Isogeny classes of cubic spaces

TL;DR

This paper classifies non-degenerate cubic spaces of countable dimension up to isogeny, showing that their classes are determined by a residual rank invariant and form a discrete set with a well-behaved order structure.

Contribution

It introduces the residual rank invariant to classify non-degenerate cubic spaces up to isogeny and establishes the structure of their isogeny classes.

Findings

Isogeny classes are determined by residual rank.

The set of classes is discrete.

The classes satisfy the descending chain condition.

Abstract

A cubic space is a vector space equipped with a symmetric trilinear form. Two cubic spaces are isogeneous if each embeds into the other. A cubic space is non-degenerate if its form cannot be expressed as a finite sum of products of linear and quadratic forms. We classify non-degenerate cubic spaces of countable dimension up to isogeny: the isogeny classes are completely determined by an invariant we call the residual rank, which takes values in $\mathbf{N} \cup \{\infty\}$. In particular, the set of classes is discrete and (under the partial order of embedability) satisfies the descending chain condition.