Tropical covers, tropical abelian varieties and Prym varieties

1. Problem Statement

The paper addresses the gap in tropical geometry regarding the systematic definition and investigation of Prym varieties associated with unramified Galois cyclic covers of tropical curves (metric graphs).

While the classical theory of Prym varieties for Riemann surfaces is well-established (where the Prym variety is an isogeny factor of the Jacobian of a cover, defined via the kernel of the norm map), the tropical analog for cyclic covers of degree $n > 2$ had not been fully developed. Specifically, the author aims to:

• Define the tropical Prym variety for general cyclic covers ($\mathbb{Z}_n$-covers).
• Establish its structure as a principally polarized tropical abelian variety (pptav).
• Define the Abel-Prym map for these covers, generalizing previous work on double covers.
• Compute the volume of the tropical Prym variety, specifically for triple covers ($\mathbb{Z}_3$), and relate this volume to the volumes of the Jacobians of the base and covering graphs.

2. Methodology

The author employs a combination of combinatorial graph theory, algebraic topology, and tropical algebraic geometry. The methodology proceeds through the following steps:

• Harmonic Morphisms and Galois Covers: The paper begins by formalizing the concept of harmonic morphisms between metric graphs. It defines Galois covers where a finite group $G$ (specifically cyclic groups $\mathbb{Z}_n$) acts on the covering graph $\tilde{\Gamma}$, preserving the harmonic structure. The concept of "strongly unramified" covers is introduced to ensure uniformity in local degrees.
• Tropical Abelian Varieties: The author defines tropical abelian varieties as real tori $\mathbb{R}^g/\Lambda$ equipped with a principal polarization (a positive definite symmetric bilinear form). The Jacobian of a metric graph, $Jac(\Gamma)$, is constructed as a tropical abelian variety using the homology $H_1(\Gamma, \mathbb{Z})$ and the space of harmonic 1-forms $\Omega(\Gamma)$.
• Group Actions on Jacobians: The core of the construction relies on the action of the Galois group $G = \mathbb{Z}_n$ on the Jacobian $Jac(\tilde{\Gamma})$. The author utilizes the norm map ($Nm_\phi$) and the pull-back map ($\phi^*$) to decompose the Jacobian of the cover.
• Definition of Prym Variety: The tropical Prym variety is defined as the connected component of the identity of the kernel of the norm map:
  $$Prym(\tilde{\Gamma}/\Gamma) := (\ker Nm_\phi)^\circ$$
  It is shown that this is equivalent to the image of the operator $(1 - \tau)$, where $\tau$ is the generator of the Galois group acting on $Jac(\tilde{\Gamma})$.
• Volume Computation via Linear Algebra: To compute the volume of the Prym variety, the author constructs explicit bases for the homology groups $H_1(\tilde{\Gamma}, \mathbb{Z})$ and $H_1(\Gamma, \mathbb{Z})$. By analyzing the Gram matrices of these bases under the intersection pairing, the author derives a relationship between the volumes of the Jacobians and the Prym variety.
• Ihara Zeta Functions: As an alternative verification method, the paper utilizes the Ihara Zeta function and the Artin-Ihara L-function. By decomposing the zeta function of the cover into L-functions associated with irreducible representations of the Galois group, the author relates the leading coefficients of the Taylor expansions (which correspond to the orders of the Jacobians) to the volume of the Prym variety.

3. Key Contributions

A. Definition of Tropical Prym Varieties for Cyclic Covers

The paper successfully generalizes the definition of tropical Prym varieties from double covers ($\mathbb{Z}_2$) to arbitrary cyclic covers ($\mathbb{Z}_n$). It proves that for an unramified cyclic cover $\phi: \tilde{\Gamma} \to \Gamma$, the Prym variety is a principally polarized tropical abelian variety (pptav).

