Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori
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2026
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| _version_ | 1866918515809714176 |
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| author | Shin, Henry |
| author_facet | Shin, Henry |
| contents | 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, $\operatorname{char}\mathbb{F}_q\ne2,3$, 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 $γ\in B^\times$, and every smooth fiber $\operatorname{Tr}(γh)=s$, $s^3\ne27\operatorname{N}(γ)$, we prove $N_{gH,B}(s;γ)=m^{-1}N_B(s,\operatorname{N}γ)+E_{gH,B}(s;γ)$, with $|E_{gH,B}(s;γ)|\le3(1-1/m)\sqrt q$. 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 $s^3=27\operatorname{N}(γ)$, 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 $\operatorname{Tr}_A(γη^n)=c$ over finite étale 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21939 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori Shin, Henry Number Theory 11T24, 11G20, 11S80, 11B37, 14F20, 14G15 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, $\operatorname{char}\mathbb{F}_q\ne2,3$, 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 $γ\in B^\times$, and every smooth fiber $\operatorname{Tr}(γh)=s$, $s^3\ne27\operatorname{N}(γ)$, we prove $N_{gH,B}(s;γ)=m^{-1}N_B(s,\operatorname{N}γ)+E_{gH,B}(s;γ)$, with $|E_{gH,B}(s;γ)|\le3(1-1/m)\sqrt q$. 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 $s^3=27\operatorname{N}(γ)$, 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 $\operatorname{Tr}_A(γη^n)=c$ over finite étale 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. |
| title | Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori |
| topic | Number Theory 11T24, 11G20, 11S80, 11B37, 14F20, 14G15 |
| url | https://arxiv.org/abs/2605.21939 |