On the rank of universal quadratic forms over real quadratic fields

Valentin Blomer; Vítězslav Kala

doi:10.25537/dm.2018v23.15-34

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arXiv · 1705.03671

On the rank of universal quadratic forms over real quadratic fields

Valentin Blomer·Vítězslav Kala

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Abstract

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\mathbb Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.

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Valentin Blomer, Vítězslav Kala. 2017-11-07. On the rank of universal quadratic forms over real quadratic fields. https://doi.org/10.25537/dm.2018v23.15-34

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