Local densities of diagonal integral ternary quadratic forms at odd primes

Local Densities of 2-adic Quadratic Forms

In this paper, we give an explicit from formula for the local density number of representing a two by two 2-integral matrix T by a quadratic 2-integral lattice L over Z2. The non-dyadic case was dealt in a previous paper. The special case when L is a (maximal) lattice in the space of trace zero elements in a quaternion algebra over Q2 yields a clean and interesting formula, which matches up per...

2-universal Positive Definite Integral Quinary Diagonal Quadratic Forms

As a generalization of the famous four square theorem of Lagrange, Ramanujan found all positive definite integral quaternary diagonal quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary diagonal quadratic forms that represent all positive definite integral binary quadratic forms. §

An Explicit Formula for Local Densities of Quadratic Forms

Let S and T be two positive definite integral matrices of rank m and n respectively. It is an ancient but still very challenging problem to determine how many times S can represent T , i.e., the number of integral matrices X with XSX = T . However, Siegel proved in his celebrated paper ([Si1]) that certain weighted averages of these numbers over the genus of S can be expressed as the Euler prod...

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