical abelian varieties

The Jacobian variety of a metric graph $\Gamma $ is a tropical principally polarised abelian variety (tropical ppav for short). We review the theory of tropical ppavs, following [Reference Foster, Rabinoff, Shokrieh and SotoFRSS18] and [Reference Len and UlirschLU19], 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 $\Lambda '$ be finitely generated free abelian groups of the same rank and let $[\cdot ,\cdot ]:\Lambda '\times \Lambda \to \mathbb {R}$ be a nondegenerate pairing. The triple $(\Lambda ,\Lambda ',[\cdot ,\cdot ])$ defines a real torus with integral structure $\Sigma =\operatorname {\mathrm {Hom}}(\Lambda ,\mathbb {R})/\Lambda '$ , where the ‘integral structure’ refers to the lattice $\operatorname {\mathrm {Hom}}(\Lambda ,\mathbb {Z})\subset \operatorname {\mathrm {Hom}}(\Lambda ,\mathbb {R})$ and where $\Lambda '$ is embedded in $\operatorname {\mathrm {Hom}}(\Lambda ,\mathbb {R})$ via the assignment $\lambda '\mapsto [\lambda ',\cdot ]$ . The transposed data $(\Lambda ',\Lambda ,[\cdot ,\cdot ]^t)$ define the dual torus $\Sigma '=\operatorname {\mathrm {Hom}}(\Lambda ',\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 $f_*:\Lambda _1'\to \Lambda _2'$ and $f^*:\Lambda _2\to \Lambda _1$ be a pair of maps satisfying

(3) $$ \begin{align} [\lambda^{\prime}_1,f^*(\lambda_2)]_1=[f_*(\lambda^{\prime}_1),\lambda_2]_2 \end{align} $$

for all $\lambda ^{\prime }_1\in \Lambda ^{\prime }_1$ and $\lambda _2\in \Lambda _2$ . The map $f^*$ defines a dual map $\overline {f}:\operatorname {\mathrm {Hom}}(\Lambda _1,\mathbb {R})\to \operatorname {\mathrm {Hom}} (\Lambda _2,\mathbb {R})$ , 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 $f':\Sigma ^{\prime }_2\to \Sigma ^{\prime }_1$ .

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 $(\operatorname {\mathrm {Ker}} f)_0$ , and the cokernel $\operatorname {\mathrm {Coker}} f$ . It is easy to see that $(\operatorname {\mathrm {Ker}} f)_0$ and $\operatorname {\mathrm {Coker}} f$ also have integral structures and the natural maps $i:(\operatorname {\mathrm {Ker}} f)_0\to \Sigma _1$ and $p:\Sigma _2\to \operatorname {\mathrm {Coker}} f$ are homomorphisms of real tori with integral structure.

Indeed, let $K=(\operatorname {\mathrm {Coker}} f^*)^{tf}$ be the quotient of $\operatorname {\mathrm {Coker}} f^*$ by its torsion subgroup (equivalently, the quotient of $\Lambda _1$ by the saturation of $\operatorname {\mathrm {Im}} f^*$ ) and let $K'=\operatorname {\mathrm {Ker}} f_*$ . Then $\operatorname {\mathrm {Hom}}(K,\mathbb {R})$ is naturally identified with the kernel of the map $\operatorname {\mathrm {Hom}}(\Lambda _1,\mathbb {R})\to \operatorname {\mathrm {Hom}} (\Lambda _2,\mathbb {R})$ dual to $f^*$ , and therefore $(\operatorname {\mathrm {Ker}} f)_0=(K,K',[\cdot ,\cdot ]_K)$ , where $[\cdot ,\cdot ]_K:K'\times K\to \mathbb {R}$ is the pairing induced by $[\cdot ,\cdot ]_1$ . We note that this pairing is well-defined: given $\lambda ^{\prime }_1\in K'$ and $\lambda _2\in \Lambda _2$ , equation (3) implies that

$$ \begin{align*}[\lambda^{\prime}_1,f^*(\lambda_2)]_1=[f_*(\lambda^{\prime}_1),\lambda_2]_2=[0,\lambda_2]_2=0. \end{align*} $$

Therefore, for $\lambda '\in K'$ and $\lambda \in K$ , the pairing $[\lambda ',\lambda ]_K=[\lambda ',\lambda ]_1$ does not depend on a choice of representative for $\lambda \in K$ . The natural maps $i^*:\Lambda _1\to K$ and $i_*:K'\to \Lambda _1'$ define $(\operatorname {\mathrm {Ker}} f)_0$ as an integral subtorus of $\Sigma _1$ . Similarly, $\operatorname {\mathrm {Coker}} f=(C,C',[\cdot ,\cdot ]_C)$ , where $C=\operatorname {\mathrm {Ker}} f^*$ , $C'=(\operatorname {\mathrm {Coker}} f_*)^{tf}$ , the pairing $[\cdot ,\cdot ]_C$ is induced by $[\cdot ,\cdot ]_2$ and p is given by the natural maps $p_*:\Lambda ^{\prime }_2\to C'$ 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 $K'$ are trivial; in other words, if $f_*$ is injective (equivalently, if $\operatorname {\mathrm {Im}} f^*$ has finite index in $\Lambda _1$ ).

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

$$ \begin{align*}(\cdot,\cdot):\Lambda'\times \Lambda'\to \mathbb{R},\quad (\lambda',\mu')=[\lambda',\xi(\mu')] \end{align*} $$

is symmetric and positive definite. Given a polarisation $\xi $ on $\Sigma $ , the pair $(\xi ,\xi )$ defines a homomorphism $\eta :\Sigma \to \Sigma '$ to the dual, whose finite kernel is identified with $\Lambda /\operatorname {\mathrm {Im}} \xi $ . The pair $(\Sigma ,\xi )$ is called a tropical polarised abelian variety. The map $\eta $ is an isomorphism if and only if $\xi $ is an isomorphism, in which case we say that the polarisation $\xi $ is principal.

