About the ABC Conjecture and an alternative

2011 ◽  
pp. 169-180 ◽  
Author(s):  
Machiel van Frankenhuijsen
Keyword(s):  
Abc Conjecture

scholarly journals A weak effective $abc$-conjecture

2008 ◽  
Vol 39 (1) ◽  
pp. 103-111 ◽  
Author(s):  
Jerzy Browkin
Keyword(s):  
Abc Conjecture

scholarly journals On some polynomials allegedly related to the abc conjecture

1998 ◽  
Vol 84 (2) ◽  
pp. 109-128
Author(s):  
Alexandr Borisov
Keyword(s):  
Abc Conjecture

scholarly journals On Baker's explicit $abc$-conjecture

2019 ◽  
Vol 94 (3-4) ◽  
pp. 435-453
Author(s):  
Kwok Chi Chim ◽  
Tarlok Nath Shorey ◽  
Sneh Bala Sinha
Keyword(s):  
Abc Conjecture

The abc Conjecture

1999 ◽  
pp. 18-31
Author(s):  
Serge Lang
Keyword(s):  
Abc Conjecture

Squarefree Values of Polynomials and the abc-Conjecture

1997 ◽  
pp. 65-86 ◽  
Author(s):  
J. Browkin ◽  
M. Filaseta ◽  
G. Greaves ◽  
A. Schinzel
Keyword(s):  
Abc Conjecture

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.

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