The number of representations of squares by integral quaternary quadratic forms

2019/03/06 by Kim, Kyoungmin
#11E12 #11E20 #11E45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1903.02248

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 x²+2y²+3z²+10w², which is, in fact, of class number 2 (see Lemma 5.5 and Proposition 5.6).

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• A classification of regular diagonal quadratic forms