Refined invariants for Abelian surfaces: between polynomiality and modularity

TL;DR

This paper explores refined invariants for abelian surfaces, demonstrating their polynomiality and modularity properties, and providing explicit formulas involving quasi-modular forms, extending previous work on toric surfaces.

Contribution

It extends the polynomiality and modularity results of tropical refined invariants from toric surfaces to abelian surfaces, with explicit formulas involving quasi-modular forms.

Findings

Refined invariants for abelian surfaces exhibit polynomiality.

Explicit formulas involving quasi-modular forms are derived.

Results resonate with small genus cases in the toric setting.

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.