Local-global principles for hermitian spaces over semi-global fields Público
Guhan, Jayanth (Summer 2022)
Published
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
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