scholarly journals Local-global principles for Weil–Châtelet divisibility in positive characteristic

2017 ◽  
Vol 163 (2) ◽  
pp. 357-367 ◽  
Author(s):  
BRENDAN CREUTZ ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curve ◽  
Rational Point ◽  
Positive Characteristic ◽  
Number Fields ◽  
Global Field ◽  
Global Fields ◽  
Characteristic 2 ◽  
Global Principles ◽  
Global Principle

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.

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 On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

scholarly journals A comparison between obstructions to local-global principles over semi-global fields

2021 ◽  
pp. 135-146
Author(s):  
David Harbater ◽  
Julia Hartmann ◽  
Valentijn Karemaker ◽  
Florian Pop
Keyword(s):  
Global Fields ◽  
Global Principles

scholarly journals Elliptic Curves over the Perfect Closure of a Function Field

2010 ◽  
Vol 53 (1) ◽  
pp. 87-94
Author(s):  
Dragos Ghioca
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Function Field ◽  
Positive Characteristic ◽  
Rational Points ◽  
Finitely Generated

AbstractWe prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

scholarly journals Points of low height on elliptic surfaces with torsion

2010 ◽  
Vol 13 ◽  
pp. 370-387
Author(s):  
Sonal Jain
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Open Subset ◽  
Function Field ◽  
Rational Point ◽  
Torsion Point ◽  
Integral Points ◽  
Elliptic Surfaces ◽  
Canonical Height

AbstractWe determine the smallest possible canonical height$\hat {h}(P)$for a non-torsion pointPof an elliptic curveEover a function field(t) of discriminant degree 12nwith a 2-torsion point forn=1,2,3, and with a 3-torsion point forn=1,2. For eachm=2,3, we parametrize the set of triples (E,P,T) of an elliptic curveE/with a rational pointPandm-torsion pointTthat satisfy certain integrality conditions by an open subset of2. We recover explicit equations for all elliptic surfaces (E,P,T) attaining each minimum by locating them as curves in our projective models. We also prove that forn=1,2 , these heights are minimal for elliptic curves over a function field of any genus. In each case, the optimal (E,P,T) are characterized by their patterns of integral points.

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