There are no universal ternary quadratic forms over biquadratic fields

TL;DR

This paper proves that no classical totally positive definite ternary quadratic forms over the ring of integers in a totally real biquadratic field are universal, supporting the conjecture of finitely many such fields.

Contribution

It establishes the non-existence of universal classical ternary quadratic forms over biquadratic fields, advancing understanding of quadratic forms in algebraic number theory.

Findings

No classical ternary forms are universal over biquadratic fields

New properties of additively indecomposable elements in these fields

Supports conjecture of finitely many fields with universal forms

Abstract

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all non-diagonal coefficients in $2\mathcal{O}_K$) and prove that no such forms in three variables are universal (i.e., represent all totally positive elements of $\mathcal{O}_K$). This provides further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable elements of $\mathcal{O}_K$; we prove several new results about their properties.