#### Date of Award

12-1-2016

#### Degree Name

Doctor of Philosophy

#### Department

Mathematics

#### First Advisor

Earnest, Andrew

#### Second Advisor

Ban, Dubravka

#### Abstract

An integral quadratic form is said to be strictly regular if it primitively represents all integers that are primitively represented by its genus. The goal of this dissertation is to extend the systematic investigation of the positive deﬁnite ternary primitive integral quadratic forms and lattices that are candidates for strict regularity. An integer that is primitively represented by a genus, but not by some speciﬁc form in that genus, is called a primitive exception for that form. So, the strictly regular forms are those forms for which there are no primitive exceptions. Our computations of primitive exceptions for each of the 119 positive deﬁnite regular ternary forms which lie in multiple-class genera, and of the companion forms in their genera, show that there are 45 inequivalent such forms that are candidates for strict regularity. We provide a proof of the strict regularity of one of these candidates, bringing the total number of forms for which such proofs are known to 15, and prove partial results on the integers primitively represented by the other form in its genus. The theory of primitive spinor exceptional integers is used to analyze the primitive exceptions for the forms in two other genera known to contain a regular ternary form. In these cases, results are obtained relating the primitive representation of certain integers c by a given form in one of these genera to the primitive representation of the integers 4c and 9c by the forms in the genus.

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