The number of representations of squares by integral ternary quadratic forms (II)

Let f be a positive definite ternary quadratic form. We assume that f is non-classic integral, that is, the norm ideal of f is Z. We say f is strongly s-regular if the number of representations of squares of integers by f satisfies the condition in Cooper and Lam's conjecture in [3]. In this article, we prove that there are only finitely many strongly s-regular ternary forms up to isometry if the minimum of the nonzero squares that are represented by the form is fixed. In particular, we show that there are exactly 207 non-classic integral strongly s-regular ternary forms that represent one (see Tables 1 and 2). This result might be considered as a complete answer to a natural extension of Cooper and Lam's conjecture.