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

Abstract

The divided generalized Bernoulli values $b_{\chi,j} = fB_{1,\chi\omega^{-j}} \bmod p$, for $\chi$ an odd primitive Dirichlet character of conductor $f$ and order $d$ with $d \mid p-1$, control (for $p \nmid \varphi(f)$) 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.