准线性时间下的扭曲伯努利零点：分布、深度与显式希尔伯特类分量
Twisted Bernoulli Zeros in Quasi-Linear Time: Distribution, Depth, and Explicit Hilbert Class Components
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中文总结 AI 辅助
本文研究准线性时间下的扭曲伯努利零点分布与深度，将零点转换为显式希尔伯特类分量生成元，扩展了相关目录范围，验证了零点性质并记录了非简单零点线。
中文摘要 AI 辅助
分划广义伯努利值 $b_{\chi,j} = fB_{1,\chi\omega^{-j}} \bmod p$，其中 $\chi$ 是导子为 $f$、阶为 $d$ 的奇本原狄利克雷特征，且满足 $d \mid p-1$，当 $p \nmid \varphi(f)$ 时，该值通过按特征划分的阿贝尔主猜想控制 $Q(\zeta_{fp})$ 的 $p$-类群的奇同型分量；这类零点是扭曲非正则对，是具有正岩泽 lambda 不变量的分支，已被 Ernvall、Holden、Delbourgo-Knospe 和 Knospe 等人研究。本文在现有制表互补的范围内对这些零点进行了考察：固定小导子和大素数（$f = 3,5$，对应 $p < 10^5$；所有导子不超过 20 的奇本原特征，对应 $p < 2 \cdot 10^4$），通过剩余类权重公式和 $F_p$ 上的一次 Bluestein 卷积计算每个谱，这是一种直接的有限域方法，其准线性阶与标准幂级数方法相同。本次考察记录了 55121 个特征-素数对中的 27508 条零点线，每条线都通过两条独立代码路径以精确的消失阶进行了验证；计数结果与随机模型一致，其中 8 条线为非简单线，包括一条深度为 3 的线，对应 $(f,p,j) = (19,37,16)$，其类分量的阶恰好为 $p^2$ 和 $37^3$。主要贡献在于将零点转换为显式可验证生成元：将 arXiv:2607.23177 的导子 3 目录从 $p < 500$ 扩展到 $p < 10^5$，所有 2441 条零点线均为简单线，每条投影的圆单位都通过新的分裂素阿廷证书被证明生成其完整的阶为 $p$ 的希尔伯特类场分量，该分量是次数为 199980 的域 $Q(\zeta_{299973})$ 中的最大分量。辅助文件包含所有表格、证书和验证程序。
英文摘要
The divided generalized Bernoulli values $b_{χ,j} = fB_{1,χω^{-j}} \bmod p$, for $χ$ an odd primitive Dirichlet character of conductor $f$ and order $d$ with $d \mid p-1$, control (for $p \nmid φ(f)$) the odd isotypic components of the $p$-class group of $Q(ζ_{fp})$ through the characterwise abelian Main Conjecture; a zero is a twisted irregular pair, a branch with positive Iwasawa lambda-invariant, in the tradition studied by Ernvall, Holden, Delbourgo-Knospe and Knospe. We survey these zeros in the regime complementary to existing tabulations: fixed small conductor and large $p$ ($f = 3, 5$ to $p < 10^5$; all odd primitive characters of conductor at most 20 to $p < 2 \cdot 10^4$), computing each spectrum by a residue-class weight formula and one Bluestein convolution over $F_p$, a direct finite-field alternative of the same quasi-linear order as the standard power-series method. The survey records 27,508 zero lines over 55,121 character-prime pairs, each verified by two independent code paths with an exact order of vanishing; the counts and digits are consistent with the random model, and eight lines are non-simple, including one of depth three at $(f,p,j) = (19,37,16)$, giving class components of order exactly $p^2$ and $37^3$. The main contribution converts zeros into explicit certified generators: the conductor-three catalogue of arXiv:2607.23177 is extended from $p < 500$ to $p < 10^5$, all 2,441 zero lines simple, each projected circular unit proven to generate its complete order-$p$ Hilbert class field component by a fresh split-prime Artin certificate -- the largest in the degree-199,980 field $Q(ζ_{299973})$. Ancillary files contain all tables, certificates, and a verification program.