Let $\Sigma =(\Lambda ,\Lambda ',[\cdot ,\cdot ])$ be a g-dimensional tropical polarised abelian variety. The associated bilinear form $(\cdot ,\cdot )$ on $\Lambda '$ extends to an inner product on the universal cover $V=\operatorname {\mathrm {Hom}}(\Lambda ,\mathbb {R})$ , 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 $\Lambda '$ , then

$$ \begin{align*}\operatorname{\mathrm{Vol}}^2(\Sigma)=\det(\lambda^{\prime}_i,\lambda^{\prime}_j). \end{align*} $$

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 polarisation $\xi _2:\Lambda ^{\prime }_2\to \Lambda _2$ on $\Sigma _2$ with associated bilinear form $(\cdot ,\cdot )_2$ , we define the induced polarisation $\xi _1:\Lambda ^{\prime }_1\to \Lambda _1$ by $\xi _1=f^*\circ \xi _2\circ f_*$ . This is indeed a polarisation, because by (3) the associated bilinear form $(\cdot ,\cdot )_1$ on $\Lambda ^{\prime }_1$ is given by

$$ \begin{align*}(\lambda^{\prime}_1,\mu^{\prime}_1)_1=[\lambda^{\prime}_1,\xi_1(\mu^{\prime}_1)]_1=[\lambda^{\prime}_1,f^*(\xi_2(f_*(\mu^{\prime}_1)))]_2= [f_*(\lambda^{\prime}_1),\xi_2(f_*(\mu^{\prime}_1))]_2=(f_*(\lambda^{\prime}_1),f_*(\mu^{\prime}_1))_2, \end{align*} $$

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

2.4 The Jacobian of a metric graph

We now construct the Jacobian variety $\operatorname {\mathrm {Jac}}(\Gamma )$ of a metric graph $\Gamma $ of genus g as a tropical ppav, following [Reference Baker and FaberBF11] and [Reference Len and UlirschLU19]. 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

$$ \begin{align*}d_A:C_1(G,A)\to C_0(G,A), \quad \sum_{e\in E(G)}a_e e\mapsto \sum_{e\in E(G)}a_e[t(e)-s(e)], \end{align*} $$

and the first simplicial homology group of G with coefficients in A is $H_1(G,A)=\operatorname {\mathrm {Ker}} d_A$ . 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 $\{de:e\in E(G)\}$ :

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

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

$$ \begin{align*}[\cdot,\cdot]:C_1(G,A) \times \Omega(G,A) \to \mathbb{R} \end{align*} $$

by

$$ \begin{align*}[\gamma, \omega]=\int_{\gamma}\omega=\sum_{e\in E(G)}\gamma_e \omega_e \ell(e),\quad \gamma=\sum_{e\in E(G)}\gamma_ee,\quad \omega=\sum_{e\in E(G)}\omega_ede. \end{align*} $$

By Lemma 2.1 in [Reference Baker and FaberBF11], the integration pairing restricts to a perfect pairing on $H_1(G,A)\times \Omega (G,A)$ .

Let $G'$ 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 $C_1(G,A)\to C_1(G',A)$ sending e to $e_1+e_2$ restricts to an isomorphism $H_1(G,A)\to H_1(G',A)$ . Similarly, the groups $\Omega (G,A)$ and $\Omega (G',A)$ 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 $\Lambda '=H_1(\Gamma ,\mathbb {Z})$ , let $[\cdot ,\cdot ]:\Lambda '\times \Lambda \to \mathbb {R}$ 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 $\Omega ^*(\Gamma )=\operatorname {\mathrm {Hom}}(\Omega (\Gamma ,\mathbb {Z}),\mathbb {R})$ , and by the universal coefficient theorem the group $\operatorname {\mathrm {Hom}}(H_1(\Gamma ,\mathbb {Z}),\mathbb {R})$ is canonically isomorphic to $H^1(\Gamma ,\mathbb {R})$ . The Jacobian variety and the Albanese variety of $\Gamma $ are the dual tropical ppavs

$$ \begin{align*}\operatorname{\mathrm{Jac}}(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb{Z}),\quad \operatorname{\mathrm{Alb}}(\Gamma)=H^1(\Gamma,\mathbb{R})/\Omega(\Gamma,\mathbb{Z}). \end{align*} $$

The group $H_1(\Gamma ,\mathbb {Z})$ carries an intersection form

(4) $$ \begin{align} (\cdot,\cdot)=[\cdot,\xi(\cdot)]:H_1(\Gamma,\mathbb{Z})\times H_1(\Gamma,\mathbb{Z})\to \mathbb{R},\quad \left(\sum_{e\in E(G)}\gamma_ee,\sum_{e\in E(G)}\delta_ee\right)=\sum_{e\in E(G)} \gamma_e\delta_e \ell(e) \end{align} $$

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 $\gamma '(q,p)$ defines the same element modulo $H_1(\Gamma ,\mathbb {Z})\subset \Omega (\Gamma )^*$ . Hence, we have a well-defined Abel–Jacobi map $\Phi _q:\Gamma \to \operatorname {\mathrm {Jac}}(\Gamma )$ with base point q:

(5) $$ \begin{align} \Phi_q:\Gamma\to \operatorname{\mathrm{Jac}}(\Gamma),\quad p\mapsto\left(\omega\mapsto \int_{\gamma(q,p)}\omega\right). \end{align} $$

The map $\Phi _q$ extends by linearity to $\operatorname {\mathrm {Div}}(\Gamma )$ , and its restriction to $\operatorname {\mathrm {Div}}^0(\Gamma )$ does not depend on the choice of base point q. The tropical analogue of the Abel–Jacobi theorem (see [Reference Mikhalkin and ZharkovMZ08], Theorem 6.3) states that $\Phi _q$ descends to a canonical isomorphism $\operatorname {\mathrm {Pic}}^0(\Gamma )\simeq \operatorname {\mathrm {Jac}}(\Gamma )$ . Since any $\operatorname {\mathrm {Pic}}^k(\Gamma )$ is a torsor over $\operatorname {\mathrm {Pic}}^0(\Gamma )$ , we can define $\operatorname {\mathrm {Vol}}(\operatorname {\mathrm {Pic}}^k(\Gamma ))=\operatorname {\mathrm {Vol}}(\operatorname {\mathrm {Jac}}(\Gamma ))$ .

