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arXiv · 1909.05422

There are no universal ternary quadratic forms over biquadratic fields

Jakub Krásenský·Magdaléna Tinková·Kristýna Zemková
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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.

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Jakub Krásenský, Magdaléna Tinková, Kristýna Zemková. 2019-09-12. There are no universal ternary quadratic forms over biquadratic fields. https://doi.org/10.1017/s001309152000022x

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