勒让德多项式与复乘法，II：二次域的类数与亏格2超奇异多项式
Legendre polynomials and complex multiplication, II: class numbers of quadratic fields and genus 2 supersingular polynomials
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中文总结 AI 辅助
该文研究亏格2的两类超奇异多项式在有限域上的因式分解，用二次变换结合勒让德多项式、类方程性质及已有结果，证明其一次因式个数由虚二次域类数刻画，部分核心结论由人工智能首次发现。
中文摘要 AI 辅助
本文研究了Ibukiyama、Katsura和Oort在1986年论文中讨论的、亏格2曲线的两个超奇异多项式$h_p(x)$和$g_p(x)$在$\u2115_p$上的因式分解。这两个多项式模$p$分别同余于雅可比多项式$P_n^{(α,0)}(1-2x)$，对应$α= \pm 1/4$、$\pm 1/6$。其模$p$一次因式的个数由虚二次域$\u211a(\sqrt{-dp})$的类数确定，其中$d \in \{1,2,3\}$。证明用到的工具包括：将这些多项式与勒让德多项式$P_n(x)$关联的二次变换；Brillhart和Morton此前证明的关于$P_{(p-e)/4}(x)$和$P_{(p-\bar e)/3}(x)$的一次与二项式二次因式的结果；以及类方程$H_{-3p}(X)$和$H_{-12p}(X)$模$p$的不可约二次因式的性质。其中，针对$h_p(x) \equiv P_n^{(\pm1/4,0)}(1-2x)$的二次变换与一次因式结论，是借助人工智能首次发现的。
英文摘要
The factorizations over $\mathbb{F}_p$ of two supersingular polynomials $h_p(x)$ and $g_p(x)$ for genus $2$ curves, discussed by Ibukiyama, Katsura and Oort in their 1986 paper, are investigated. These polynomials are congruent modulo $p$ to the Jacobi polynomials $P_n^{(α,0)}(1-2x)$, for $α= \pm 1/4, \pm 1/6$, respectively. The number of their linear factors (mod $p$) is determined in terms of class numbers of the imaginary quadratic fields $\mathbb{Q}(\sqrt{-dp})$, where $d \in \{1,2,3\}$. The proofs use a quadratic transformation relating these polynomials to the Legendre polynomials $P_n(x)$; previous results on linear and binomial quadratic factors of $P_{(p-e)/4}(x)$ and $P_{(p-\bar e)/3}(x)$ proved by Brillhart and Morton; and properties of the irreducible quadratic factors of the class equations $H_{-3p}(X)$ and $H_{-12p}(X)$ (mod $p$). The quadratic transformation and linear factor results for $h_p(x) \equiv P_n^{(\pm1/4,0)}(1-2x)$ were first discovered using artificial intelligence.
