On Kitaoka's conjecture and lifting problem for universal quadratic forms

Abstract

For a totally positive definite quadratic form over the ring of integers of a totally real number field 𝐾, we show that there are only finitely many totally real field extensions of 𝐾 of a fixed degree over which the form is universal (namely, those that have a short basis in a suitable sense). Along the way we give a general construction of a universal form of rank bounded by 𝐷(log 𝐷) 𝑑−1 , where 𝑑 is the degree of 𝐾 over ℚ and 𝐷 is its discriminant. Furthermore, for any fixed degree we prove (weak) Kitaoka's conjecture that there are only finitely many totally real number fields with a universal ternary quadratic form.

Lifting problem for universal quadratic forms

“…While the asymptotic local-global principle of Hsia-Kitaoka-Kneser [HKK78] implies that universal quadratic forms exist over every (totally real) number field, it does not provide any information about their ranks. Still unproven is even the influential Kitaoka's conjecture from the 1990's that there are only finitely many totally real number fields with a ternary universal quadratic form, despite interesting results for quadratic [CKR96] and biquadratic [KTZ20] fields, for fields of odd degree [EK97], and a weaker statement in every fixed degree [KY23]. Further, Kim-Kim-Park [Kim00,KKP22] studied real quadratic fields with universal forms of ranks up to eight.…”

Lifting problem for universal quadratic forms over totally real cubic number fields

“…The results in the paper of Hsia, Kitaoka, and Kneser [13] imply that there exists a universal quadratic form over $${\mathcal {O}_K}$$ for any totally real number field $$K$$. Subsequently, numerous results concerning them have been studied, see, for example, [4–12, 14, 16–23, 26, 29, 34]. We refer the interested readers to a survey paper of Kala [15] for the recent developments on universal quadratic forms (and lattices) over the rings of integers of totally real number fields, and the references therein.…”

“…The question whether there exists a universal $${\mathbb {Z}}$$‐form over a totally real number field of degree $$\geqslant 4$$ other than real biquadratic fields still remains open. Thanks to Kala and Yatsyna who studied the so‐called ‘weak lifting problem’ and proved some finiteness results in a more general setting (see [19, Theorem 2] for the general statement), we know that for $$d,r\in {\mathbb {N}}$$, there are only finitely many totally real number fields $$K$$ of degree $$d=[K:{\mathbb {Q}}]$$ such that there is a $${\mathbb {Z}}$$‐form of rank $$r$$ that is universal over $$\mathcal {O}_K$$ (see [19, Corollary 9]).…”