Refined invariants for Abelian surfaces: between polynomiality and modularity

Abstract: Tropical refined invariants for toric surfaces, introduced Block and G{ö}ttsche, are obtained couting tropical curves with a Laurent polynomial multiplicity. Brugall{é} 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.