An explicit formula for the extended Gross-Keating datum of a quadratic form

TL;DR

This paper provides an explicit formula for the extended Gross-Keating datum of quadratic forms over certain local fields and applies it to derive formulas for Siegel series, including algorithms for computation.

Contribution

It introduces a new explicit formula for the extended Gross-Keating datum and develops algorithms for computing Siegel series in relevant local fields.

Findings

Explicit formula for extended Gross-Keating datum

Algorithm implementation in Mathematica for computations

Application to Siegel series calculation

Abstract

In this paper, we give a formula for the extended Gross-Keating datum of a quadratic form defined over a finite extension of $\mathbb{Z}_p$ (for $p>2$) or a finite unramified extension of $\mathbb{Z}_2$. As an application, we describe an explicit formula for the Siegel series for $\mathbb{Z}_p$. We also present the details of algorithms implemented in a Mathematica package to compute the extended Gross-Keating datum and the Siegel series.