Braid Monodromy Rigidity on One-Dimensional Symmetric Cyclic Prym Slices: Adjoint $\mathfrak{sl}_4$-Dichotomy, Ramanujan-Sum Trace Averages, and Length-Three Cyclotomic Separation
Year: 2026
Type: preprint
Abstract: Let $m\ge 3$ be an odd integer and let $S=\mathbf{P}^1_\lambda\setminus\{0,\infty,1,-1\}\subset\mathcal{M}_{0,6}$ be the one-dimensional symmetric slice parametrizing degree-$m$ cyclic covers $C_\lambda\to\mathbf{P}^1$ branched at $(-1,-\lambda,0,\lambda,1,\infty)$ with a $(2,2,1,1)$-symmetric branch weight tuple $\mathbf{a}$ under an involution $h\in(\mathbf{Z}/m\mathbf{Z})^\times$ ($H=\{1,h\}$). Unlike on $\mathcal{M}_{0,6}$, every puncture of $S$ is a simultaneous multi-point collision whose local monodromy $g_p(t)$ on $V_t=H_1(C_\lambda,\mathbf{C})_t$ satisfies $\operatorname{rank}(g_p(t)-cI_4)\ge 2$ for all $c\in\mathbf{C}^\times$, and for composite $m=a\cdot m'$ ($\gcd(a,m')=1$) the CRT multiplier $v\equiv(-1,+1)\pmod{(a,m')}\notin\pm H$ shares identical projective puncture spectra with $t=1$ and collides on $g_1g_0$ and $g_1g_{-1}$. We establish an unconditional global braid-monodromy rigidity and cyclotomic separation theory on $S$: (1) via cyclotomic linear disjointness in the $g_1(t)$-eigenbasis, any $\operatorname{Ad}(\langle g_1(t),g_0(t)\rangle)$-invariant Lie subalgebra of $\mathfrak{sl}_4(\mathbf{C})$ is $\{0\}$ or $\mathfrak{sl}_4(\mathbf{C})$ (Theorem A); (2) for every $m=15N$ with $\gcd(N,30)=1$, closed-form Chebyshev-Laurent and Ramanujan-sum Galois trace averages of $(g_1g_0)^2$ prove that $\langle g_1(t),g_0(t)\rangle$ is Zariski-dense in $\operatorname{SU}(J(t))\simeq\operatorname{SU}(d_t-1,5-d_t)$ up to $\mu_{15N}$ scalars, while expanding $\tau_t(g_1^2g_0)$ in the dual basis $\mathcal{B}_{15}$ over $\mathbf{Q}(\zeta_N)$ separates all non-trivial primitive cosets $[t]\neq[\pm 1]$ (Theorem B); (3) $g_1^2g_0$ uniformly separates $v\equiv(-1,+1)\pmod{(a,m')}$ from $\pm H$ for arbitrary coprime odd $a,m'\ge 3$ (Theorem C); and (4) four commuting Galois involution projectors on $\mathbf{Q}(\zeta_a)\otimes\mathbf{Q}(\zeta_b)\otimes\mathbf{Q}(\zeta_N)$ extend Zariski density and length-$3$ separation to the three-factor family $m=a\cdot b\cdot N$ ($\gcd(N,30)=1$), with four method-boundary counterexamples (Theorem D). By Goursat's lemma, this determines the generic derived Mumford-Tate group $\operatorname{Hg}(B_{\eta,\mathrm{prim}})_{\mathrm{der}}=\prod_{[t]}\operatorname{SU}(J(t))$ and endomorphism algebra $\mathbf{Q}(\zeta_m)^H$ over $S$; while unbalanced families can carry cross-orbit determinant Hodge classes (such as $D_{[8]}\otimes D_{[-13]}$ at $m=105,N=7$), on characterwise $(2,2)$-balanced subfamilies ($d_t=3$) the central torus acts by the identity on every determinant line, giving $\operatorname{Hg}(B_{\eta,\mathrm{prim}})=\prod_{[t]}\operatorname{SU}(2,2)$ and reducing all-powers Hodge algebraicity to the separate determinant-cycle input on individual factors (Corollary E). Archival copy of version 1 of this paper, published on rStation on 6 November 2026 (rStation hodge_conjecture paper 114, v1). paper.md is the manuscript exactly as published there; the PDF is typeset from it with the published title and abstract; supplement.zip is the supplement as published there. Files: paper.md — SHA-256 007f503b19cf142888994bd78bc929f0cd15a752c4fda97dfa1cc32f1132d3ee braid-monodromy-rigidity-on-one-dimensional-symmetric-cyclic-prym-slices.pdf — SHA-256 fd4f5887a7b23a7f87281c5b04ce189dd9922d7b45a9617be730a53f8d9ec069 supplement.zip — SHA-256 807d3c37f3081f980efc6afebeaefb0dcd4e9efd392dd1b5fe8b1b5ea76c46f0
Author: Li, Hongmin
Institutions: Institute of Science Tokyo, The University of Tokyo
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225569386