Hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics

TL;DR

This paper explores the structure of quadratic forms over function fields of quadrics, proposing a conjecture on their dimensions relative to anisotropic parts, and proves it in specific cases, extending Hoffmann's separation theorem.

Contribution

It introduces a conjecture generalizing Hoffmann's separation theorem and proves it for cases where the anisotropic part's dimension is less than 2^{s-1}.

Findings

Conjecture relates form dimensions to anisotropic parts over function fields.

Proved the conjecture when the anisotropic part's dimension is less than 2^{s-1}.

Shows forms in the kernel of the restriction homomorphism have dimensions divisible by 2^{s+1}.

Abstract

Let $p$ and $q$ be anisotropic quadratic forms over a field $F$ of characteristic $\neq 2$, let $s$ be the unique non-negative integer such that $2^s < \mathrm{dim}(p) \leq 2^{s+1}$, and let $k$ denote the dimension of the anisotropic part of $q$ after scalar extension to the function field $F(p)$ of $p$. We conjecture that $\mathrm{dim}(q)$ must lie within $k$ of a multiple of $2^{s+1}$. This can be viewed as a direct generalization of Hoffmann's separation theorem. Among other cases, we prove that the conjecture is true if $k<2^{s-1}$. When $k=0$, this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism $W(F)\rightarrow W(F(p))$ has dimension divisible by $2^{s+1}$.