Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Hensel's lemma for the norm principle for spinor groups

Amin Soofiani

TL;DR

The paper proves a Hensel-type lifting result for the norm principle of spinor groups Spin(h) associated to regular skew-hermitian forms over quaternion algebras. It shows that, assuming the H^1-norm principle holds for Spin(đ”„) over all regular skew-hermitian forms on quaternion algebras defined over finite extensions of the residue field k, the H^1-norm principle also holds for Spin(h) over quaternion algebras over K, the complete discretely valued field, by a careful reduction to the residue field case and a detailed case analysis. Central to the argument are reductions to simply connected covers, the use of Larmour’s decomposition, and leveraging Gille, Knebusch, and Scharlau norm principles to control spinor-norm obstructions through quadratic and unramified ramified extensions. The work culminates in a full proof of the main theorem by treating unramified/ramified cases and employing Hensel’s lemma to lift local cohomological information to global conclusions, thereby generalizing earlier results for quadratic forms to skew-hermitian forms over quaternions. The results have implications for norm principles of type D_n groups over valued fields and connect obstruction theory, cohomology, and ramification in a unified framework.

Abstract

Let $K$ be a complete discretely valued field with residue field $k$ with $char \ k \ne 2$. Assuming that the norm principle holds for spinor groups $Spin(\mathfrak{h})$ for every regular skew-hermitian form $\mathfrak{h}$ over every quaternion algebra $\mathfrak{D}$ (with respect to the canonical involution on $\mathfrak{D}$) defined over any finite extension of $k$, we show that the norm principle holds for spinor groups $Spin(h)$ for every regular skew-hermitian form $h$ over every quaternion algebra $D$ (with respect to the canonical involution on $D$) defined over $K$.

Hensel's lemma for the norm principle for spinor groups

TL;DR

The paper proves a Hensel-type lifting result for the norm principle of spinor groups Spin(h) associated to regular skew-hermitian forms over quaternion algebras. It shows that, assuming the H^1-norm principle holds for Spin(đ”„) over all regular skew-hermitian forms on quaternion algebras defined over finite extensions of the residue field k, the H^1-norm principle also holds for Spin(h) over quaternion algebras over K, the complete discretely valued field, by a careful reduction to the residue field case and a detailed case analysis. Central to the argument are reductions to simply connected covers, the use of Larmour’s decomposition, and leveraging Gille, Knebusch, and Scharlau norm principles to control spinor-norm obstructions through quadratic and unramified ramified extensions. The work culminates in a full proof of the main theorem by treating unramified/ramified cases and employing Hensel’s lemma to lift local cohomological information to global conclusions, thereby generalizing earlier results for quadratic forms to skew-hermitian forms over quaternions. The results have implications for norm principles of type D_n groups over valued fields and connect obstruction theory, cohomology, and ramification in a unified framework.

Abstract

Let be a complete discretely valued field with residue field with . Assuming that the norm principle holds for spinor groups for every regular skew-hermitian form over every quaternion algebra (with respect to the canonical involution on ) defined over any finite extension of , we show that the norm principle holds for spinor groups for every regular skew-hermitian form over every quaternion algebra (with respect to the canonical involution on ) defined over .

Paper Structure

This paper contains 24 sections, 50 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Equivalence of the two variants and reduction to simply connected groups
  3. Equivalence of $H^0$ and $H^1$ variants of the norm principle
  4. Reduction to the center, and to simply connected covers
  5. Gille's, Knebusch's, and Scharlau's norm principles
  6. Rationality, Gille's norm principle, and Knebusch's norm principle
  7. Scharlau's norm principle
  8. Obstruction to the norm principle for the spinor group, and reduction to quadratic field extensions
  9. Obstruction to the norm principle for $Spin(A,\sigma)$
  10. Reduction to quadratic field extensions
  11. Reduction to anisotropic skew-hermitian forms
  12. Larmour's theorem
  13. Notations and some explicit formulas
  14. Notations
  15. Explicit formulas for the maps $f$ and $g$
  16. ...and 9 more sections

Key Result

Theorem 1.4

4 Let $K$ be a complete discretely valued field with residue field $k$ with $char(k) \ne 2$. Assume that the $H^1$-variant of the norm principle holds for $Spin(q)$ for every regular quadratic form $q$ of even dimension defined over any finite extension of $k$. Then the $H^1$-variant of the norm pri

Theorems & Definitions (116)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 106 more