二次亏格分裂雅可比轨迹的次数与洪伯特型重构
Degrees of Genus-Two Split-Jacobian Loci and Humbert-Form Reconstruction
AI总结：
该研究推导了二次亏格分裂雅可比轨迹相关西格尔模形式的次数关系，确定洪伯特曲面相关模形式的限制与系数，将多项式计算转化为线性问题并完成n=5的实例计算。
AI中文摘要：
设$\u0393_n \u2282 \u039c_2$为具有极大次数$n$椭圆子覆盖的二次亏格曲线的轨迹，由不可约加权齐次多项式$F_n \u2208 \u0396[J_2,J_4,J_6,J_{10}]$在$\u2119(2,4,6,10)$中截出。设$\u03bd(n)$为$X_1(n) \u2192 X(1)$的次数，$G_{n^2}$为水平一的西格尔模形式，其除子为洪伯特曲面$H_{n^2}$，$k(H_{n^2})$为其权。我们证明：从$F_n$得到的亚纯西格尔模形式$F_n(\u03c4)$沿乘积轨迹恰有$\u03bd(n)$阶极点；$\u03c7_{10}^{\u03bd(n)} F_n(\u03c4)$是$G_{n^2}$的常数倍；对所有$n \u2265 2$（无论奇偶），$°_w F_n = k(H_{n^2}) - 10\u03bd(n)$。我们确定$G_{n^2}$在乘积轨迹上的限制为模多项式的显式乘积；对$n \u2265 3$，其首项傅里叶-雅可比系数为精确$n$阶挠点上的theta函数乘积；并将$G_{n^2}$在标量倍数意义下刻画为其权下在单个挠除子上消失的唯一形式。这些数据将$F_n$的计算从消元问题转化为公式所规定规模的线性问题，我们对$n=5$完成了该计算。
英文摘要：
Let $\mathcal{L}_n \subset \mathcal{M}_2$ be the locus of genus-two curves admitting a maximal degree-$n$ elliptic subcover, cut out in $\mathbb{P}(2,4,6,10)$ by an irreducible weighted-homogeneous polynomial $F_n \in \mathbb{Z}[J_2,J_4,J_6,J_{10}]$. Let $ν(n)$ be the degree of $X_1(n) \to X(1)$, let $G_{n^2}$ be the Siegel modular form of level one with divisor the Humbert surface $H_{n^2}$, and let $k(H_{n^2})$ be its weight. We prove that the meromorphic Siegel modular form $F_n(τ)$ obtained from $F_n$ has a pole of order exactly $ν(n)$ along the product locus, that $χ_{10}^{ν(n)} F_n(τ)$ is a constant multiple of $G_{n^2}$, and that $°_w F_n = k(H_{n^2}) - 10ν(n)$ for every $n \geq 2$, even or odd. We determine the restriction of $G_{n^2}$ to the product locus as an explicit product of modular polynomials and, for $n \geq 3$, its leading Fourier-Jacobi coefficient as a product of theta functions over the torsion points of exact order $n$, and we characterize $G_{n^2}$, up to scalar, as the unique form of its weight vanishing on a single torsion divisor. These data convert the computation of $F_n$ from elimination into a linear problem of the size the formula prescribes, which we carry out for $n=5$.
