On the geometry of the Humbert surface of square discriminant
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909290616324096 |
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| author | Frengley, Sam |
| author_facet | Frengley, Sam |
| contents | For every positive integer $N$ we determine the Enriques--Kodaira type of the Humbert surface of discriminant $N^2$ which parametrises principally polarised abelian surfaces that are $(N,N)$-isogenous to a product of elliptic curves. A key step in the proof is to analyse the fixed point locus of a Fricke-like involution on the Hilbert modular surface of discriminant $N^2$ which was studied by Hermann and by Kani and Schanz. To this end, we construct certain "diagonal" Hirzebruch--Zagier divisors which are fixed by this involution. In our analysis we obtain a genus formula for these divisors, which includes the case of modular curves associated to (any) extended Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ and which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09830 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the geometry of the Humbert surface of square discriminant Frengley, Sam Algebraic Geometry Number Theory 14G35, 14H10, 11G15, 11G05 For every positive integer $N$ we determine the Enriques--Kodaira type of the Humbert surface of discriminant $N^2$ which parametrises principally polarised abelian surfaces that are $(N,N)$-isogenous to a product of elliptic curves. A key step in the proof is to analyse the fixed point locus of a Fricke-like involution on the Hilbert modular surface of discriminant $N^2$ which was studied by Hermann and by Kani and Schanz. To this end, we construct certain "diagonal" Hirzebruch--Zagier divisors which are fixed by this involution. In our analysis we obtain a genus formula for these divisors, which includes the case of modular curves associated to (any) extended Cartan subgroup of $\mathrm{GL}_2(\mathbb{Z}/N\mathbb{Z})$ and which may be of independent interest. |
| title | On the geometry of the Humbert surface of square discriminant |
| topic | Algebraic Geometry Number Theory 14G35, 14H10, 11G15, 11G05 |
| url | https://arxiv.org/abs/2408.09830 |