Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori

Abstract

Prescribed trace/norm estimates and Soto-Andrade-type sums control whole fibers or related global character sums. We prove a coset-refined trace theorem for cubic norm-one tori. Let $B/{\mathbb{F}}_q$ be finite étale cubic, $\mathrm{char}{\mathbb{F}}_q\ne 2,3$, and let $T_B=\ker (N_{B/{\mathbb{F}}_q}:{\mathrm{Res}}_{B/{\mathbb{F}}_q}{\mathbb{G}}_m\to {\mathbb{G}}_m)$. For every subgroup $H\subset T_B({\mathbb{F}}_q)$ of index $m$, every coset $gH$, every $\gamma \in B^{\times }$, and every smooth fiber $\mathrm{Tr}(\gamma h)=s$, $s^3\ne 27N(\gamma )$, we prove $N_{gH,B}(s;\gamma )=m^{-1}{N}_B(s,N\gamma )+E_{gH,B}(s;\gamma )$, with $|E_{gH,B}(s;\gamma )|\le 3(1-1/m)\sqrt{q}$. The geometric input is a Picard-Kummer kernel calculation: no nontrivial torus character becomes geometrically constant on a smooth trace/norm curve, so nontrivial coset character sums have square-root cancellation. On the nodal boundary $s^3=27N(\gamma )$, the kernel degenerates exactly to a cyclic cubic Kummer kernel. Its Frobenius-fixed part is the sole source of order-$q$ bias; after removing that explicit projection, remaining characters again have square-root cancellation up to bounded normalization/node correction. The same geometry gives local branch theory for ${\mathrm{Tr}}_A(\gamma {\eta }^n)=c$ over finite étale cubic ${\mathbb{Z}}_p$-algebras, $p\ge 5$. The logarithmic tangent and trace-dual codifferent coordinates identify singular branches: nondegenerate classes have quadratic Hensel models, while the genuinely affine degenerate class has a cubic first-obstruction model; in full norm-fiber orbits singular branch counting reduces to one cubic norm equation.

79 pages