Finally, we recall the principal results [Reference An, Baker, Kuperberg and ShokriehABKS14], which concern the tropical Jacobi inversion problem. Consider the degree g Abel–Jacobi map

$$ \begin{align*}\Phi:\operatorname{\mathrm{Sym}}^g(\Gamma)\to \operatorname{\mathrm{Pic}}^g(\Gamma),\quad \Phi(p_1,\ldots,p_g)=p_1+\cdots+p_g. \end{align*} $$

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

$$ \begin{align*}\operatorname{\mathrm{Sym}}^g(\Gamma)=\bigcup_{F\in \operatorname{\mathrm{Sym}}^g(E(G))}C(F), \end{align*} $$

where for a multiset $F=\{e_1,\ldots ,e_g\}\in \operatorname {\mathrm {Sym}}^g(E(G))$ of g edges of G the cell $C(F)$ consists of divisors supported on F:

$$ \begin{align*}C(F)=\{p_1+\cdots+p_g:p_i\in e_i\}. \end{align*} $$

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

   (6) $$ \begin{align} \operatorname{\mathrm{Vol}}(\Phi(C(F)))=\frac{w(F)}{\operatorname{\mathrm{Vol}}(\operatorname{\mathrm{Jac}}(\Gamma))},\quad w(F)=\operatorname{\mathrm{Vol}} (C(F))=\ell(e_1)\cdots \ell(e_g). \end{align} $$

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

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 $\operatorname {\mathrm {Pic}}^g(\Gamma )$ with no overlaps in the interior of cells and summing together their volumes gives $\operatorname {\mathrm {Vol}}(\operatorname {\mathrm {Jac}}(\Gamma ))=\operatorname {\mathrm {Vol}}(\operatorname {\mathrm {Pic}}^g(\Gamma ))$ .

Theorem 2.4 Theorem 1.5 of [Reference An, Baker, Kuperberg and ShokriehABKS14]

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

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

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

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 Subsection 2.6), finite groups do not 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 $\operatorname {\mathrm {Nm}}:\operatorname {\mathrm {Div}}(\widetilde {G})\to \operatorname {\mathrm {Div}}(G)$ and $\iota :\operatorname {\mathrm {Div}}(\widetilde {G})\to \operatorname {\mathrm {Div}}(\widetilde {G})$ given by

$$ \begin{align*}\operatorname{\mathrm{Nm}}\left(\sum a_v v\right)=\sum a_v p(v),\quad \iota\left(\sum a_v v\right)=\sum a_v \iota(v) \end{align*} $$

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

A divisor in the kernel $D\in \operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}\subset \operatorname {\mathrm {Div}}(\widetilde {G})$ has degree 0 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

$$ \begin{align*}\epsilon(D)=\deg E\bmod 2. \end{align*} $$

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

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 $D\in \operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}\subset \operatorname {\mathrm {Div}}(\widetilde {G})$ is of the form

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

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,

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

which is clearly preserved by addition.

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

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

on $\widetilde {G}$ . Its norm is $\operatorname {\mathrm {Nm}}(D)=\sum (c_{\widetilde {v}^+}+c_{\widetilde {v}^-})D_v\in \operatorname {\mathrm {Div}}(G)$ , 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 $\operatorname {\mathrm {Nm}}(D)=0$ in $\operatorname {\mathrm {Div}}(G)$ , then setting $a_v=c_{\widetilde {v}^+}-c=-c_{\widetilde {v}^-}$ , we see that

$$\begin{align*}D=cD^+ + \sum_{v\in V(G)} a_v (D_{\widetilde{v}^+} - D_{\widetilde{v}^-}), \end{align*}$$

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 $b_u$ as the number of edges between u and v in $E(G)\backslash S$ and S, respectively. Then

$$\begin{align*}(D_{\widetilde{v}^+} - D_{\widetilde{v}^-})(\widetilde{u}^{{\kern1.5pt}\pm}) = \pm(a_u-b_u), \end{align*}$$

and

$$\begin{align*}(D_{\widetilde{v}^+} - D_{\widetilde{v}^-}) (\widetilde{v}^{{\kern1.5pt}\pm})= -\sum_{u\neq v} \mp{}(a_u + b_u). \end{align*}$$

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

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

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

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

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

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

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

Example 2.9. Consider the free double cover $p:\widetilde {G}\to G$ shown in Figure 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 $|\operatorname {\mathrm {Jac}} (G')|=64$ and $|\operatorname {\mathrm {Jac}}(G)|=4$ ; therefore, $\operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}$ and $\operatorname {\mathrm {Prym}}(\widetilde {G}/G)$ have orders $16$ and $8$ , respectively. The group $\operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}$ 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 give a complete set of relations on the $D_i$ :

$$ \begin{align*}2D_1=0,\quad 8D_2=0,\quad D_4=D_1+4D_2,\quad D_3=3D_2. \end{align*} $$

It follows that $\operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}\simeq \mathbb {Z}/2\mathbb {Z}\oplus \mathbb {Z}/8\mathbb {Z}$ with generators $D_1$ and $D_2$ and hence $\operatorname {\mathrm {Prym}}(\widetilde {G}/G)\simeq \mathbb {Z}/8\mathbb {Z}$ with generator $D_1+D_2$ .

