Refined invariants for Abelian surfaces: between polynomiality and modularity
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911418005061632 |
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| author | Blomme, Thomas Mével, Gurvan |
| author_facet | Blomme, Thomas Mével, Gurvan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03170 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Refined invariants for Abelian surfaces: between polynomiality and modularity Blomme, Thomas Mével, Gurvan Algebraic Geometry Combinatorics 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. |
| title | Refined invariants for Abelian surfaces: between polynomiality and modularity |
| topic | Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2602.03170 |