Unit lattices of D₄-quartic number fields with signature (2,1)

2025/07/08 by Sergio Ricardo Zapata Ceballos, Sara Chari, Ceballos, Sergio Ricardo Zapata +12 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11R27 #Secondary 11R33

paper · pdf · doi:10.48550/arxiv.2507.06130

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There has been a recent surge of interest on distributions of shapes of unit lattices in number fields, due to both their applications to number theory and the lack of known results. In this work we focus on D₄-quartic fields with signature (2,1); such fields have a rank 2 unit group. Viewing the unit lattice as a point of GL₂(ℤ)\backslash \mathfrakh, we prove that every lattice which arises this way must correspond to a transcendental point on the boundary of a certain fundamental domain of GL₂(ℤ)\backslash \mathfrakh. Moreover, we produce three explicit (algebraic) points of GL₂(ℤ)\backslash \mathfrakh which are limit points of the set of (points associated to) unit lattices of D₄-quartic fields with signature (2,1).

Citations

• Unboundedness of shapes of unit lattices in totally real cubic fields
• Density of shapes of periodic tori in the cubic case
• Shapes of unit lattices in D_p-number fields
• Lower Bounds for Regulators of Number Fields in terms of their Discriminants
• Square-free values of reducible polynomials
• Provably weak instances of Ring-LWE
• On lattices, learning with errors, random linear codes, and cryptography

Cited by

• Classification of totally real number fields via their zeta function, regulator, and log unit lattice

Related

• Shapes of unit lattices in D_p-number fields
• On the distribution of shapes of pure quartic number fields
• The number of quartic D₄-fields ordered by conductor
• The shapes of Galois quartic fields
• An improved error term for counting D₄-quartic fields