Computation of Gross-Keating invariants

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.

This paper has been merged with arXiv:1709.02772. The package will be maintained at https://github.com/chlee-0/computeGK