On the genera of certain integral lattices in ternary quadratic spaces

TL;DR

This paper explores the relationship between integral lattices in ternary quadratic spaces and associated algebraic orders, establishing a correspondence between their genera through Clifford algebra structures.

Contribution

It introduces a novel method linking the genus of certain integral lattices to the genus of associated algebraic orders via Clifford algebra representations.

Findings

Established a correspondence between lattice genus and order genus.

Connected integral lattices with algebraic orders in Clifford algebras.

Provided a framework for classifying lattices using algebraic structures.

Abstract

This paper treats certain integral lattices with respect to ternary quadratic forms, which are obtained from the data of a non-zero element and a maximal lattice in a quaternary quadratic space. Such a lattice can be described by means of an order associated with the lattice in the even Clifford algebra of the ternary form. This provides a correspondence between the genus of the lattice and that of the order.