Simultaneous representation by adjoint quadratic forms

Representation by Ternary Quadratic Forms

The problem of determining when an integral quadratic form represents every positive integer has received much attention in recent years, culminating in the 15 and 290 Theorems of Bhargava-Conway-Schneeberger and Bhargava-Hanke. For ternary quadratic forms, there are always local obstructions, but one may ask whether there are ternary quadratic forms which represent every locally represented in...

Representation of Quadratic Forms by Integral Quadratic Forms

Representation of Quadratic forms

Gauss Sums & Representation by Ternary Quadratic Forms

This paper specifies some conditions as to when an integer m is locally represented by a positive definite diagonal integer-matrix ternary quadratic form Q at a prime p. We use quadratic Gauss sums and a version of Hensel’s Lemma to count how many solutions there are to the equivalence Q(~x) ≡ m (mod p) for any k ≥ 0. Given that m is coprime to the determinant of the Hessian matrix of Q, we can...

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...