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Integral Springer Theorem for Quadratic Lattices under Base Change of Odd Degree

Yong Hu, Jing Liu, Fei Xu

Abstract

A quadratic lattice $M$ over a Dedekind domain $R$ with fraction field $F$ is defined to be a finitely generated torsion-free $R$-module equipped with a non-degenerate quadratic form on the $F$-vector space $F\otimes_{R}M$. Assuming that $F\otimes_{R}M$ is isotropic of dimension $\geq 3$ and that $2$ is invertible in $R$, we prove that a quadratic lattice $N$ can be embedded into a quadratic lattice $M$ over $R$ if and only if $S\otimes_{R}N$ can be embedded into $S\otimes_{R}M$ over $S$, where $S$ is the integral closure of $R$ in a finite extension of odd degree of $F$. As a key step in the proof, we establish several versions of the norm principle for integral spinor norms, which may be of independent interest.

Integral Springer Theorem for Quadratic Lattices under Base Change of Odd Degree

Abstract

A quadratic lattice over a Dedekind domain with fraction field is defined to be a finitely generated torsion-free -module equipped with a non-degenerate quadratic form on the -vector space . Assuming that is isotropic of dimension and that is invertible in , we prove that a quadratic lattice can be embedded into a quadratic lattice over if and only if can be embedded into over , where is the integral closure of in a finite extension of odd degree of . As a key step in the proof, we establish several versions of the norm principle for integral spinor norms, which may be of independent interest.

Paper Structure

This paper contains 15 sections, 26 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Some applications of O'Meara's local embedding criterion
  3. Embedding criterion and integral Springer theorem in the local case
  4. Uniqueness of Jordan splittings and cancellation law
  5. Local analyses of isometries, transporters and integral spinor norms
  6. Observations on local integral orthogonal groups
  7. Sets of transporters and their spinor norms
  8. Norm principles for integral spinor norms
  9. Global theory for embeddings of quadratic lattices
  10. Adelic language and genus theory
  11. Spinor genera and classes containing a given sublattice
  12. Proof of the main theorem and some examples
  13. Norm principles for global integral and adelic spinor norms
  14. Proof of the main theorem
  15. Examples

Key Result

Theorem 1.2

Assume that the quadratic space $F\otimes_{R}M$ is isotropic of dimension $\geq 3$ and that $2$ is invertible in $R$. Then Question Spring-question has a positive answer.

Theorems & Definitions (66)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1: O'Meara
  • proof
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 56 more