Lower order terms in the shape of cubic fields

2024/09/12 by Robert Hough, Eun Hye Lee, Hough, Robert +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.08417

openalex publication_date 2024/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We demonstrate equidistribution of the lattice shape of cubic fields when ordered by discriminant, giving an estimate in the Eisenstein series spectrum with a lower order main term. The analysis gives a separate discussion of the contributions of reducible and irreducible binary cubic forms, following a method of Shintani. Our work answers a question posed at the American Institute of Math by giving a precise geometric and spectral description of an evident barrier to equidistribution in the lattice shape.

Related

• The shapes of pure cubic fields
• The shape of quartic fields
• Unboundedness of shapes of unit lattices in totally real cubic fields
• Quadratic units and cubic fields
• Lower bound estimates for the rank of universal quadratic forms in some families of real cubic fields with density one