DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

Abstract

We provide evidence for this conclusion: given a finite Galois cover f : X -> P-Q(1) of group G, almost all (in a density sense) realizations of G over 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 Q of given group and bounded discriminant. This widely extends a result of Granville on the lack of Q-rational points on quadratic twists of hyperelliptic curves over Q with large genus, under the abc-conjecture (a diophantine reformulation of the case G = Z/2Z 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 P-Q(1) of group G. We also introduce a local{global principle for specializations of Galois covers f : X -> P-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 P-Q(1). On the other hand, it allows to generate conditionally 'many' curves over Q failing the Hasse principle, thus generalizing a recent result of Clark and Watson devoted to the hyperelliptic case.