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

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 $\ge 3$ 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.

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