Multiplicities of non-arithmetic ternary quadratic forms and elliptic curves of positive rank (Q1291046)

A quadratic form \(q\) on \({\mathbb R}^n\) is said to be totally non-arithmetic if there is no \({\mathbb Q}\)-rational subspace \(U\) of \({\mathbb R}^n\), \(\dim U > 1\), such that the \({\mathbb Q}\)-subspace of \({\mathbb R}\) generated by the coefficients of \(q\) with respect to a \(\mathbb Q\)-rational basis of \(U\) has dimension 1. The author shows that in the case \(n = 3\) the following conditions on such a form are equivalent: (i) \(q\) integrally represents nonzero numbers with unbounded multiplicities, (ii) the coefficients of \(q\) with respect to a basis of \({\mathbb Q}^3\) generate a 2-dimensional \(\mathbb Q\)-subspace of \(\mathbb R\). The difficult part of this theorem is that (ii) implies (i). To prove this one has to investigate certain pairs of arithmetic ternary quadratic forms. The author's interesting method is to associate a family of elliptic curves with such a pair, and to show that at least one of them has a rational point of infinite order.