Specialization of Quadratic and Symmetric Bilinear Forms

Preface

Contents

1 Fundamentals of Specialization Theory

1.1 Introduction: on the Problem of Specialization of Quadratic and Bilinear Forms

1.2 An Elementary Treatise on Symmetric Bilinear Forms

1.3 Specialization of Symmetric Bilinear Forms

1.4 Generic Splitting in Characteristic 2

1.5 An Elementary Treatise on Quadratic Modules

1.6 Quadratic Modules over Valuation Rings

1.7 Weak Specialization

1.8 Good Reduction

2 Generic Splitting Theory

2.1 Generic Splitting of Regular Quadratic Forms

2.2 Separable Splitting

2.3 Fair Reduction and Weak Obedience

2.4 Unified Theory of Generic Splitting

2.5 Regular Generic Splitting Towers and Base Extension

2.6 Generic Splitting Towers of a Specialized Form

3 Some Applications

3.1 Subforms which have Bad Reduction

3.2 Some Forms of Height 1

3.3 The Subform Theorem

3.4 Milnor's Exact Sequence

3.5 A Norm Theorem

3.6 Strongly Multiplicative Forms

3.7 Divisibility by Pfister Forms

3.8 Pfister Neighbours and Excellent Forms

3.9 Regular Forms of Height 1

3.10 Some Open Problems in Characteristic 2

3.11 Leading Form and Degree Function

3.12 The Companion Form of an Odd-dimensional Regular Form

3.13 Definability of the Leading Form over the Base Field

4 Specialization with Respect to Quadratic Places

4.1 Quadratic Places; Specialization of Bilinear Forms

4.2 Almost Good Reduction with Respect to Extensions of Quadratic Places

4.3 Realization of Quadratic Places; Generic Splitting of Specialized Forms in Characteristic 2

4.4 Stably Conservative Reduction of Quadratic Forms

4.5 Generic Splitting of Stably Conservative Specialized Quadratic Forms

References

Index