Refined invariants for Abelian surfaces: between polynomiality and modularity
Algebraic Geometry
2026-02-04 v1 Combinatorics
Abstract
Tropical refined invariants for toric surfaces, introduced Block and G{\"o}ttsche, are obtained couting tropical curves with a Laurent polynomial multiplicity. Brugall{\'e} and Jaramillo-Puentes then exhibited a polynomial behavior of the coefficients of this Laurent polynomial, seen as function on the curve degree. The authors provided explicit formula for small genus, involving quasi-modular forms. Inspired by the toric setting, the first-named author defined refined invariants for abelian surfaces and extended the polynomiality result. In this paper, we further study this regularity for abelian surfaces, providing explicit formulas involving quasi-modular forms. This resonates with the small genus cases of the toric setting.
Cite
@article{arxiv.2602.03170,
title = {Refined invariants for Abelian surfaces: between polynomiality and modularity},
author = {Thomas Blomme and Gurvan Mével},
journal= {arXiv preprint arXiv:2602.03170},
year = {2026}
}