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

Abstract

Abstract. 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 [2]. In this article, we prove that there are only finitely many strongly s-regular ternary forms up to equivalence if the minimum of the non zero squares that are represented by the form is fixed. In particular, we show that there are exactly 207 nonclassic 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.

Quadratic forms with a strong regularity property on the representations of squares

“…As a natural generalization of the Cooper and Lam's conjecture, one may consider the problem to classify all strongly s-regular quadratic forms. Related to this question, we [15] proved that there are only finitely many isometry classes of strongly s-regular ternary quadratic forms if the minimum of the nonzero squares that are represented by the form is fixed, and we classified all strongly s-regular ternary quadratic forms that represent 1. Recently, the first author [13] proved that there are only finitely many isometry classes of strongly s-regular quaternary quadratic forms if the minimum of the nonzero squares that are represented by them is fixed, and classifies all strongly s-regular diagonal quaternary quadratic forms that represent 1.…”

The number of representations of squares by integral ternary quadratic forms

“…It was proved in [8] that every strongly s-regular ternary quadratic form represents all squares that are represented by its genus, and there are only finitely many strongly s-regular ternary quadratic forms up to isometry if m s pf q " min nPZ`t n : rpn 2 , f q ‰ 0u is fixed. Furthermore, it was proved that there are exactly 207 strongly s-regular ternary quadratic forms that represent one.…”