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arXiv · 2302.12080

Real quadratic fields with a universal quadratic form of given rank have density zero

Vitezslav Kala·Pavlo Yatsyna·Błażej Żmija
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Abstract

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients.

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Vitezslav Kala, Pavlo Yatsyna, Błażej Żmija. 2023-02-23. Real quadratic fields with a universal quadratic form of given rank have density zero. https://arxiv.org/abs/2302.12080

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