Kitaoka's Conjecture for quadratic fields

TL;DR

The paper resolves Kitaoka's Conjecture for real quadratic fields by proving that only $13$ discriminants $D$ yield a real quadratic field ${\mathbb{Q}}(\sqrt{D})$ admitting a ternary universal lattice, and it identifies candidate lattices for each case. The authors develop a precise framework (MAINTHEOREM1 and its refined version) to bound discriminants via local obstructions controlled by least quadratic non-residues $\gamma_p$, enabling explicit discriminant bounds tied to lattice rank and representation of multiples. They extend linear-independence techniques to ranks up to $8$, using Gram-matrix decompositions $A+B\sqrt{\Delta_D}$ and positive-definiteness criteria, and they supplement the theory with extensive Magma computations to verify universality in the finite candidate set. The explicit list of thirteen discriminants, together with conjectural universal lattices for each, sharpens our understanding of how local and global arithmetic constraints govern universal lattices over real quadratic fields and informs minimal-rank invariants of universal lattices more broadly.

Abstract

We prove that there are at most 13 real quadratic fields that admit a ternary universal quadratic lattice, thus establishing a strong version of Kitaoka's Conjecture for quadratic fields. More generally, we obtain explicit upper bounds on the discriminants of real quadratic fields with a quadratic lattice of rank at most 7 that represents all totally positive multiples of a fixed integer.

Theorems & Definitions (72)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: Hadamard's inequality
• proof
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 2.4
• proof