On the local-global principle for embedding problems over global fields

AbstractWe extend existing results characterizing Weil-Châtelet divisibility of locally trivial torsors over number fields to global fields of positive characteristic. Building on work of González-Avilés and Tan, we characterize when local-global divisibility holds in such contexts, providing examples showing that these results are optimal. We give an example of an elliptic curve over a global field of characteristic 2 containing a rational point which is locally divisible by 8, but is not divisible by 8 as well as examples showing that the analogous local-global principle for divisibility in the Weil-Châtelet group can also fail.

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MILD PRO-p-GROUPS AS GALOIS GROUPS OVER GLOBAL FIELDS

International Journal of Number Theory ◽

10.1142/s1793042109002377 ◽

2009 ◽

Vol 05 (05) ◽

pp. 779-795 ◽

Cited By ~ 1

Author(s):

LANDRY SALLE

Keyword(s):

Galois Group ◽

Cohomology Group ◽

Function Fields ◽

Number Fields ◽

Galois Groups ◽

Global Field ◽

Global Fields ◽

Global Principle

This paper is devoted to finding new examples of mild pro-p-groups as Galois groups over global fields, following the work of Labute ([6]). We focus on the Galois group [Formula: see text] of the maximal T-split S-ramified pro-p-extension of a global field k. We first retrieve some facts on presentations of such a group, including a study of the local-global principle for the cohomology group [Formula: see text], then we show separately in the case of function fields and in the case of number fields how it can be used to find some mild pro-p-groups.

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Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0053 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Diego Izquierdo ◽

Giancarlo Lucchini Arteche

Keyword(s):

Laurent Series ◽

Homogeneous Spaces ◽

Function Fields ◽

Independent Interest ◽

Complex Numbers ◽

Brauer Group ◽

Two Dimensional ◽

Global Fields ◽

Global Principles ◽

Global Principle

Abstract In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field ℂ ⁢ ( ( x , y ) ) {\mathbb{C}((x,y))} of Laurent series in two variables over the complex numbers and over function fields of curves over ℂ ⁢ ( ( t ) ) {\mathbb{C}((t))} . We give examples that prove that the Brauer–Manin obstruction with respect to the whole Brauer group is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.

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