Automorphisms of even unimodular lattices and equivariant Witt groups

TL;DR

This paper characterizes automorphisms of even unimodular lattices via their characteristic polynomials and introduces a Witt group criterion for G-stable lattices in bilinear forms over discretely valued fields.

Contribution

It generalizes Gross and McMullen's theorem by classifying characteristic polynomials of lattice automorphisms and provides a new Witt group-based criterion for G-stable lattices.

Findings

Identifies which irreducible polynomials occur as automorphism characteristic polynomials.

Provides a Witt group criterion for the existence of G-stable lattices.

Generalizes previous results on lattice automorphisms.

Abstract

We characterize the irreducible polynomials that occur as a characteristic polynomial of an automorphism of an even unimodular lattice of given signature, generalizing a theorem of Gross and McMullen. As part of the proof, we give a general criterion in terms of Witt groups for a bilinear form equipped with an action of a group G over a discretely valued field to contain a unimodular G-stable lattice.