Spinor representations of positive definite ternary quadratic forms

For a positive definite integral ternary quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. The famous Minkowski–Siegel formula implies that if the class number of [Formula: see text] is one, then [Formula: see text] can be written as a constant multiple of a product of local densities which are easily computable. In this paper, we consider the case when the spinor genus of [Formula: see text] contains only one class. In this case the above also holds if [Formula: see text] is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the recent results on both the representations of generalized Bell ternary forms and the representations of ternary quadratic forms with some congruence conditions.

Ternary quadratic forms and Brandt matrices

In a recent paper [9] the author showed (among other results) estimates on the asymptotic behaviour of the representation numbers of positive definite integral ternary quadratic forms, in particular, that for n in a fixed square class tZ2 and lattices L, K in the same spinor genus one has . The main tool utilized for the proof was the theory of modular forms of weight 3/2, especially Shimura’s lifting from the space of cusp forms of weight 3/2 to the space of modular forms of weight 2.

Representation of spinor exceptional integers by ternary quadratic forms

The existence and basic properties of what are now referred to as spinor exceptional integers for a genus of integral ternary quadratic forms were first observed in the 1950’s by Jones and Watson [7] and Kneser [8] in the context of indefinite forms. The study of these integers and their generalizations has been undertaken by a number of authors in recent years, and has contributed significantly to the understanding of representation properties unique to ternary forms. In this direction, the present author proved in a previous paper [4] that if c is a primitive spinor exceptional integer for a genus of integral ternary quadratic forms and f is some form in this genus, then a form in the spinor genus of f primitively represents c if and only if f primitively represents an integer of the type ct2, for some odd positive integer t, relatively prime to the discriminant d, which satisfies the condition that the Jacobi symbol (–cd/t) equals 1.

On spinor exceptional representations

Let f(x1 …, xm) be a quadratic form with integer coefficients and c ∈ Z. If f(x) = c has a solution over the real numbers and if f(x) ≡ c (mod N) is soluble for every modulus N, then at least some form h in the genus of f represents c. If m ≧ 4 one may further conclude that h belongs to the spinor genus of f. This does not hold when m = 3.