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Level structures on tropical abelian varieties
Abstract
We introduce level structures on tropical abelian varieties and give a modular interpretation of Ag[m]trop, the tropicalization of the moduli space 𝒜g[m] of principally polarized abelian varieties with level m structure. We study the case of abelian surfaces in greater depth. The link of A2[m]trop is an explicit simplicial complex whose vertices are the primitive vectors of (ℤ/mℤ)4 up to sign. As a topological space, this link is homotopic to a wedge sum of closed orientable surfaces and circles; we compute the number of each of these and the genera of the surfaces. We deduce the weight zero compactly supported rational cohomology of 𝒜2[m], completing a calculation of Oda-Schwermer from 1990.
Related
- Realization of tropical curves in Abelian surfaces
- Dynamical degrees of automorphisms on abelian varieties
- A holomorphic (2, 2)-Theorem for Abelian varieties of CM-type
- Arithmetic jet spaces and the unramified Zilber--Pink conjecture for subvarieties of abelian varieties
- A Poincaré-Lefschetz Theorem for Cellular Cosheaves and an Application to the Tropical Homology of Orbifold Toric Varieties
BibTeX
@misc{assaf2026level,
title = {{Level structures on tropical abelian varieties}},
author = {Eran Assaf and Madeline Brandt and Juliette Bruce and Melody Chan and Raluca Vlad},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14895},
eprint = {2609.14895},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14895},
}
RIS
TY - UNPB
TI - Level structures on tropical abelian varieties
AU - Eran Assaf
AU - Madeline Brandt
AU - Juliette Bruce
AU - Melody Chan
AU - Raluca Vlad
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.14895
UR - https://arxiv.org/abs/2609.14895
AB - We introduce level structures on tropical abelian varieties and give a modular interpretation of A_g[m]^trop, the tropicalization of the moduli space 𝒜_g[m] of principally polarized abelian varieties with level m structure. We study the case of abelian surfaces in greater depth. The link of A_2[m]^trop is an explicit simplicial complex whose vertices are the primitive vectors of (ℤ/mℤ)^4 up to sign. As a topological space, this link is homotopic to a wedge sum of closed orientable surfaces and circles; we compute the number of each of these and the genera of the surfaces. We deduce the weight zero compactly supported rational cohomology of 𝒜_2[m], completing a calculation of Oda-Schwermer from 1990.
ER -