Hilbert surfaces, modular forms, and Siegel-Veech constants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918350863466496 |
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| author | Nguyen, Duc-Manh |
| author_facet | Nguyen, Duc-Manh |
| contents | We give the values of the Siegel-Veech constants associated with saddle connections having distinct endpoints on translation surfaces in Prym eigenform loci in $Ω\mathcal{M}_3(2,2)^{\rm odd}$. In particular, we show that these constants are actually the same for all of these loci. As a by-product, we show that the Euler characteristic of the Hilbert modular surfaces which parametrize Abelian surfaces with $(1,2)$-polarization admitting a real multiplication and the Euler characteristic of their product locus are related by a simple formula. For principally polarized Abelian surfaces, a similar phenomenon has been observed by Bainbridge. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_19901 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hilbert surfaces, modular forms, and Siegel-Veech constants Nguyen, Duc-Manh Algebraic Geometry Complex Variables Dynamical Systems Number Theory 30F30, 11F03, 37F34 We give the values of the Siegel-Veech constants associated with saddle connections having distinct endpoints on translation surfaces in Prym eigenform loci in $Ω\mathcal{M}_3(2,2)^{\rm odd}$. In particular, we show that these constants are actually the same for all of these loci. As a by-product, we show that the Euler characteristic of the Hilbert modular surfaces which parametrize Abelian surfaces with $(1,2)$-polarization admitting a real multiplication and the Euler characteristic of their product locus are related by a simple formula. For principally polarized Abelian surfaces, a similar phenomenon has been observed by Bainbridge. |
| title | Hilbert surfaces, modular forms, and Siegel-Veech constants |
| topic | Algebraic Geometry Complex Variables Dynamical Systems Number Theory 30F30, 11F03, 37F34 |
| url | https://arxiv.org/abs/2602.19901 |