Figure 1

An example of a free double cover.

We note that the Abel–Prym map $\widetilde {G}=\operatorname {\mathrm {Sym}}^1(\widetilde {G})\to \operatorname {\mathrm {Prym}}^1(\widetilde {G}/G)$ 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 [Reference Jensen and LenJL18] and [Reference Len and UlirschLU19]). As in the finite case, the cover $\pi $ induces a surjective norm map

$$ \begin{align*}\operatorname{\mathrm{Nm}}:\operatorname{\mathrm{Pic}}^0(\widetilde{\Gamma})\to \operatorname{\mathrm{Pic}}^0(\Gamma),\quad \operatorname{\mathrm{Nm}}\left(\sum a_i \widetilde{p}_i\right) = \sum a_i\pi(\widetilde{p}_i), \end{align*} $$

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

The kernel $\operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}$ 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 $\operatorname {\mathrm {Nm}}(\widetilde {D})\simeq 0$ . Then $\operatorname {\mathrm {Nm}}(\widetilde {D}) +\operatorname {\mathrm {div}}{f} = 0$ 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 $\operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}$ has two connected components corresponding to the parity of E [Reference Jensen and LenJL18, Proposition 6.1] (note that, in the more general case when $\pi $ is a dilated unramified double cover, $\operatorname {\mathrm {Ker}} \operatorname {\mathrm {Nm}}$ has only one connected component).

The Prym variety $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ has the structure of a tropical ppav, which we now describe. Denote $\widetilde {\Lambda }=\Omega (\widetilde {\Gamma },\mathbb {Z})$ , $\widetilde {\Lambda }'=H_1(\widetilde {\Gamma },\mathbb {Z})$ , $\Lambda =\Omega (\Gamma ,\mathbb {Z})$ and $\Lambda '=H_1(\Gamma ,\mathbb {Z})$ . 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

$$ \begin{align*}\pi_*\left[\sum_{\widetilde{e}\in E(\widetilde{G})} a_{\widetilde{e}} \widetilde{e}\right]=\sum_{\widetilde{e}\in E(\widetilde{G})} a_{\widetilde{e}} \pi(\widetilde{e}),\quad \pi^* \left[\sum_{e\in E(G)}a_e de\right]= \sum_{e\in E(G)}a_e (d\widetilde{e}^{{\kern2pt}+}+d\widetilde{e}^{{\kern2pt}-}). \end{align*} $$

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 $\operatorname {\mathrm {Nm}}:\operatorname {\mathrm {Jac}}(\widetilde {\Gamma })\to \operatorname {\mathrm {Jac}}(\Gamma )$ of real tori with integral structure (by Proposition 2.2.3 in [Reference Len and UlirschLU19], this homomorphism is identified with the norm homomorphism $\operatorname {\mathrm {Nm}}:\operatorname {\mathrm {Pic}}^0(\widetilde {\Gamma })\to \operatorname {\mathrm {Pic}}^0(\Gamma )$ under the Abel–Jacobi isomorphism, justifying the notation). Hence, $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ is in fact the real torus with integral structure $(\operatorname {\mathrm {Ker}}\operatorname {\mathrm {Nm}})_0=(K,K',[\cdot ,\cdot ]_K)$ , where $K=(\operatorname {\mathrm {Coker}} \pi ^*)^{tf}$ , $K'=\operatorname {\mathrm {Ker}} \pi _*$ and $[\cdot ,\cdot ]_K$ is the pairing induced by the integration pairing on $\widetilde {\Gamma }$ . Alternatively, we can describe $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ as the quotient

$$ \begin{align*}\operatorname{\mathrm{Prym}}(\widetilde{\Gamma}/\Gamma)=\frac{\operatorname{\mathrm{Ker}} \overline{\pi}:\Omega^*(\widetilde{\Gamma})\to \Omega^*(\Gamma)}{\operatorname{\mathrm{Ker}} \pi_*:H_1(\widetilde{\Gamma},\mathbb{Z})\to H^1(\Gamma,\mathbb{Z})}, \end{align*} $$

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

The polarisation $\widetilde {\xi }:H_1(\widetilde {\Gamma },\mathbb {Z})\to \Omega (\widetilde {\Gamma },\mathbb {Z})$ on $\operatorname {\mathrm {Jac}}(\widetilde {\Gamma })$ induces a polarisation $i^*\widetilde {\xi }:K'\to K$ on $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ , and Theorem 2.2.7 in [Reference Len and UlirschLU19] states that there exists a principal polarisation $\psi :K'\to K$ on $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ such that $i^*\widetilde {\xi }=2\psi $ . Hence, $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ is a tropical ppav. We note that the inner product $(\cdot ,\cdot )_P$ on $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ induced by the principal polarisation $\psi $ is half of the restriction of the inner product $(\cdot ,\cdot )_{\widetilde {\Gamma }}$ from $\operatorname {\mathrm {Jac}}(\widetilde {\Gamma })$ . In other words, for $\gamma , \delta \in \operatorname {\mathrm {Ker}} \pi _*$ we have

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

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

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

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

for $\operatorname {\mathrm {Ker}} \pi _*:H_1(\widetilde {\Gamma },\mathbb {Z})\to H_1(\Gamma ,\mathbb {Z})$ . The principal polarisation $\psi =\frac {1}{2}\widetilde {\xi }$ gives a corresponding basis of the second lattice $(\operatorname {\mathrm {Coker}} \pi ^*)^{tf}$ :

$$ \begin{align*}\omega_j=\psi(\widetilde{\gamma}_j)=\frac{1}{2}\sum_{\widetilde{e}\in E(\widetilde{G})} \widetilde{\gamma}_{j,\widetilde{e}}d\widetilde{e},\quad j=1,\ldots,g-1. \end{align*} $$

