# Quaternary quadratic lattices over number fields

## TL;DR

This paper establishes a correspondence between isometry classes of maximal lattices in definite quaternary quadratic spaces over number fields and ideal classes in quaternion algebras, providing an effective enumeration algorithm.

## Contribution

It introduces a novel method linking lattice classes to quaternion algebra ideals, enabling efficient classification and enumeration of lattices.

## Key findings

- Established a correspondence between lattice classes and quaternion ideal classes.
- Developed an algorithm for enumerating lattice representatives.
- Applied the method to classify lattices in specific quadratic spaces.

## Abstract

We relate proper isometry classes of maximal lattices in a totally definite quaternary quadratic space (V,q) with trivial discriminant to certain equivalence classes of ideals in the quaternion algebra representing the Clifford invariant of (V,q). This yields a good algorithm to enumerate a system of representatives of proper isometry classes of lattices in genera of maximal lattices in (V,q).