# The number of representations of squares by integral quaternary quadratic forms

## TL;DR

This paper investigates the finiteness and classification of strongly s-regular integral quaternary quadratic forms, establishing a finite list under fixed minimal nonzero square, and explicitly identifying 34 such diagonal forms representing one.

## Contribution

It proves finiteness of strongly s-regular forms with fixed minimal nonzero square and classifies all diagonal forms representing one, using eta-quotients for specific cases.

## Key findings

- Finitely many strongly s-regular forms exist with fixed minimal nonzero square.
- Exactly 34 diagonal strongly s-regular forms represent one.
- Eta-quotients confirm s-regularity of specific forms.

## 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 $5.6$).