The number of representations of squares by integral ternary quadratic forms

TL;DR

This paper investigates how many ways integers can be represented as squares by positive definite integral ternary quadratic forms, identifying certain genera that are indistinguishable by squares and resolving a related conjecture.

Contribution

It identifies non-trivial genera of ternary quadratic forms indistinguishable by squares and proves a conjecture by Cooper and Lam.

Findings

Found non-trivial genera indistinguishable by squares

Established relations between indistinguishable genera and the conjecture

Resolved the conjecture of Cooper and Lam completely

Abstract

Let $f$ be a positive definite integral ternary quadratic form and let $r(k,f)$ be the number of representations of an integer $k$ by $f$. In this article we study the number of representations of squares by $f$. We say the genus of $f$, denoted by $\text{gen}(f)$, is indistinguishable by squares if for any integer $n$, $r(n^2,f)=r(n^2,f')$ for any quadratic form $f' \in \text{gen}(f)$. We find some non trivial genera of ternary quadratic forms which are indistinguishable by squares. We also give some relation between indistinguishable genera by squares and the conjecture given by Cooper and Lam, and we resolve their conjecture completely.