Spinor representations of positive definite ternary quadratic forms

Abstract

For a positive definite integral ternary quadratic form $f$, let $r(k,f)$ be the number of representations of an integer $k$ by $f$. The famous Minkowski-Siegel formula implies that if the class number of $f$ is one, then $r(k,f)$ can be written as a constant multiple of a product of local densities which are easily computable. In this article, we consider the case when the spinor genus of $f$ contains only one class. In this case the above also holds if $k$ is not contained in a set of finite number of square classes which are easily computable (see, for example, \cite{sp1} and \cite {sp2}). By using this fact, we prove some extension of the results given in both \cite {cl} on the representations of generalized Bell ternary forms and \cite {be} on the representations of ternary quadratic forms with some congruence conditions.