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

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.

💡 Research Summary

The paper establishes a relative version of Springer’s theorem for quadratic lattices over general Dedekind domains, extending the classical result that an anisotropic quadratic form over a field remains anisotropic after any odd‑degree extension. Let R be a Dedekind domain with fraction field F, assume that 2 is invertible in R, and let M be a quadratic lattice whose scalar extension V = F⊗_R M is isotropic of dimension at least three. For any quadratic lattice N over R and any finite odd‑degree extension E/F, denote by S the integral closure of R in E. The main theorem (Theorem 1.2) asserts that N embeds into M over R if and only if the base‑changed lattice S⊗_R N embeds into S⊗_R M over S.

The proof proceeds in two major stages. First, a local version is proved for complete discrete valuation rings (with 2 invertible). Using O’Meara’s theory of Jordan splittings and his embedding criterion (Theorem 2.1), the authors reduce the embedding problem to a family of conditions on the “scaled” sublattices M≤i and N≤i for all integers i. By invoking the classical Springer theorem for fields, they show that these local conditions are equivalent before and after the odd‑degree base change, yielding the local integral Springer theorem (Theorem 2.2).

The second stage lifts the result to the global setting. The authors introduce adelic language for orthogonal groups and develop a genus‑theoretic framework in which isomorphism classes of lattices that admit an embedding of N into M acquire the structure of a finite 2‑torsion abelian group. They define a natural homomorphism φ between the corresponding groups for R and for its odd‑degree extension S. The crux is to prove that φ is injective. This injectivity rests on a series of “norm principles for integral spinor norms”. Section 3 establishes these principles locally: for modular lattices L the orthogonal group O(L) coincides with O(F⊗_R L) when the underlying quadratic space is anisotropic, and various lemmas control the behavior of transporters (sets X⁺(M/N)) under scaling. The authors then prove an adelic norm principle (Section 5.1) by patching the local results and using the strong approximation property for the spin group Spin(V), which holds under the isotropy and dimension hypotheses (Remark 4.9).

With φ injective, the global embedding problem reduces to the spinor‑genus level: Theorem 5.8 shows that N embeds into M within a given spinor genus precisely when the same holds after odd‑degree base change. Finally, strong approximation yields the original statement of Theorem 1.2.

The paper also discusses the necessity of its hypotheses. Counterexamples are provided showing that the isotropy condition, the dimension ≥ 3 requirement, and the invertibility of 2 cannot be dropped in general. Moreover, the authors note that while the theorem is proved under the assumption that 2∈R×, the methods suggest it may hold without this restriction.

Overall, the work delivers a substantial generalization of Springer’s theorem to integral quadratic lattices over arbitrary Dedekind domains, introduces new norm‑principle techniques for integral spinor norms, and connects local lattice theory with global adelic methods. These contributions are likely to influence further research in arithmetic of quadratic forms, integral representations, and the study of algebraic groups over rings.