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.
Dynamical uniform boundedness and the abc-conjecture