Let $\omega ^*_j$ denote the basis of $\operatorname {\mathrm {Ker}} \overline {\pi }:\Omega ^*(\widetilde {\Gamma })\to \Omega ^*(\Gamma )$ dual to the $\omega _j$ , so that $\omega ^*_j(\omega _k)=\delta _{jk}$ ; then elements of $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ 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 $\operatorname {\mathrm {Vol}}(\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma ))=\sqrt {\det G}$ , 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 $\operatorname {\mathrm {Ker}} \overline {\pi }$ , 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

(9) $$ \begin{align} \operatorname{\mathrm{Vol}}(C(\omega^*_1,\ldots,\omega^*_{g-1}))=\det (\omega^*_i,\omega^*_j)=\det G^{-1} \det(\widetilde{\gamma}_i,\widetilde{\gamma}_j)\det G^{-1}=\frac{1}{\det G}=\frac{1}{\operatorname{\mathrm{Vol}}(\operatorname{\mathrm{Prym}}(\widetilde{\Gamma}/\Gamma))}. \end{align} $$

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

2.7 Polyhedral spaces and harmonic morphisms

The spaces $\operatorname {\mathrm {Sym}}^d(\Gamma )$ , $\operatorname {\mathrm {Jac}}(\Gamma )$ and $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ are examples of rational polyhedral spaces, which are topological spaces locally modelled 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 [Reference Mikhalkin and ZharkovMZ14], [Reference Gross and ShokriehGS19] 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 1 if the complement in P of all cells of codimension 2 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 [Reference Len and RanganathanLR18]] Let $f:P\to Q$ be a map of equidimensional polyhedral spaces of the same dimension and let $\deg _f$ be a nonnegative 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 1 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 0 cell N of Q adjacent to D, the sum

(10) $$ \begin{align} \deg_f(C)=\sum_{M\subset f^{-1}(N),\, M\supset C} \deg_f (M) \end{align} $$

of the degrees $\deg _f M$ over all codimension 0 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 1 cell of P and, in addition, if $f(C)=0$ on a codimension 0 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 1 cells of P. If Q is connected through codimension 1, 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 fibres: given $p\in P$ and an open neighbourhood $V\ni f(p)$ , there exists an open neighbourhood $U\ni p$ 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 1 is surjective and has a well-defined global degree, which is the sum of the degrees over all points of any fibre.

The structure of a rational polyhedral space on a tropical abelian variety, such as $\operatorname {\mathrm {Jac}}(\Gamma )$ and $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ , 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 $\operatorname {\mathrm {Sym}}^d(\Gamma )$ , however, requires some care. The space $\operatorname {\mathrm {Sym}}^d(\Gamma )$ 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 equation (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 $\operatorname {\mathrm {Sym}}^d(\Gamma )$ is given in the paper [Reference Brandt and UlirschBU18]. However, neither of the aforementioned issues arise in our article. First, we can always choose a loopless model for any tropical curve. Second, and more important, all of the calculations in this article 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 $\Psi ^d:\operatorname {\mathrm {Sym}}(\widetilde {\Gamma })\to \operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ associated to a double cover $\widetilde {\Gamma }\to \Gamma $ , and one of our first results (Theorem 4.1 part (1)) states that $\Psi ^d$ contracts all of those cells $C(F)$ where F has repeated edges. Hence, all top-dimensional cells of the symmetric product $\operatorname {\mathrm {Sym}}^d(\Gamma )$ can be assumed to be parallelotopes.

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 [Reference TerrasTer10] 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; that is, 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

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

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 [Reference BassBas92] (see also [Reference TerrasTer10]), expresses the reciprocal $\zeta (s,G)^{-1}$ as an explicit polynomial

$$ \begin{align*}\zeta(s,G)^{-1}=(1-s^2)^{g-1}\det(I_n-As+(Q-I_n)s^2), \end{align*} $$

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 [Reference NorthshieldNor98] shows that the leading Taylor coefficient computes the complexity; that is, the order of the Jacobian of G:

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

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 [Reference Stark and TerrasST96] and [Reference Stark and TerrasST00]) 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

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

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).

Similar 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 $nd\times nd$ Artinised valency and Artinised adjacency matrices as

$$ \begin{align*}Q_{\rho}=Q\otimes I_d,\quad (A_{\rho})_{uv}=\sum\rho(F(e)), \end{align*} $$

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

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

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 respectively at the right regular and trivial representations $\rho _K$ and $1_K$ :

$$ \begin{align*}\zeta(s,\widetilde{G})=L(s,\rho_K,\widetilde{G}/G),\quad \zeta(s,G)=L(s,1_K,\widetilde{G}/G). \end{align*} $$

Furthermore, for a reducible representation $\rho =\rho _1\oplus \rho _2$ the L-function factors as

$$ \begin{align*}L(s,\rho,\widetilde{G}/G)=L(s,\rho_1,\widetilde{G}/G)L(s,\rho_2,\widetilde{G}/G). \end{align*} $$

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

(13) $$ \begin{align} \zeta(s,\widetilde{G})=\zeta(s,G)\prod_{\rho}L(s,\rho,\widetilde{G}/G)^{d(\rho)}, \\[-15pt]\nonumber \end{align} $$

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

3.2 The order of the Prym group

We now specialise 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 $s=1$ respectively compute the orders $|\operatorname {\mathrm {Jac}}(\widetilde {G})|$ and $|\operatorname {\mathrm {Jac}}(G)|$ . 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 [Reference An, Baker, Kuperberg and ShokriehABKS14], the Jacobian group of a graph G of genus g (and, by extension, the Jacobian variety of a metric graph) admits a noncanonical 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 $\operatorname {\mathrm {Prym}}(\widetilde {G}/G)$ and control the geometry of the Prym varieties of double covers of metric graphs.

