Kitaoka's Conjecture for quadratic fields

TL;DR

This paper proves that only up to 13 real quadratic fields can have a ternary universal quadratic lattice, providing explicit bounds on discriminants for such fields and advancing understanding of Kitaoka's Conjecture.

Contribution

It establishes a strong version of Kitaoka's Conjecture for quadratic fields by proving an upper limit on such fields and deriving explicit discriminant bounds.

Findings

At most 13 real quadratic fields admit a ternary universal quadratic lattice.

Explicit upper bounds on discriminants of quadratic fields with certain universal lattices.

Progress towards confirming Kitaoka's Conjecture for quadratic fields.

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.