Strong u-invariant and Period-Index Bounds Open Access

Abstract

Let K be a field. The u-invariant of K is the maximal dimension of anisotropic quadratic forms over K. For example, the u-invariant of C is 1, for F a non-real global or local field the u-invariant of F is 1, 2, 4, or 8, etc. Considerable progress has also been made, particularly in the computation of the u-invariant of function fields of p-adic curves due to Parimala and Suresh, and by Harbater, Hartmann, and Krashen regarding the u-invariants in the case of function fields of curves over complete discretely valued fields. Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2^{m+2} variables has a nontrivial zero was shown by Leep.

For a central simple algebra A over a field K, there are two major invariants, viz., period and index. For a field K, the Brauer-l-dimension of K for a prime number l, is the smallest natural number d such that for every finite field extension L/K and every central simple L-algebra A (of period a power of l), we have that index(A) divides period(A)^d.

If K is a number field or a local field, then classical results from class field theory tell us that the Brauer-l-dimension of K is 1. This invariant is expected to grow under a field extension, bounded by the transcendence degree. Some recent works in this area include that of Harbater-Hartmann-Krashen for K, a complete discretely valued field, in the good characteristic case. In the bad characteristic case, for such fields K, Parimala-Suresh have given some bounds.

In this dissertation, I will present similar bounds for the strong u-invariant and the Brauer-l-dimension of a complete non-Archimedean valued field K with residue field k and for function fields of curves over such fields.

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 The Initiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Severi-Brauer varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Patching and Local-Global Principles . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1 Introduction to field patching . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Necessary definitions . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.2 Patching problem . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Patching over Berkovich curves . . . . . . . . . . . . . . . . . . . . . 19

3.2.1 Necessary definitions . . . . . . . . . . . . . . . . . . . . . . . 19

3.2.2 A local-global principle over Berkovich curves . . . . . . . . . 27

4 Complete Ultrametric Fields . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Strong u-invariant . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.2 Review of literature: Complete discretely valued fields . . . . . . . . . 37

5.3 Review of literature: Complete non-Archimedean valued fields . . ... …. 38

5.4 Results regarding strong u-invariant . . . . . . . . . . . . . . . . . . . 38

6 Brauer l-dimension . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.2 Review of literature: Complete discretely valued fields . . . . . . . . . 43

6.3 Necessary results from literature . . . . . . . . . . . . . . . . . . . . . 44

6.4 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.5 Results regarding Brauer l-dimension . . . . . . . . . . . . . . . . . . 48

6.6 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Appendix A Embedding Problem. . . . . . . . . . . . . . . . . . . . . . . . .  51

A.0.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 54

Appendix B Non-Archimedean Differential Algebraic Geometry .. . . . . . . . . 55

B.0.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . 56

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 57

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	u-invariant quadratic forms non-Archimedean valued fields Brauer l-dimension central simple algebras
Committee Chair / Thesis Advisor	Venapally, Suresh, Emory University
Committee Members	Raman, Parimala, Emory University Borthwick, David, Emory University

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