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arXiv · 1903.02248

The number of representations of squares by integral quaternary quadratic forms

Kyoungmin Kim
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Abstract

Let $f$ be a positive definite (non-classic) integral quaternary quadratic form. We say $f$ is strongly $s$-regular if it satisfies a regularity property on the number of representations of squares of integers. In this article, we prove that there are only finitely many strongly $s$-regular quaternary quadratic forms up to isometry if the minimum of the nonzero squares that are represented by the quadratic form is fixed. Furthermore, we show that there are exactly $34$ strongly $s$-regular diagonal quaternary quadratic forms representing one (see Table $1$). In particular, we use eta-quotients to prove the strongly $s$-regularity of the quaternary quadratic form $x^2+2y^2+3z^2+10w^2$, which is, in fact, of class number $2$ (see Lemma $5.5$ and Proposition $5.6$).

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Kyoungmin Kim. 2019-03-06. The number of representations of squares by integral quaternary quadratic forms. https://arxiv.org/abs/1903.02248

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