On locally primitively universal quadratic forms

TL;DR

This paper investigates properties of quadratic forms that nearly represent all positive integers, establishing that primitively universal forms represent zero p-adically and that almost universal forms in five or more variables are also almost primitively universal.

Contribution

It proves that primitively universal forms always represent zero over p-adic integers and that in five or more variables, almost universal forms are also almost primitively universal.

Findings

Primitively universal forms nontrivially represent zero p-adically.

Almost universal forms in ≥5 variables are almost primitively universal.

Abstract

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than almost universality. The two main results of this paper are: 1) every primitively universal form nontrivially represents zero over every ring of p-adic integers, and 2) every almost universal form in five or more variables is almost primitively universal.