On the Gross-Keating invariant of a quadratic form over a non-archimedean local field

TL;DR

This paper studies the properties of the Gross-Keating invariant for half-integral symmetric matrices over non-archimedean local fields, clarifying its structure for matrices of any size.

Contribution

It establishes fundamental properties of the Gross-Keating invariant for matrices of arbitrary size over non-archimedean local fields, extending previous understanding.

Findings

Basic properties of the Gross-Keating invariant are characterized.

The invariant's behavior is clarified for matrices of size n ≥ 4.

Results apply to arbitrary non-archimedean local fields of characteristic zero.

Abstract

Let $B$ be a half-integral symmetric matrix of size $n$ defined over $\mathbb{Q}_p$. The Gross-Keating invariant of $B$ was defined by Gross and Keating, and has important applications to arithmetic geometry. But the nature of the Gross-Keating invariant was not understood very well for $n\geq 4$. In this paper, we establish basic properties of the Gross-Keating invariant of a half-integral symmetric matrix of general size over an arbitrary non-archimedean local field of characteristic zero.