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

TL;DR

This paper classifies strongly s-regular positive definite ternary quadratic forms with fixed minimal non-zero square representations, identifying exactly 207 such forms that represent one, thus extending Cooper and Lam's conjecture.

Contribution

It proves finiteness of strongly s-regular forms with fixed minimal non-zero square and explicitly enumerates all such forms representing one.

Findings

Finiteness of strongly s-regular forms with fixed minimal non-zero square.

Exact count of 207 non-classic integral strongly s-regular forms representing one.

Complete classification related to Cooper and Lam's conjecture.

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 {\it strongly $s$-regular } if the number of representations of squares of integers by $f$ satisfies the condition in Cooper and Lam's conjecture in \cite {cl}. 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 $207$ 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.