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

2023/02/23 by Kala, Vitezslav, Yatsyna, Pavlo, Żmija, Błażej · 1 citation
#11A55 #11E12 #11E20 #11H55 #11R11 #11R80 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2302.12080

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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• Asymptotics of n-universal lattices over number fields

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