scholarly journals Local-global principle for Witt equivalence of function fields over global fields

2002 ◽  
Vol 91 (2) ◽  
pp. 293-302 ◽  
Author(s):  
Przemyslaw Koprowski
Keyword(s):  
Function Fields ◽  
Global Fields ◽  
Witt Equivalence ◽  
Global Principle

scholarly journals Witt equivalence of function fields over global fields

2017 ◽  
Vol 369 (11) ◽  
pp. 7861-7881 ◽  
Author(s):  
Paweł Gładki ◽  
Murray Marshall
Keyword(s):  
Function Fields ◽  
Global Fields ◽  
Witt Equivalence

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

2009 ◽  
Vol 05 (05) ◽  
pp. 779-795 ◽  
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.

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

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.

scholarly journals Witt equivalence of function fields of conics

2020 ◽  
Vol 30 (1) ◽  
pp. 63-78
Author(s):  
P. Gladki ◽  
◽  
M. Marshall
Keyword(s):  
Quadratic Form ◽  
Function Fields ◽  
Local Fields ◽  
Bilinear Forms ◽  
Conic Sections ◽  
Witt Rings ◽  
Global Fields ◽  
Witt Equivalence ◽  
Form Theory ◽  
And Function

Two fields are Witt equivalent if, roughly speaking, they have the same quadratic form theory. Formally, that is to say that their Witt rings of symmetric bilinear forms are isomorphic. This equivalence is well understood only in a few rather specific classes of fields. Two such classes, namely function fields over global fields and function fields of curves over local fields, were investigated by the authors in their earlier works [5] and [6]. In the present work, which can be viewed as a sequel to the earlier papers, we discuss the previously obtained results in the specific case of function fields of conic sections, and apply them to provide a few theorems of a somewhat quantitive flavour shedding some light on the question of numbers of Witt non-equivalent classes of such fields.

scholarly journals Witt equivalence of function fields of curves over local fields

2017 ◽  
Vol 45 (11) ◽  
pp. 5002-5013
Author(s):  
Paweł Gładki ◽  
Murray Marshall
Keyword(s):  
Function Fields ◽  
Local Fields ◽  
Witt Equivalence

scholarly journals Local–global principle for reduced norms over function fields of -adic curves

2017 ◽  
Vol 154 (2) ◽  
pp. 410-458 ◽  
Author(s):  
R. Parimala ◽  
R. Preeti ◽  
V. Suresh
Keyword(s):  
Local Field ◽  
Function Field ◽  
Function Fields ◽  
Residue Field ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Hasse Principle ◽  
Central Simple ◽  
Global Principle ◽  
Reduced Norm

Let $K$ be a (non-archimedean) local field and let $F$ be the function field of a curve over $K$. Let $D$ be a central simple algebra over $F$ of period $n$ and $\unicode[STIX]{x1D706}\in F^{\ast }$. We show that if $n$ is coprime to the characteristic of the residue field of $K$ and $D\cdot (\unicode[STIX]{x1D706})=0$ in $H^{3}(F,\unicode[STIX]{x1D707}_{n}^{\otimes 2})$, then $\unicode[STIX]{x1D706}$ is a reduced norm from $D$. This leads to a Hasse principle for the group $\operatorname{SL}_{1}(D)$, namely, an element $\unicode[STIX]{x1D706}\in F^{\ast }$ is a reduced norm from $D$ if and only if it is a reduced norm locally at all discrete valuations of $F$.

scholarly journals The Farey density of norm subgroups of global fields (II)

1977 ◽  
Vol 18 (1) ◽  
pp. 57-67
Author(s):  
S. D. Cohen ◽  
R. W. K. Odoni
Keyword(s):  
Algebraic Number ◽  
Function Fields ◽  
Rational Functions ◽  
Number Fields ◽  
Ground Field ◽  
Algebraic Number Fields ◽  
Global Fields ◽  
Finite Constant

In this paper we shall derive for function fields in one variable over finite constant fields results analogous to [1], where algebraic number fields were considered. The ground field P will be the set of all rational functions in a given transcendent X, with coefficients in k = GF(q), q = pr, p a prime; thus P = k(X).

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