Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

TL;DR

This paper establishes a local-global principle for quadratic forms over the fraction fields of two-dimensional henselian domains, extending understanding of quadratic form behavior in algebraic geometry and number theory.

Contribution

It proves the local-global principle for quadratic forms of ranks 3, 4, and at least 5 under specific conditions over certain two-dimensional henselian domains.

Findings

Validates the local-global principle for rank 3 and 4 quadratic forms.

Extends the principle to rank ≥ 5 forms when the domain has a specific structure.

Applicable when the residue field is C_1 or satisfies similar conditions.

Abstract

Let $R$ be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let $L$ and $k$ be respectively its fraction field and residue field. Let $\Omega_R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $\Spec R$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to $\Omega_R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A[y]$, where $A$ is a complete discrete valuation ring with a not too restrictive condition on the residue field $k$, which is satisfied when $k$ is $C_1$.