Anisotropic modules over artinian principal ideal rings

TL;DR

This paper generalizes the concept of anisotropic bilinear forms from vector spaces to modules over artinian principal ideal rings, providing multiple equivalent definitions and exploring related notions like quasi-anisotropy and radical root, with applications in algebraic number theory.

Contribution

It introduces a comprehensive framework for anisotropy in modules over artinian principal ideal rings, extending classical bilinear form concepts and establishing new definitions and properties.

Findings

Multiple equivalent definitions of anisotropy for modules.

Characterization of anisotropy via forms on vector spaces.

Introduction of quasi-anisotropy and radical root concepts.

Abstract

Let V be a finite-dimensional vector space over a field k and let W be a 1-dimensional k-vector space. Let < , >: V x V \to W be a symmetric bilinear form. Then < , > is called anisotropic if for all nonzero v \in V we have <v,v> \neq 0. Motivated by a problem in algebraic number theory, we come up with a generalization of the concept of anisotropy to symmetric bilinear forms on finitely generated modules over artinian principal ideal rings. We will give many equivalent definitions of this concept of anisotropy. One of the definitions shows that one can check if a form is anisotropic by checking if certain forms on vector spaces are anisotropic. We will also discuss the concept of quasi-anisotropy of a symmetric bilinear form, which has no useful vector space analogue. Finally we will discuss the radical root of a symmetric bilinear form, which doesn't have a useful vector space analogue either. All three concepts have applications in algebraic number theory.