Tropical refined invariants for toric surfaces, introduced Block and Göttsche, are obtained couting tropical curves with a Laurent polynomial multiplicity. Brugallé 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.
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Blomme et al. (Mon,) studied this question.
Thomas Blomme
Gurvan Mével
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