Local-global principles for norm one tori and multinorm tori over semi-global fields Open Access
Mishra, Sumit Chandra (Summer 2020)
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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