B. Structural Decomposition

The author establishes a fundamental isogeny decomposition for the Jacobian of the covering graph:
$$Jac(\tilde{\Gamma}) \sim Prym(\tilde{\Gamma}/\Gamma) \times \mu^* Jac(\Gamma)$$
where $\mu^*$ is the pull-back map. This confirms that the Prym variety acts as the complement of the pull-back of the base Jacobian within the cover's Jacobian.

C. The Abel-Prym Map

A new map, the Abel-Prym map $\Psi: \tilde{\Gamma} \to Prym(\tilde{\Gamma}/\Gamma)$, is defined as the composition of the tropical Abel-Jacobi map and the projection $(1-\tau)$. This extends the known theory for double covers to higher-degree cyclic covers.

D. Volume Formula for Triple Covers

A significant technical result is the explicit computation of the volume of the Prym variety for free $\mathbb{Z}_3$-covers. The author proves the following relation:
$$Vol_2(Prym(\tilde{\Gamma}/\Gamma)) = \frac{Vol_2(Jac(\tilde{\Gamma}))}{3 \cdot Vol_2(Jac(\Gamma))}$$
This generalizes the known formula for double covers (where the factor is 2).

E. Connection to Zeta Functions

The paper provides a novel link between the volume of the tropical Prym variety and the Artin-Ihara L-functions. It demonstrates that the ratio of the Jacobian volumes can be computed via the determinants of matrices derived from the L-functions associated with the non-trivial irreducible representations of the Galois group.

4. Key Results

• Proposition 4.3 & 4.4: The Prym variety is identified as $\ker(Nm_G)^0 = \text{Im}(1-\tau)$ and can be expressed as the quotient of the anti-invariant part of the 1-forms by the anti-invariant part of the homology lattice:
  $$Prym(\tilde{\Gamma}/\Gamma) = \Omega(\tilde{\Gamma})_-^* / H_1(\tilde{\Gamma}, \mathbb{Z})_-$$
• Proposition 4.7 (Volume Formula): For a free $\mathbb{Z}_3$-cover, the volume of the Prym variety is exactly one-third of the ratio of the volume of the cover's Jacobian to the base's Jacobian.
• Example 4.8: A concrete calculation is performed on a genus 2 graph with a triple cover.
  • $|Jac(\tilde{\Gamma})| = 588$
  • $|Jac(\Gamma)| = 4$
  • Calculated Volume: $588 / (3 \times 4) = 49$.
  • This result is independently verified using the Artin-Ihara L-function, yielding the same value (49).
• Lemma 4.9: Characterizes the genus of points in the fiber of an unramified $\mathbb{Z}_3$-cover, showing that points in the dilation cycle $\gamma(\phi)$ have genus $\geq \frac{1}{2}(\deg_{\gamma_2}(\phi) + \deg_{\gamma_3}(\phi) - (d_x - 1))$.

5. Significance

• Bridging Classical and Tropical Geometry: The paper solidifies the analogy between classical algebraic geometry (Riemann surfaces) and tropical geometry. By successfully defining Prym varieties for cyclic covers, it confirms that deep structural properties of Jacobians (like isogeny decompositions) hold in the tropical setting.
• Computational Tools: The derivation of the volume formula using both linear algebra (Gram matrices) and analytic number theory (Zeta functions) provides robust tools for computing invariants of tropical curves. This is particularly useful for enumerative geometry problems involving tropical curves.
• Foundation for Further Research: The definition of the Abel-Prym map for $\mathbb{Z}_n$-covers opens the door to studying tropical analogs of the Torelli theorem and other moduli problems related to cyclic covers.
• Generalization: The results move beyond the specific case of double covers (which are often treated as special cases due to the involution symmetry) to general cyclic groups, offering a more comprehensive theory of tropical covers.

In summary, Mohajer's work provides a rigorous framework for understanding Prym varieties in the context of tropical cyclic covers, establishing their geometric structure, and providing explicit computational formulas for their volumes, thereby enriching the intersection of tropical geometry and arithmetic combinatorics.