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 1, then F consists of $g-1$ edges and a genus 1 decomposition cannot have rank greater than g.

A genus 1 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 1 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 $\operatorname {\mathrm {Prym}}(\widetilde {G}/G)$ is equal to

(14) $$ \begin{align} |\operatorname{\mathrm{Prym}}(\widetilde{G}/G)|=\frac{1}{2}|\operatorname{\mathrm{Ker}} \operatorname{\mathrm{Nm}}|=\sum_{r=1}^g 4^{r-1}C_r, \\[-15pt]\nonumber\end{align} $$

where $C_r$ is the number of odd genus 1 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 $\mathbb {Z}/2\mathbb {Z}$ :

$$ \begin{align*}\zeta(s,\widetilde{G})^{-1}=\zeta(s,G)^{-1}L(s,\widetilde{G}/G,\rho)^{-1}.\\[-15pt] \end{align*} $$

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

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

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

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

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

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

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

We now calculate this $n\times n$ determinant. First of all, $Q_{\rho }=Q$ since $\rho $ is 1-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:

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

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 [Reference Reiner and TsengRT14]), and its determinant is computed using a standard argument involving an appropriate factorisation 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 factorise the signed Laplacian as $Q_\rho -A_\rho =B_S(G){}^tB_S(G)$ , where

$$ \begin{align*}(B_S(G))_{ve}=\left\{\begin{array}{c@{\quad}c} 1, & t(e)=v\ \text{and}\ s(e)\neq v, \text{or}\ s(e)=v, t(e)\neq v\ \text{and}\ e\in S,\\ -1, & s(e)=v, t(e)\neq v\ \text{and}\ e\notin S,\\ 2, & s(e)=t(e)=v\ \text{and}\ e\in S,\\ 0, & \text{otherwise}\end{array}\right. \end{align*} $$

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

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

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

(16) $$ \begin{align} \det B_S(G\backslash F)^2=\prod_{k=0}^{r-1} \det B_S(G_k)^2. \end{align} $$

The quantity $\det B_S(G_k)^2$ for a genus 1 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 1 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 1 decomposition. The edges $e_1$ , $e_2$ , $e_4$ and $e_5$ are decompositions of rank 1, while $e_3$ is a decomposition of rank 2. Hence, by (14)

$$ \begin{align*}|\operatorname{\mathrm{Prym}}(\widetilde{G}/G)|=4+1\cdot 4=8, \end{align*} $$

which agrees with the calculations in Example 2.9.

3.3 The volume of the tropical Prym variety

In this subsection, 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 $ . Similar to the discrete case, an odd genus 1 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 1, 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

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

where the sum is taken over all odd genus 1 decompositions F of $\Gamma $ .

We first establish the relationship between the volumes of the three tropical ppavs $\operatorname {\mathrm {Jac}}(\widetilde {\Gamma })$ , $\operatorname {\mathrm {Jac}}(\Gamma )$ and $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ . 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

(18) $$ \begin{align} \langle\cdot,\cdot\rangle:C_1(G,\mathbb{Z})\times C_1(G,\mathbb{Z})\to \mathbb{Z},\quad \left\langle \sum_{e\in E(G)} a_e e,\sum_{e\in E(G)} b_e e\right\rangle=\sum_{e\in E(G)}a_eb_e. \end{align} $$

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

$$ \begin{align*}\mathcal{B}=\{\gamma_0,\ldots,\gamma_{g-1}\} \end{align*} $$

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:

$$ \begin{align*}\gamma=\langle\gamma,e_1\rangle \gamma_1+\cdots+\langle \gamma,e_g\rangle \gamma_g. \end{align*} $$

Similarly, let $\widetilde {\gamma }_0\in H_1(\widetilde {G},\mathbb {Z})$ and $\widetilde {\gamma }_i^{{\kern1.5pt}\pm }\in H_1(\widetilde {G},\mathbb {Z})$ for $i=1,\ldots ,g-1$ denote the unique cycle respectively of $\widetilde {T}\cup \{\widetilde{e}^{{\kern2pt}-}_0\}$ and $\widetilde {T}\cup \{\widetilde {e}_i^{{\kern1.5pt}\pm }\}$ such that $\langle \widetilde {\gamma }_0,\widetilde{e}^{{\kern2pt}-}_0\rangle =1$ and $\langle \widetilde {\gamma }^{\pm }_i,\widetilde {e}^{{\kern1.5pt}\pm }_i\rangle =1$ respectively for $i=1,\ldots ,g-1$ . Then

$$ \begin{align*}\widetilde{\mathcal{B}}=\{\widetilde{\gamma}_0,\widetilde{\gamma}_1^{{\kern1.5pt}\pm},\ldots,\widetilde{\gamma}_{g-1}^{{\kern1.5pt}\pm}\} \end{align*} $$

is a basis of $H_1(\widetilde {G},\mathbb {Z})$ and we similarly have

$$ \begin{align*}\widetilde{\gamma}=\langle \widetilde{\gamma},\widetilde{e}^{{\kern2pt}-}_0\rangle\widetilde{\gamma}_0+\langle\widetilde{\gamma},\widetilde{e}^{{\kern2pt}+}_1\rangle \widetilde{\gamma}^+_1+\cdots+\langle \widetilde{\gamma},\widetilde{e}^{{\kern2pt}+}_{g-1}\rangle \widetilde{\gamma}^+_{g-1}+\langle\widetilde{\gamma},\widetilde{e}^{{\kern2pt}-}_1\rangle \widetilde{\gamma}^-_1+\cdots+\langle \widetilde{\gamma},\widetilde{e}^{{\kern2pt}-}_{g-1}\rangle \widetilde{\gamma}^-_{g-1} \end{align*} $$

