There are no universal ternary quadratic forms over biquadratic fields

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.

To appear in Proc. Edinburgh Math. Soc