Twisted Bernoulli Zeros in Quasi-Linear Time: Distribution, Depth, and Explicit Hilbert Class Components

Abstract

The divided generalized Bernoulli values $b_{\chi ,j}=f{B}_{1,\chi {\omega }^{-j}}\mathrm{mod}p$, for $\chi$ an odd primitive Dirichlet character of conductor $f$ and order $d$ with $d\mid p-1$, control (for $p\nmid \phi (f)$) the odd isotypic components of the $p$-class group of $Q({\zeta }_{fp})$ 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 $Q({\zeta }_{299973})$. Ancillary files contain all tables, certificates, and a verification program.

13 pages, 3 figures, 5 tables. Ancillary files: complete zero-line tables (27,508 lines), 2,441 split-prime Artin certificates, and a verification program. Companion to arXiv:2607.23177 and arXiv:2607.27503. Substantially revised framing relative to a withdrawn earlier version; the relationship to work of Delbourgo-Knospe and Knospe is discussed in Section 1