Tropical Prym-Torelli Morphism

• The tropical Prym–Torelli morphism is defined as the moduli assignment sending an unramified cyclic cover of a tropical curve to the connected component of the kernel of the norm map, forming a principally polarized tropical abelian variety.
• It leverages the structure of tropical Jacobians, norm maps, and Galois symmetries to convert covering data into computable combinatorial invariants and volume formulas.
• The theory addresses injectivity issues and boundary pathologies via refined combinatorial models and piecewise-polynomial corrections, extending classical Prym theory to multiple cyclic cover settings.

The tropical Prym–Torelli morphism is the moduli-theoretic assignment that sends a tropical covering to its tropical Prym variety, viewed as a principally polarized tropical abelian variety. In its most general form currently available for unramified Galois cyclic covers of tropical curves or metric graphs, one starts with a cover

$\varphi:\widetilde{\Gamma}\to \Gamma$

and assigns

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$

the connected component of the identity in the kernel of the norm map. In the earlier double-cover literature, the same idea appears as a map

$\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$

sending a harmonic double cover to its principally polarized tropical Prym. The subject lies at the intersection of tropical Jacobians, norm maps, Abel–Prym geometry, signed-matroid combinatorics, and tropicalization of algebraic Prym theory; its central issues are the construction of the target ppav, the extent to which the covering is recoverable from that target, and the failure of naively defined Prym data to behave well under all degenerations (Mohajer, 2 Apr 2026, Len, 2022, Len et al., 2019).

1. Moduli-theoretic formulation

For an unramified cyclic cover of degree $n$,

$\varphi:\widetilde{\Gamma}\to\Gamma,$

the tropical Jacobian is

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$

with its natural polarization coming from the integration pairing on harmonic $1$-forms and cycles. The norm map

$\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$

induces the tropical Prym variety

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$

Recent work formulates the tropical Prym–Torelli-type morphism precisely as

$(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$

with source the moduli of unramified cyclic covers of tropical curves or metric graphs and target the moduli of principally polarized tropical abelian varieties up to lattice equivalence preserving the polarization (Mohajer, 2 Apr 2026).

In the double-cover setting, the same moduli idea is stated as

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$0

where $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$1 denotes the moduli space of harmonic double covers of graphs of genus $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$2, and $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$3 denotes the moduli space of principally polarized tropical abelian varieties of dimension $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$4. This formulation is accompanied by a divisor-theoretic realization of the kernel of the norm map and by the observation that, unlike the classical Jacobian Torelli map, the tropical Prym map is not expected to be uniformly well-behaved on the boundary (Len, 2022).

The cyclic-cover framework is explicitly group-theoretic. If $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$5 is a principally polarized tropical abelian variety, then a finite group $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$6 acts on $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$7 if and only if there is a monomorphism

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$8

This ambient viewpoint is essential because the Prym is constructed as the anti-invariant part of the Jacobian under the Galois action (Mohajer, 2 Apr 2026).

2. Construction from norm maps and Galois symmetries

If $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ,$9 is generated by an automorphism $\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$0 of $\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$1, then $\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$2 acts on $\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$3, and the group-ring operator

$\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$4

encodes the invariant part of the tropical Jacobian. The key identities are

$\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$5

together with

$\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$6

For cyclic covers one has the fundamental formula

$\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$7

so the Prym is exactly the connected anti-invariant factor of the Jacobian (Mohajer, 2 Apr 2026).

The same object admits the quotient description

$\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$8

and, equivalently,

$\mathcal{R}_g^{\trop}\to \mathcal{A}_{g-1}^{\trop},$9

where $n$0 denotes the non-invariant part under the $n$1-action. Up to isogeny,

$n$2

These formulas make the morphism representation-theoretic rather than merely divisor-theoretic (Mohajer, 2 Apr 2026).

The polarization is subtle. For cyclic covers, the tropical Prym is again a principally polarized tropical abelian variety, and the induced inner product is scaled by

$n$3

For unramified double covers in the skeleton framework, the tropical norm construction exhibits a dichotomy absent from the classical statement: if the dilation cycle is trivial, then $n$4 has two connected components and the induced polarization satisfies

$n$5

if the dilation cycle is non-trivial, then $n$6 is connected and the induced polarization is already principal (Len et al., 2019). For harmonic double covers with dilation, later work emphasizes that the induced polarization on $n$7 is not necessarily principal, even though there exists a natural principal polarization on the Prym torus (Ghosh et al., 2023). This divergence of formalisms is central to the modern literature.

