Mathematika

Table of Contents - June 1979 - Volume 26, Issue 01  

Research Article

Stable, fragile and absolutely symmetric quadratic forms

K. L. Fieldsa1

a1 Rider College, Lawrenceville, New Jersey

This paper originated with the observation that while all of the known stable lattice packings of spheres are highly symmetric, it is futile to try to prove a converse statement: the ordinary integer-lattice provides a distinctly unstable packing of spheres, but admits a large group of orthogonal symmetries nonetheless. The integerlattice is in fact very unstable—the slightest perturbation places the spheres in a more efficient configuration. We will call such a lattice fragile. The purpose of this note is to prove that a highly symmetric lattice must be either stable or fragile.

(Received September 29 1978)

Key Words:

  • 10C05: NUMBER THEORY; Forms; Quadratic forms;
  • 10E25: NUMBER THEORY; Geometry of numbers; Quadratic forms