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

TL;DR

This paper proves an integral Springer theorem for quadratic lattices under base change of odd degree, establishing conditions for embeddings over Dedekind domains with applications to spinor norms.

Contribution

It introduces a new integral Springer theorem for quadratic lattices under odd degree base change and develops norm principles for integral spinor norms.

Findings

Embedding criteria for quadratic lattices under base change

Development of norm principles for spinor norms

Extension of Springer theorem to integral lattices

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.