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Well-rounded ideal lattices from totally definite quaternion algebras

Yuan Xiang Chew, Frédérique Oggier

TL;DR

The paper develops a framework for constructing well-rounded lattices from ideals in orders of totally definite quaternion algebras via a positive definite trace-form. It proves a sharp WR criterion: an ideal lattice (Λ, b_α) is well-rounded precisely when the reduced norm-one subgroup Λ^1 spans the quaternion algebra A over Q, and it classifies WR lattices in the base-field case K = Q, showing only Z^4, D_4, and A_2 ⊕ A_2 arise from Z-orders. Explicit constructions and examples (including the Hurwitz order and degree-3 field cases) demonstrate these WR lattices and the role of Λ^1 in the geometry of the lattice. The work also discusses extensions to higher degree fields, the interplay between orders and ideals, and poses open questions about a full classification when [K:Q] ≥ 2. Overall, it bridges WR lattice theory with the arithmetic of quaternion algebras, providing both exact results and constructive methods with potential applications in coding and lattice-based cryptography.

Abstract

We study well-rounded ideal lattices from totally definite quaternion algebras. We prove existence and classification results, and illustrate our methods with examples.

Well-rounded ideal lattices from totally definite quaternion algebras

TL;DR

The paper develops a framework for constructing well-rounded lattices from ideals in orders of totally definite quaternion algebras via a positive definite trace-form. It proves a sharp WR criterion: an ideal lattice (Λ, b_α) is well-rounded precisely when the reduced norm-one subgroup Λ^1 spans the quaternion algebra A over Q, and it classifies WR lattices in the base-field case K = Q, showing only Z^4, D_4, and A_2 ⊕ A_2 arise from Z-orders. Explicit constructions and examples (including the Hurwitz order and degree-3 field cases) demonstrate these WR lattices and the role of Λ^1 in the geometry of the lattice. The work also discusses extensions to higher degree fields, the interplay between orders and ideals, and poses open questions about a full classification when [K:Q] ≥ 2. Overall, it bridges WR lattice theory with the arithmetic of quaternion algebras, providing both exact results and constructive methods with potential applications in coding and lattice-based cryptography.

Abstract

We study well-rounded ideal lattices from totally definite quaternion algebras. We prove existence and classification results, and illustrate our methods with examples.

Paper Structure

This paper contains 10 sections, 19 theorems, 49 equations, 1 table.

Table of Contents

  1. Introduction
  2. Ideal Lattices from Quaternion Algebras
  3. Totally definite quaternion algebras
  4. Ideals and orders
  5. Lattices
  6. Well-Rounded Lattices from Quaternion Algebras
  7. Existence of well-rounded lattices
  8. Characterization of well-rounded lattices
  9. More properties
  10. Examples and Discussions

Key Result

Proposition 2.3.3

hou2017construction Let $A$ be a totally definite quaternion algebra over the number field $K$, and $\alpha \in K^\times$ be totally positive. The map is a positive definite symmetric bilinear form satisfying where $\alpha$ is understood as $\alpha \mathbin{\mathop{\otimes}\displaylimits_{}}1$

Theorems & Definitions (48)

  • Definition 1.0.1
  • Definition 2.1.1
  • Definition 2.1.2
  • Definition 2.2.1
  • Definition 2.2.2
  • Definition 2.2.3
  • Definition 2.3.1
  • Definition 2.3.2
  • Proposition 2.3.3
  • Definition 2.3.4
  • ...and 38 more