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 $\widetilde {T}^-$ and 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 $t(\widetilde {e}_0^{{\kern1.5pt}+})=\widetilde {t(e_0)}^-$ on $T^-$ via $+\widetilde {e}_0^{{\kern1.5pt}+}$ and then back to $\widetilde {s(e_0)}^-$ via a unique path in $\widetilde {T}^-$ . In other words,

$$ \begin{align*}\widetilde{\gamma}_0=\widetilde{e}_0^++\widetilde{e}_0^-+\mbox{edges of }\widetilde{T}^{\pm},\quad \iota_*(\widetilde{\gamma}_0)=\widetilde{e}_0^++\widetilde{e}_0^-+\mbox{edges of }\widetilde{T}^{\pm},\quad p_*(\widetilde{\gamma}_0)=2e_0+\mbox{edges of }T; \end{align*} $$

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

$$ \begin{align*}\iota_*(\widetilde{\gamma}_0)=\widetilde{\gamma}_0,\quad p_*(\widetilde{\gamma}_0)=2\gamma_0. \end{align*} $$

Now consider the cycle $\widetilde {\gamma }_i^+$ for $e_i\in S\backslash \{e_0\}$ . We introduce the index

$$ \begin{align*}\sigma_i=\left\{\begin{array}{c@{\quad}c} +1,& s(\widetilde{e}_i^+)=s(e_i)^+,\\ -1, & s(\widetilde{e}_i^+)=s(e_i)^-.\end{array}\right. \end{align*} $$

If $\sigma _i=1$ , then the cycle $\widetilde {\gamma }_i^+$ starts at $s(\widetilde {e}_i^{{\kern1.5pt}+})=\widetilde {s(e_i)}^+$ on $\widetilde {T}^+$ and then moves to $t(\widetilde {e}_i^{{\kern1.5pt}+})=\widetilde {t(e_i)}^-$ on $\widetilde {T}^-$ via $\widetilde {e}_i^{{\kern1.5pt}+}$ 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 $-\widetilde {e}_0^{{\kern1.5pt}+}$ . If $\sigma _i=-1$ , then $\widetilde {\gamma }_i^+$ crosses from $\widetilde {T}^+$ to $\widetilde {T}^-$ and hence contains $\widetilde {e}_0^{{\kern1.5pt}+}$ . Similarly, we calculate that the cycle $\widetilde {\gamma }_i^-$ contains the edge $\sigma _i\widetilde {e}_0^{{\kern1.5pt}+}$ . In other words, for $e_i\in S\backslash \{e_0\}$ we have

$$ \begin{align*}\widetilde{\gamma}_i^{{\kern1.5pt}\pm}=\widetilde{e}_i^{{\kern1.5pt}\pm}\mp \sigma_i\widetilde{e}_0^++\mbox{edges of }\widetilde{T}^{\pm},\quad \iota_*(\widetilde{\gamma}_i^{{\kern1.5pt}\pm})=\widetilde{e}_i^{\mp}\mp \sigma_i\widetilde{e}_0^-+\mbox{edges of }\widetilde{T}^{\pm},\ \ p_*(\widetilde{\gamma}^{\pm}_i)=e_i\mp \sigma_ie_0+\mbox{edges of }T; \end{align*} $$

hence, computing the intersection numbers, we see that

$$ \begin{align*}\iota_*(\widetilde{\gamma}_i^{{\kern1.5pt}\pm})=\widetilde{\gamma}_i^{\mp}\mp \sigma_i\widetilde{\gamma}_0,\quad p_*(\widetilde{\gamma}^{\pm}_i)=\gamma_i\mp \sigma_i\gamma_0,\quad e_i\in S\backslash\{e_0\}. \end{align*} $$

Finally, for $e_i\notin S$ the cycle $\gamma _i^{\pm }$ is contained in $\widetilde {T}^{\pm }\cup \{\widetilde {e}_i^{{\kern1.5pt}\pm }\}$ and hence does not contain the edge $\widetilde {e}_0^{{\kern1.5pt}+}$ . It follows that

$$ \begin{align*}\iota_*(\widetilde{\gamma}_i^{{\kern1.5pt}\pm})=\widetilde{e}_i^{\mp}+\mbox{edges of }\widetilde{T}^{\pm},\quad p_*(\widetilde{\gamma}_i^{{\kern1.5pt}\pm})=e_i+\mbox{edges of }T,\quad e_i\notin S, \end{align*} $$

and therefore

$$ \begin{align*}\iota_*(\widetilde{\gamma}_i^{{\kern1.5pt}\pm})=\widetilde{\gamma}_i^{\mp},\quad p_*(\widetilde{\gamma}_i^{{\kern1.5pt}\pm})=\gamma_i,\quad e_i\notin S. \end{align*} $$

It is now clear that

(19) $$ \begin{align} \widetilde{\mathcal{B}}^{\prime}_2=\{\widetilde{\gamma}_i^+-\iota_*(\widetilde{\gamma}_i^+)\}_{i=1}^{g-1}= \{\widetilde{\gamma}_i^+-\widetilde{\gamma}_i^-+\sigma_i\widetilde{\gamma}_0\}_{e_i\in S\backslash\{e_0\}}\cup \{\widetilde{\gamma}_i^+-\widetilde{\gamma}_i^-\}_{e_i\notin S} \end{align} $$

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

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 $\operatorname {\mathrm {Jac}}(\widetilde {\Gamma })$ , $\operatorname {\mathrm {Jac}}(\Gamma )$ and $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ are related as

$$ \begin{align*}\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Prym}}(\widetilde{\Gamma}/\Gamma))=\frac{\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\widetilde{\Gamma}))}{2\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\Gamma))}, \end{align*} $$

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

Proof. We first introduce the following alternative basis $\mathcal {B}'$ for $H_1(G,\mathbb {Z})$ :

