Isometries of lattices and Hasse principles

TL;DR

This paper characterizes when certain polynomials can be characteristic polynomials of lattice isometries, explores related Hasse principle questions, and applies these results to knot signatures.

Contribution

It provides necessary and sufficient conditions for polynomials to be characteristic polynomials of lattice isometries and extends Hasse principle analysis to this context and knot signatures.

Findings

Criteria for polynomials to be lattice isometry characteristic polynomials

Hasse principle results for signatures of knots

Generalized answers to Hasse principle questions

Abstract

We give necessary and sufficient conditions for an integral polynomial without linear factors to be the characteristic polynomial of an isometry of some even, unimodular lattice of given signature. This gives rise to Hasse principle questions, which we answer in a more general setting. As an application, we prove a Hasse principle for signatures of knots.