Elliptic curves and positive definite ternary forms (Q1609605)

Let \(f(X,Y,Z)\) and \(g(X,Y,Z)\) be two positive definite ternary forms of the same genus, and let \(r(f,n)\), \(r(g,n)\) be the number of representations of the natural number \(n\) represented by \(f\) and \(g\) respectively. The authors of the paper under review give, for some special forms \(f\) and \(g\), criteria when these two numbers \((r(f,n)\) and \(r(g,n))\) are equal or not. The idea proceeds as follows: The difference of the theta series of the forms \(f\) and \(g\) is a cusp of weight \(3/2\) which corresponds, via Shimura lift, to an elliptic curve \(E\), and then one uses a famous theorem of Waldspurger.