Local-global principles for hermitian spaces over semi-global fields Público

Abstract

This dissertation studies the Hasse principle for projective homogenous spaces under unitary groups over semi-global fields and obtains partial results. We show that a local-global principle holds for the isotropy of hermitian forms over 2-dimensional complete local fields under certain conditions. We also prove a theorem for isotropy of hermitian spaces over odd degree extensions of function fields of p-adic curves.

Table of Contents

1 Preliminaries 1

1.1 Central Simple Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Involutions and Hermitian Forms . . . . . . . . . . . . . . . . . . . . 3

1.3 Linear Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Projective Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . 10

1.5 Morita Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Hasse Principle 15

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Division Algebras with an involution of the first kind over two dimensional local fields . . . . . . . . . . . . . . . . 17

2.3 2-torsion division algebras with an involution of the second kind over

two dimensional local fields . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Maximal Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 A local global principle for hermitian forms over two dimensional local

fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Behavior under blowups . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 Springer’s problem for odd degree extensions 43

3.1 Complete discretely valued fields . . . . . . . . . . . . . . . . . . . . 44

3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Bibliography 47

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Palavra-chave	Algebraic Number Theory Algebraic Geometry
Committee Chair / Thesis Advisor	Suresh Venapally, Emory University
Committee Members	David Zureick-Brown, Emory University Raman Parimala, Emory University

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