Tropical covers, tropical abelian varieties and Prym varieties

Abstract: We define and investigate the tropical Prym varieties associated to unramified Galois cyclic covers of tropical curves (or equivalently metric graphs) $\tildeΓ\to Γ$. Our approach here is to study the tropical Prym varieties using group actions on tropical abelian varieties induced by the cyclic Galois group of the cover of tropical curves. We also define and conider the Abel-Prym map for tropical cyclic covers extending that for double covers. As a special case we consider free $\Z_3$-covers of tropical curves and their associated tropical Prym variety and compute its volume generalizing the case of double covers.

• The paper develops a tropical theory for Prym varieties using unramified cyclic covers on metric graphs and group actions to extend classical results.
• It demonstrates the isogeny decomposition of tropical Jacobians and derives explicit volume formulas for tropical Prym varieties in the triple cover case.
• The work bridges tropical geometry with arithmetic invariants through harmonic morphisms, Ihara zeta functions, and Artin-Ihara L-functions for deeper geometric insights.

Tropical Covers, Tropical Abelian Varieties, and Prym Varieties: A Technical Analysis

Introduction and Motivation

This paper develops a robust theory for Prym varieties in the setting of tropical geometry, concentrating on unramified Galois cyclic covers of tropical curves or, equivalently, metric graphs. Whereas the classical theory associates Prym varieties to unramified cyclic covers between algebraic curves—Prym varieties appearing as isogeny factors of Jacobians with precise control via the covering data—the tropical analog is constructed using group actions on tropical abelian varieties induced by the Galois group of the cover. The paper also generalizes the Abel-Prym map to arbitrary cyclic covers and conducts explicit computations for triple (degree 3) covers, most notably deriving exact volume formulas for the associated tropical Prym varieties, which extend the case of double covers found in earlier literature.

Harmonic Morphisms and Galois Covers in Tropical Settings

A foundational part of the paper is the framework of harmonic morphisms between metric graphs. These morphisms generalize the classical covering theory for Riemann surfaces to tropical curves, respecting combinatorial structure and edge lengths modulo specified dilation factors. Unramified harmonic morphisms of local degree one correspond exactly to (topological) covering spaces in the metric graph category.

Galois covers in the tropical setting are formalized by equipping the covering graph $\widetilde{\Gamma}$ with a transitive group action by a finite group $G$ consistent with the covering map. This induces natural properties on fibers, and the structure of the covering can be understood in terms of group actions, analogous to classical monodromy representations.

A rigorous discussion of ramification and unramified (free) covers is provided. Notably, the paper singles out the notion of "strongly unramified" covers, where local degree is constant across fibers over each base point, a property always satisfied for harmonic double covers and playing a crucial role in the algebraic structure of the induced Prym varieties.

Tropical Abelian Varieties and Group Actions

The paper reviews the construction of tropical abelian varieties as real tori endowed with positive-definite bilinear forms (tropical polarizations). The automorphism group structure, isomorphisms, and group actions (by finite Galois groups) on these tropical tori are developed to support the descent theory needed in the construction of tropical Prym varieties.

Tropical Prym Varieties and the Abel-Prym Map

Central to the paper is the definition of the Prym variety $\py(\widetilde{\Gamma}/\Gamma)$ associated to an unramified cyclic cover $\varphi: \widetilde{\Gamma} \to \Gamma$. The Prym is defined as the identity component of the kernel of the norm map induced by the covering, following the classical model, but realized in the tropical category:

$\py(\widetilde{\Gamma}/\Gamma) = (\ker \nm_\varphi)^\circ$

The paper shows that the Prym variety inherits a principal polarization and admits an explicit description in terms of the tropical Jacobian and group action: it coincides with the fixed part under the difference operator $1 - \tau$ for $\tau$ a generator of the cyclic Galois group.

A key technical result is the isogeny decomposition:

$\jac(\widetilde{\Gamma}) \sim \py(\widetilde{\Gamma}/\Gamma) \times \mu^*\jac(\Gamma)$

where $\mu^*$ is the induced map on Jacobians, generalizing the splitting of Jacobians in the classical setting of cyclic covers.

The Abel-Prym map in the tropical context is defined by composing the Abel-Jacobi map of the covering graph with the difference operator $1 - \tau$, situating divisors in the Prym subspace of the Jacobian.

Explicit Case: Tropical Triple Covers

The paper provides a comprehensive treatment of unramified degree 3 (triple) covers, generalizing known results for double covers. The construction of bases for homology at the graph and cover level permits effective calculation of the Prym sublattice and its polarization.

A major quantitative result is the precise volume formula for the Prym variety in the triple cover case:

$G$0

This formula, derived via explicit Gram matrix computations and change-of-basis determinants, confirms that the analogous volume relations known for genus 2 double covers extend to higher cyclic covers, subject to the structure of norm maps and the principal polarization on the Prym.

The methodology is further illustrated with explicit combinatorial computations for concrete graphs, efficiently verifying the formula in worked examples.

Ihara Zeta Functions and Artin-Ihara L-functions

Extending the combinatorial perspective, the paper connects the complexity and arithmetic of tropical curves to Ihara-type graph zeta functions and Artin-Ihara L-functions. These functions encode cycles and path equivalence classes, with their leading Laurent coefficients at $G$1 reflecting the order of the Jacobian (complexity) of the graph.

The Artin-Ihara L-function associated to the cover and a representation of the Galois group decomposes the zeta function of the cover as a product over irreducible representations, mirroring number-theoretic analogies. The paper gives determinant formulas for these L-functions in terms of Artinized adjacency and valency matrices. In the specific example of the triple cover, direct computation using representations and roots of unity corroborates the previously obtained volume result.

Implications and Future Directions

The construction of tropical Prym varieties through group actions on tropical Jacobians and the effective computation of their volumes provide substantial tools for tropical Brill-Noether theory, the arithmetic of tropical curves, and connections to moduli spaces. The extension to arbitrary cyclic covers suggests imminent generalizations to more complicated group actions and ramification data in the tropical setting.

The explicit use of graph zeta functions and their connections with combinatorial invariants of tropical abelian varieties indicate possible intersections with arithmetic geometry, particularly for the investigation of special subvarieties in tropical moduli spaces and their relations to Shimura varieties in the non-archimedean and tropical worlds.

Further research may extend these constructions to ramified covers, non-cyclic Galois groups, and their applications in tropical Hodge theory, as well as explore tropical analogues of classical theorems involving Prym–Torelli and Schottky loci. The approach using explicit bases and volumes could be ported to practical algorithms for combinatorial and computational applications.

Conclusion

This paper delivers a systematic account of Prym varieties for unramified cyclic covers in tropical geometry, generalizing classical constructions with explicit computation of invariants such as volume. Through group actions on tropical Jacobians, the work establishes essential correspondence between classical and tropical theories, opening avenues for combinatorial, arithmetic, and geometric applications in tropical and non-archimedean settings.