Computation of Gross-Keating invariants

TL;DR

This paper introduces computeGK, a Mathematica package for calculating Gross-Keating invariants and related quantities of half-integral matrices over p-adic integers, aiding research in quadratic forms and arithmetic geometry.

Contribution

The paper presents a new computational tool for efficiently calculating Gross-Keating invariants and associated series, with applications to arithmetic intersection numbers.

Findings

Successful implementation of computeGK in Mathematica

Explicit calculations of arithmetic intersection numbers

Enhanced understanding of quadratic forms over p-adic rings

Abstract

The Gross-Keating invariant of a half-integral matrix over a $p$-adic integer ring is a fundamental concept in the study of quadratic forms, and has important applications to Siegel modular forms and arithmetic geometry. We introduce the Mathematica package computeGK, a computer program for calculating the Gross-Keating invariant and the Siegel series of a half-integral matrix over $\mathbb{Z}_p$, as well as other related quantities. As a by-product, we obtain a table of the arithmetic intersection numbers related to the classical modular polynomials using the explicit formula of Gross and Keating.