The signature of an even symmetric form with vanishing associated linking form

TL;DR

This paper proves that the signature of an even symmetric form with a vanishing associated linking form is divisible by 8, extending classical results and applying to manifold signatures, knot theory, and Diophantine equations.

Contribution

It generalizes the divisibility of signatures by 8 for forms with vanishing linking forms, broadening classical theorems and providing new applications.

Findings

Signature divisible by 8 for forms with vanishing linking form

Applications to 4n-dimensional manifold signatures

Restrictions on solutions to certain Diophantine equations

Abstract

We prove that the signature of an even, symmetric form on a finite rank integral lattice, has signature divisible by 8, provided its associated linking form vanishes in the Witt group of linking forms. Our result generalizes the well know fact that an even, unimodular form has signature divisible by 8. We give applications to signatures of 4n-dimensional manifolds, signatures of classical knots, and provide new restrictions to solutions of certain Diophantine equations.