Density results for specialization sets of Galois covers

Abstract

We provide evidence for this conclusion: given a finite Galois cover $f:X\to {\mathbb{P}}_{\mathbb{Q}}^1$ 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 $f:X\to {\mathbb{P}}_{\mathbb{Q}}^1$ 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.

37 pages