Minkowski, Hensel, and Hasse: On the beginnings of the local-global principle

2011 ◽  
pp. 153-177
Keyword(s):  
Global Principle

scholarly journals Waring’s problem for Beatty sequences and a local to global principle

2014 ◽  
Vol 26 (1) ◽  
pp. 1-16 ◽  
Author(s):  
William D. Banks ◽  
Ahmet M. Güloğlu ◽  
Robert C. Vaughan
Keyword(s):  
Waring’S Problem ◽  
Waring's Problem ◽  
Beatty Sequences ◽  
Global Principle

On the exponential local-global principle

2013 ◽  
Vol 159 (2) ◽  
pp. 101-111 ◽  
Author(s):  
Boris Bartolome ◽  
Yuri Bilu ◽  
Florian Luca
Keyword(s):  
Global Principle

Local-global principle for annihilation of general local cohomology

2001 ◽  
Vol 87 (1) ◽  
pp. 129-136 ◽  
Author(s):  
J. Asadollahi ◽  
K. Khashyarmanesh ◽  
Sh. Salarian
Keyword(s):  
Local Cohomology ◽  
Global Principle

scholarly journals DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

2019 ◽  
pp. 1-42
Author(s):  
Joachim König ◽  
François Legrand
Keyword(s):  
Hyperelliptic Curves ◽  
Branch Points ◽  
Abc Conjecture ◽  
Galois Cover ◽  
Quadratic Twists ◽  
Large Genus ◽  
Galois Covers ◽  
The One ◽  
Almost All ◽  
Global Principle

We provide evidence for this conclusion: given a finite Galois cover $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ , almost all (in a density sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of $f$ . We show that this holds if the number of branch points of $f$ is sufficiently large, under the abc-conjecture and, possibly, the lower bound predicted by the Malle conjecture for the number of Galois extensions of $\mathbb{Q}$ of given group and bounded discriminant. This widely extends a result of Granville on the lack of $\mathbb{Q}$ -rational points on quadratic twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the abc-conjecture (a diophantine reformulation of the case $G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a few finite groups $G$ for which the above conclusion holds unconditionally for almost all covers of $\mathbb{P}_{\mathbb{Q}}^{1}$ of group $G$ . We also introduce a local–global principle for specializations of Galois covers $f:X\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ and show that it often fails if $f$ has abelian Galois group and sufficiently many branch points, under the abc-conjecture. On the one hand, such a local–global conclusion underscores the ‘smallness’ of the specialization set of a Galois cover of $\mathbb{P}_{\mathbb{Q}}^{1}$ . On the other hand, it allows to generate conditionally ‘many’ curves over $\mathbb{Q}$ failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.

A local-global principle for quadratic forms

1975 ◽  
Vol 142 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Alexander Prestel
Keyword(s):  
Quadratic Forms ◽  
Global Principle
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