(20) $$ \begin{align} \mathcal{B}'=\{\gamma_0\}\cup\{\gamma_i-\sigma_i\gamma_0\}_{e_i\in S\backslash \{e_0\} }\cup \{\gamma_i\}_{e_i\notin S}. \end{align} $$

We now compute the pullback $\widetilde {\mathcal {B}}^{\prime }_1$ of $\mathcal {B}'$ to $H_1(\widetilde {G},\mathbb {Z})$ via the map

$$ \begin{align*}p^*:H_1(G,\mathbb{Z})\to H_1(\widetilde{G},\mathbb{Z}),\quad \sum_{e\in E(G)} a_e e\mapsto \sum_{e\in E(G)} a_e (\widetilde{e}^{{\kern2pt}+}+\widetilde{e}^{{\kern2pt}-}). \end{align*} $$

Since $\gamma _i$ consists of $+e_i$ and edges of T, we have

$$ \begin{align*}p^*(\gamma_i)=\widetilde{e}^{{\kern2pt}+}_i+\widetilde{e}^{{\kern2pt}-}_i+\mbox{edges of }T^{\pm} \end{align*} $$

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

$$ \begin{align*}p^*(\gamma_0)=\widetilde{\gamma}_0,\quad p^*(\gamma_i)=\widetilde{\gamma}^+_i+\widetilde{\gamma}^-_i,\quad i=1,\ldots,g-1. \end{align*} $$

Hence,

$$\begin{align*}\widetilde{\mathcal{B}}^{\prime}_1=p^*(\mathcal{B}')=\{\widetilde{\gamma}_0\}\cup \{\widetilde{\gamma}_i^++\widetilde{\gamma}_i^--\sigma_i\widetilde{\gamma}_0\}_{e_i\in S\backslash\{e_0\}}\cup \{\widetilde{\gamma}_i^++\widetilde{\gamma}_i^-\}_{e_i\notin S}. \end{align*}$$

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 $\operatorname {\mathrm {Ker}} p_*$ corresponding to the principal polarisation on $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ . 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 $\operatorname {\mathrm {Jac}}(G)$ and $\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma )$ are given by

$$ \begin{align*}\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\Gamma))=\operatorname{\mathrm{Gram}}_G(\mathcal{B}'),\quad \operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Prym}}(\widetilde{\Gamma}/\Gamma))=\operatorname{\mathrm{Gram}}_P(\widetilde{\mathcal{B}}^{\prime}_2). \end{align*} $$

We now consider the set $\widetilde {\mathcal {B}}'=\widetilde {\mathcal {B}}^{\prime }_1\cup \widetilde {\mathcal {B}}^{\prime }_2$ , 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 $\widetilde {\mathcal {B}}'$ 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

$$\begin{align*}\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\widetilde{\Gamma}))=\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}})=2^{2-2g}\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}'). \end{align*}$$

We now compute $\operatorname {\mathrm {Gram}}(\widetilde {\mathcal {B}}')$ 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,

$$ \begin{align*}\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}')=\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_1)\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_2). \end{align*} $$

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

$$ \begin{align*}\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_2)=2^{g-1}\operatorname{\mathrm{Gram}}_P(\widetilde{\mathcal{B}}^{\prime}_2). \end{align*} $$

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

$$ \begin{align*}\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_1)=2^g\operatorname{\mathrm{Gram}}_G(\mathcal{B}'), \end{align*} $$

because $\widetilde {\mathcal {B}}^{\prime }_1$ is the pullback of $\mathcal {B}'$ . Putting all this together, we have

$$ \begin{align*}\frac{\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\widetilde{\Gamma}))}{2\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\Gamma))}=\frac{2^{2-2g}\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}')}{2\operatorname{\mathrm{Gram}}_G(\mathcal{B}')}= \frac{2^{1-2g}\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_1)\operatorname{\mathrm{Gram}}_{\widetilde{G}}(\widetilde{\mathcal{B}}^{\prime}_2)}{\operatorname{\mathrm{Gram}}_G(\mathcal{B}')}=\operatorname{\mathrm{Gram}}_P(\widetilde{\mathcal{B}}^{\prime}_2), \end{align*} $$

which is equal to $\operatorname {\mathrm {Vol}}^2(\operatorname {\mathrm {Prym}}(\widetilde {\Gamma }/\Gamma ))$ , 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 1. In this case, $\operatorname {\mathrm {Vol}}(F)=1$ for any set of edges; hence, by Kirchhoff’s theorem and (7) we have

$$ \begin{align*}\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\widetilde{\Gamma}))=|\operatorname{\mathrm{Jac}}(\widetilde{G})|,\quad \operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Jac}}(\Gamma))=|\operatorname{\mathrm{Jac}}(G)|. \end{align*} $$

It follows by Proposition 3.6 that

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

But $|\operatorname {\mathrm {Prym}}(\widetilde {G}/G)|$ can be computed using (14), which agrees with the right-hand side of (17) when all edge lengths are equal to 1. This completes the proof.

Example 3.7. Let $\Gamma $ be the genus 2 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).

Figure 2

Two free double covers of the dumbbell graph. Flipped edges are blue.

For the cover $\pi _1$ , the odd genus 1 decompositions are $\{e_1\}$ and $\{e_2\}$ of rank 1 and $\{e_3\}$ of rank 2. For $\pi _2$ , the only odd genus 1 decomposition is $\{e_2\}$ of rank 1. Hence, Theorem 3.4 states that

$$ \begin{align*}\operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Prym}}(\widetilde{\Gamma}_1/\Gamma))=x_1+x_2+4x_3,\quad \operatorname{\mathrm{Vol}}^2(\operatorname{\mathrm{Prym}}(\widetilde{\Gamma}_2/\Gamma))=x_2. \\[-15pt]\end{align*} $$

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