Primitive Representations by Spinor Genera of Ternary Quadratic Forms

Abstract

Let a be primitively represented by the genus of a ternary quadratic lattice L defined over the ring of integers of an algebraic number field F. Criteria to determine whether a is primitively represented by every spinor genus in the genus of L involve certain subgroups θ∗(Lp, a) of the multiplicative groups of the localizations Fp of F with respect to the various nonarchimedean prime spots p on F. In this paper these groups θ∗(Lp, a) are determined explicitly for nondyadic and 2-adic prime spots. Examples are given which show how this information can, in some instances, be used in combination with known results, to determine all integers primitively represented by a particular positive definite ternary quadratic form.