Representations of squares by certain ternary quadratic forms (Q2926279)

Let \(b\), \(c\) be positive integers and let \(n\) be a positive integer with prime factorization \(n = \prod_p p^{\lambda_p}\). Generalizing a classical result of \textit{A. Hurwitz} [C. R. Acad. Sci., Paris 98, 504--507 (1884; JFM 16.0155.01)] for the number of representations of \(n^2\) as a sum of three integer squares, \textit{S. Cooper} and \textit{H. Y. Lam} [J. Number Theory 133, No. 2, 719--737 (2013; Zbl 1309.11022)] conjectured that for certain specified values of \(b\) and \(c\), the number of representations of \(n^2\) by the ternary integral quadratic form \(x^2+by^2+cz^2\) can be expressed by a formula of the type \((\prod_{p\mid 2bc}g(b,c,p,\lambda_p))(\prod_{p\nmid 2bc} h(b,c,p,\lambda_p))\), with NEWLINE\[NEWLINEh(b,c,p,\lambda_p)=\frac{p^{\lambda_p+1}-1}{p-1} - \left(\frac{-bc}{p}\right)\frac{p^{\lambda_p-1}}{p-1},NEWLINE\]NEWLINE where \(\left(\frac{-bc}{p}\right)\) denotes the Legendre symbol, and \(g(b,c,p,\lambda_p)\) is determined on a case-by-case basis. The case \((b,c)=(1,1)\) is the classical formula of Hurwitz, and Cooper and Lam [loc. cit.] proved that there also exist formulas of the conjectured type for the pairs \((b,c) =(1,2), (1,3), (2,2), (3,3)\). In the paper under review, the author uses theta function identities to derive such formulas for the pairs \((1,5),(1,6),(2,3)\). As noted by the author, many additional cases of the conjecture of Cooper and Lam, including the ones covered in this paper, have now been proved \textit{X. Guo} et al. [J. Number Theory 140, 235--266 (2014; Zbl 1305.11025)] using results from the theory of modular forms of weight \(3/2\).