Local-global principles for norm one tori and multinorm tori over semi-global fields 公开

Mishra, Sumit Chandra (Summer 2020)

Permanent URL: https://etd.library.emory.edu/concern/etds/w3763797q?locale=zh
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Abstract

Let be a complete discretely valued field with the residue field . Let be the function field of a smooth, projective, geometrically integral curve over and be a regular proper model of such that the reduced special fibre is a union of regular curves with normal crossings. Suppose that the graph associated to is a tree (e.g. ). Let be a Galois extension of degree such that is coprime to . Suppose that is an algebraically closed field or a finite field containing a primitive root of unity. Then we show that the local-global principle holds for the norm one torus associated to the extension with respect to discrete valuations on , i.e., an element in is a norm from the extension if and only if it is a norm from the extensions for all discrete valuations of . We also prove that such a local-global principle holds for multinorm tori over associated to two cyclic extensions each of degree for a prime if the residue field is algebraically closed or a finite field. We prove that for finitely many quadratic cyclic extensions, the local-global principle holds for the associated multinorm tori if the residue field is algebraically closed, and the graph associated to a regular proper model is a tree.

Table of Contents

Introduction 1 Prerequisites 7

2.1. Linear Algebraic Groups ................................7

2.2. Galois cohomology and torsors.......................11

2.2.1. Profinite cohomology............................12

2.2.2. The Long exact sequence associated

to a short exact sequence of -groups ...13

2.3. Local-global principles for linear algebraic

groups...........................................................16

2.4. -equivalence and flasque tori........................16

3. Semi-global fields and Patching 21

3.1. Overfields of a semi-global field......................22

3.2. Tate-Shafarevich groups.................................22

3.3. The associated graph......................................23

4. Local-global principles for norm one tori over

semi-global fields 27

4.1. Norm one elements- completely discretely

valued fields................................................27

4.2. Two dimensional complete fields..................29

4.2.1. Structure of extensions of two

dimensional complete fields.................30

4.2.2. Norms over two dimensional complete

fields..................................................33

4.3. Sha vs Sha_X...............................................35

4.4. Local-Global Principle.................................37

5. Local-global principle for multinorm tori over

semi-global fields 41

5.1. Sha vs Sha_X...............................................42

5.2. Local-Global Principle..................................44

5.2.1. Multinorm tori associated to two

degree p cyclic extensions................... 44

5.2.2. More general multinorm tori...............48

6. Examples of failure of local-global principle for

norm one tori and multinorm tori over semi-global

fields 58

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