3. Abel–Prym maps and tropicalization compatibility

For cyclic unramified covers, the tropical Abel–Prym map is defined by composing the tropical Abel–Jacobi map

$n$8

with the anti-invariant projector: $n$9 This extends the double-cover Abel–Prym map from earlier work to arbitrary cyclic covers. Conceptually, $\varphi:\widetilde{\Gamma}\to\Gamma,$0 embeds the covering graph into its Jacobian, and $\varphi:\widetilde{\Gamma}\to\Gamma,$1 extracts the anti-invariant geometry determined by the Galois symmetry (Mohajer, 2 Apr 2026).

For free double covers, the Abel–Prym map is developed in much greater detail. In divisor notation,

$\varphi:\widetilde{\Gamma}\to\Gamma,$2

is given by

$\varphi:\widetilde{\Gamma}\to\Gamma,$3

The case $\varphi:\widetilde{\Gamma}\to\Gamma,$4 is especially rigid: $\varphi:\widetilde{\Gamma}\to\Gamma,$5 is surjective, it is harmonic, and its global degree is

$\varphi:\widetilde{\Gamma}\to\Gamma,$6

On a top-dimensional cell indexed by an odd genus one decomposition $\varphi:\widetilde{\Gamma}\to\Gamma,$7, the local degree is

$\varphi:\widetilde{\Gamma}\to\Gamma,$8

and it vanishes on the contracted cells. This gives a precise tropical analogue of the algebraic degree of the Abel–Prym map (Len et al., 2020, Len, 2022).

The strongest tropicalization theorem identifies the tropical target of the Prym–Torelli morphism. For an unramified double cover

$\varphi:\widetilde{\Gamma}\to\Gamma,$9

over a non-Archimedean field, with associated tropical cover

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$0

there is a canonical isomorphism of principally polarized tropical abelian varieties

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$1

Moreover,

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$2

and the diagram comparing algebraic and tropical Abel–Prym maps commutes after retraction to the skeleton. At the moduli level, this yields the commutative tropicalization diagram for the classical Prym map $\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$3 and its tropical counterpart (Len et al., 2019). A plausible implication is that tropical Prym–Torelli phenomena should be interpreted as the non-Archimedean shadows of algebraic Prym–Torelli phenomena, but not as automatic tropical analogues of algebraic injectivity statements.

4. Combinatorial models, matroids, and quantitative invariants

A major feature of the tropical theory is that the Prym–Torelli morphism is computable from combinatorial data. For harmonic double covers, one associates a signed graphic matroid

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$4

identified with Zaslavsky’s signed graphic matroid, together with an orientation, an index function

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$5

and the edge-length function. From these data one reconstructs the principalization

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$6

and the central theorem states that

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$7

The same paper proves that the principalized Prym is invariant under simplification of the cover or matroid (Röhrle et al., 2023).

The volume theory is equally explicit. For free double covers, there is a Kirchhoff-type formula

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$8

where the sum runs over odd genus one decompositions. For harmonic double covers with dilation, the formula becomes

$\Jac(\Gamma)=\Omega(\Gamma)^*/H_1(\Gamma,\mathbb Z),$9

and also

$1$0

The factor $1$1 records the number of connected components of the dilation subgraph and already signals that the moduli behavior of the Prym differs from that of the Jacobian (Len et al., 2020, Ghosh et al., 2023).

The recent cyclic-cover generalization preserves this computational character. For a free $1$2-cover one has

$1$3

and the polarization scaling

$1$4

leads to

$1$5

In the worked genus-$1$6 example,

$1$7

hence

$1$8

This extends the double-cover volume computations to a genuinely higher cyclic symmetry (Mohajer, 2 Apr 2026).

A further quantitative refinement is the second moment of the tropical Prym. For genus $1$9, the formula

$\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$0

contains a polynomial term $\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$1 and a piecewise-polynomial correction term $\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$2. The correction term has no Jacobian analogue and is tied to changes in the Voronoi polytope of the Prym (Zakharov, 8 Jul 2025). This suggests that the tropical Prym–Torelli morphism is controlled not only by signed-matroid data but also by wall-crossing phenomena in the cone of edge lengths.

5. Injectivity, non-injectivity, and boundary pathologies

The tropical Prym–Torelli morphism is not presently governed by a general injectivity theorem. In the cyclic-cover paper, the construction is explicitly described as “Prym–Torelli-type” and is primarily definitional and computational rather than a proof of a full Torelli theorem (Mohajer, 2 Apr 2026). The double-cover literature makes the same point in more concrete form: one can define the moduli map and analyze its fibers in special regimes, but a general tropical Torelli theorem for Pryms is not established (Len, 2022).

