Report
On the geometry of the Humbert surface of square discriminant
| العنوان: | On the geometry of the Humbert surface of square discriminant |
|---|---|
| المؤلفون: | Frengley, Sam |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Algebraic Geometry, Mathematics - Number Theory, 14G35, 14H10, 11G15, 11G05 |
| الوصف: | 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. Comment: 66 pages, 18 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2408.09830 |
| رقم الانضمام: | edsarx.2408.09830 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |