scholarly journals On the second rational K -group of an elliptic curve over global fields of positive characteristic

2011 ◽  
Vol 102 (6) ◽  
pp. 1053-1098 ◽  
Author(s):  
Satoshi Kondo ◽  
Seidai Yasuda
Keyword(s):  
Elliptic Curve ◽  
Positive Characteristic ◽  
Global Fields

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.

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 The remaining cases of the Kramer–Tunnell conjecture

2016 ◽  
Vol 152 (11) ◽  
pp. 2255-2268
Author(s):  
Kęstutis Česnavičius ◽  
Naoki Imai
Keyword(s):  
Elliptic Curve ◽  
Local Field ◽  
Positive Characteristic ◽  
Quadratic Extension ◽  
Base Change ◽  
Local Fields ◽  
Root Number ◽  
Characteristic Case ◽  
Local Root

For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the base change of $E$ to the quadratic extension in terms of a certain norm index. The formula is known in all cases except some where $K$ is of characteristic $2$, and we complete its proof by reducing the positive characteristic case to characteristic $0$. For this reduction, we exploit the principle that local fields of characteristic $p$ can be approximated by finite extensions of $\mathbb{Q}_{p}$: we find an elliptic curve $E^{\prime }$ defined over a $p$-adic field such that all the terms in the Kramer–Tunnell formula for $E^{\prime }$ are equal to those for $E$.

scholarly journals Descent obstructions and Brauer–Manin obstruction in positive characteristic

2012 ◽  
Vol 12 (3) ◽  
pp. 545-551
Author(s):  
David Harari ◽  
José Felipe Voloch
Keyword(s):  
Positive Characteristic ◽  
Integral Points ◽  
Connected Group ◽  
Group Schemes ◽  
Global Fields

AbstractWe prove that the Brauer–Manin obstruction is the only obstruction to the existence of integral points on affine varieties over global fields of positive characteristic $p$. More precisely, we show that the only obstructions come from étale covers of exponent $p$ or, alternatively, from flat covers coming from torsors under connected group schemes of exponent $p$.

scholarly journals Tight Closure and Plus Closure for Cones Over Elliptic Curves

2005 ◽  
Vol 177 ◽  
pp. 31-45 ◽  
Author(s):  
Holger Brenner
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Positive Characteristic ◽  
Primary Ideal ◽  
Coordinate Ring ◽  
Tight Closure

We characterize the tight closure of a homogeneous primary ideal in a normal homogeneous coordinate ring over an elliptic curve by a numerical condition and we show that it is in positive characteristic the same as the plus closure.

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