Hilbert-90 quotient maps, torsion defects, and symmetric monodromy

Abstract

Let $τ(z)=-1-z^{-1}$. We study the reduced rational maps $h_d:\mathbb{P}^1\to\mathbb{P}^1$ obtained by cancelling common factors in $H_d^{\rm raw}(z)=z^d(τ(z)^d-1)/(z^d-1)$. These maps arise by Hilbert-90 descent from the trace-zero maps $X^{dq}-X^d$ on $\ker\operatorname{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}$, but the principal object is the resulting $τ$-equivariant quotient-map family; nonconstant separable members are viewed as covers. We prove that cancellation is exactly a torsion-defect phenomenon. If $\ell(-)$ denotes scheme-theoretic length and $\boldsymbolμ_d=\ker([d]:\mathbb{G}_m\to\mathbb{G}_m)$, then $\mathrm{deg}(h_d)=d-\ell((1+X+Y=0)\cap\boldsymbolμ_d^2)$, and, in characteristic $p>0$ with $d=p^s d_0$ and $p\nmid d_0$, $h_d=\operatorname{Frob}_{p^s}\circ h_{d_0}$ and $\mathrm{deg}(h_d)=p^s\mathrm{deg}(h_{d_0})$. We classify the tame quotient strata of morphism degree at most one and exactly two; the maximal-defect stratum yields a characteristic-two Mersenne trace-zero permutation family. In characteristic zero we prove the main monodromy theorem: every non-linear quotient is Morse and has full symmetric geometric monodromy, $G_{h_d}=S_{\mathrm{deg}(h_d)}$; the proof rules out branch-value collisions via a cyclotomic cross-ratio equation. In positive characteristic we isolate Frobenius-sparse Kummer and Artin-Schreier quotients, a certificate-verified characteristic-19 Klein-four Galois quotient, and the first nonsparse Frobenius-lacunary tower up to its stated primitivity and wild-inertia boundary. A twisted off-diagonal fiber-square trace formula turns $2$-transitive monodromy into a uniform obstruction to $τ$-twisted exceptionality.