On integral representations by quadratic forms

Representation of Quadratic Forms by Integral Quadratic Forms

Representation by Integral Quadratic Forms - a Survey

An integral symmetric matrix S = (sij) ∈ M sym m (Z) with sii ∈ 2Z gives rise to an integral quadratic form q(x) = 12 xSx on Z. If S is positive definite, the number r(q, t) of solutions x ∈ Z of the equation q(x) = t is finite, and it is one of the classical tasks of number theory to study the qualitative question which numbers t are represented by q or the quantitative problem to determine th...

Representations of Integers by Ternary Quadratic Forms

We investigate the representation of integers by quadratic forms whose theta series lie in Kohnen’s plus space M 3/2(4p), where p is a prime. Conditional upon certain GRH hypotheses, we show effectively that every sufficiently large discriminant with bounded divisibility by p is represented by the form, up to local conditions. We give an algorithm for explicitly calculating the bounds. For smal...

On Representations of Integers by Indefinite Ternary Quadratic Forms

Let f be an indefinite ternary integral quadratic form and let q be a nonzero integer such that −qdet(f) is not a square. Let N(T, f, q) denote the number of integral solutions of the equation f(x) = q where x lies in the ball of radius T centered at the origin. We are interested in the asymptotic behavior of N(T, f, q) as T → ∞. We deduce from the results of our joint paper with Z.Rudnick that...

Multiplicative Properties of Integral Binary Quadratic Forms

In this paper, the integral binary quadratic forms for which the set of represented values is closed under k-fold products, for even positive integers k, will be characterized. This property will be seen to distinguish the elements of odd order in the form class group of a fixed discriminant. Further, it will be shown that this closure under k-fold products can always be expressed by a klinear ...