关于平方判别式Humbert曲面的几何性质
On the geometry of the Humbert surface of square discriminant
AI总结：
该研究确定了平方判别式Humbert曲面的Enriques--Kodaira型，通过构造对角Hirzebruch--Zagier除子分析Fricke型对合不动点，得到含扩张Cartan子群模曲线情形的亏格公式。
摘要：
对于每个正整数$N$，我们确定了判别式为$N^2$的Humbert曲面的Enriques--Kodaira型，该曲面参数化了与椭圆曲线乘积$(N,N)$-同源的主极化阿贝尔曲面。证明的关键步骤是分析判别式为$N^2$的Hilbert模曲面上一类Fricke型对合的不动点轨迹，该对合曾由Hermann以及Kani和Schanz研究过。为此，我们构造了被该对合固定的“对角”Hirzebruch--Zagier除子。在分析中，我们得到了这些除子的亏格公式，该公式涵盖了与$\text{GL}_2(\text{Z}/N\text{Z})$的（任意）扩张Cartan子群相关的模曲线情形，可能具有独立研究价值。
英文摘要：
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.
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