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 \'etale cubic, $\operatorname{char}\mathbb{F}_q\ne2,3$, and let $T_B=\ker(\operatorname{N}_{B/\mathbb{F}_q}:\operatorname{Res}_{B/\mathbb{F}_q}\mathbb{G}_m\to\mathbb{G}_m)$. 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 $\operatorname{Tr}(\gamma h)=s$, $s^3\ne27\operatorname{N}(\gamma)$, we prove $N_{gH,B}(s;\gamma)=m^{-1}N_B(s,\operatorname{N}\gamma)+E_{gH,B}(s;\gamma)$, with $|E_{gH,B}(s;\gamma)|\le3(1-1/m)\sqrt q$. 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 $s^3=27\operatorname{N}(\gamma)$, 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 $\operatorname{Tr}_A(\gamma\eta^n)=c$ over finite \'etale cubic $\mathbb{Z}_p$-algebras, $p\ge5$. 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.