Spinor representations of positive definite ternary quadratic forms

TL;DR

This paper explores the representation of integers by positive definite ternary quadratic forms, extending known formulas to cases where the spinor genus contains only one class, and applies these results to generalized Bell forms and forms with congruence conditions.

Contribution

It extends the Minkowski-Siegel formula applicability to forms with a single spinor genus class, including cases with certain non-represented integers, and applies this to specific classes of ternary forms.

Findings

Established conditions for representation counts in spinor genus cases.

Extended formulas to generalized Bell ternary forms.

Applied results to forms with congruence conditions.

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.