There are, however, precise non-injectivity results. The tropical analogue of Donagi’s tetragonal theorem proves that for orientable generic towers the tetragonal construction is a triality, and if the three resulting tropical curves are connected and the base is a tree, then

$\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$3

as principally polarized tropical abelian varieties. Since the three towers are distinct, the tropical Prym–Torelli morphism is not injective in general (Röhrle et al., 8 Jul 2025). The earlier tropical $\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$4-gonal construction already produced the related trigonal/Recillas isomorphism

$\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$5

for free double covers of trigonal tropical curves, showing that Pryms can coincide with Jacobians of tetragonal curves and thereby supporting the same non-uniqueness phenomenon (Röhrle et al., 2022).

Matroidal analysis reveals a second source of non-injectivity. If two double covers have isomorphic decorated signed-matroid data, then their principalized Pryms are isomorphic; however, the converse fails in strong form: different double covers can have the same Prym variety, and even non-isomorphic simple matroids can yield isomorphic Pryms (Röhrle et al., 2023). The fibers of the tropical Prym–Torelli morphism therefore contain genuinely combinatorial redundancy.

Boundary behavior is also problematic. For dilated double covers, the tropical Prym variety behaves discontinuously under deformations that change the number of connected components of the dilation subgraph. The concrete example in which

$\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$6

jumps after an edge contraction to

$\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$7

shows that the divisorial Prym does not vary continuously under all natural contractions (Ghosh et al., 2023). The continuous Prym introduced later is designed to behave well under edge contraction and in moduli, and satisfies

$\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$8

The same work relates the piecewise-polynomial correction term in the second moment to the problem of extending the Prym–Torelli map to the second Voronoi compactification, and identifies degenerations of $\Nm_\varphi:\Pic^0(\widetilde{\Gamma})\to \Pic^0(\Gamma)$9-covers as the source of indeterminacy (Zakharov, 8 Jul 2025). This suggests that any satisfactory compactified tropical Prym–Torelli theory must distinguish carefully between divisorial Prym, continuous Prym, and principalized Prym formalisms.

6. Classical context and cyclic generalizations

The tropical theory is best understood against the backdrop of classical Prym–Torelli results. For ramified double coverings of smooth projective curves, the Prym map

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$0

is generically injective in the ranges

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$1

and the paper also shows that a very general Prym variety of dimension at least $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$2 is not isogenous to a Jacobian (Marcucci et al., 2010). A stronger global theorem proves that the ramified Prym map

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$3

is an embedding for all $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$4 and all $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$5, while for $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$6 and $\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$7 the map has positive-dimensional fibers on the hyperelliptic locus (Naranjo et al., 2020). For unramified double covers, theta-duality yields the scheme-theoretic equalities

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$8

and from these one obtains a Torelli theorem recovering the covering from the Prym data and the associated Brill–Noether locus (Lahoz et al., 2010).

These classical results do not transfer verbatim to the tropical category, but they supply the conceptual model. In the tropical literature, the strongest compatibility statement is not injectivity but tropicalization compatibility of the Prym construction itself (Len et al., 2019). By contrast, the most recent tropical extension goes in a different direction: it enlarges the source from double covers to unramified Galois cyclic covers of arbitrary degree. In that setting the Prym remains the connected kernel of the norm map, the anti-invariant formula

$\Prym(\widetilde{\Gamma}/\Gamma):=(\ker \Nm_\varphi)^\circ.$9

replaces the involution-based description, and the Abel–Prym map is defined uniformly by

$(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$0

The free $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$1-cover computations indicate that the tropical Prym–Torelli morphism persists beyond degree $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$2 as a concrete and calculable assignment, even though a full tropical Torelli theorem is not yet available in that generality (Mohajer, 2 Apr 2026).

Outside the tropical category, recent algebraic work also shows that Prym–Torelli phenomena can become fully rigid for higher abelian Galois symmetries: for unramified $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$3-coverings of genus $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$4 curves, the Prym map is injective on both isotropic and non-isotropic components, and more generally the Prym map of unramified $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$5-covers of genus $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$6 is generically finite unless $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$7 is cyclic of order $(\widetilde{\Gamma}\to\Gamma)\longmapsto \Prym(\widetilde{\Gamma}/\Gamma),$8 (Borówka et al., 29 Mar 2025). A plausible implication is that the tropical cyclic-cover theory may eventually develop a finer Torelli landscape in which the behavior depends sharply on the Galois group, the cover type, and the chosen tropical compactification.