Density Results for Specialization Sets of Galois covers

Abstract

We provide evidence for this conclusion: given a finite Galois cover f : X → P 1 Q 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 1 Q of group G. We also introduce a local-global principle for specializations of Galois covers f : X → P 1 Q 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 1 Q . 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.

Hasse principle violations in twist families of superelliptic curves

“…This result generalizes Theorem 2 of [CW18.2], which proves the result when C is hyperelliptic. In [KL19], König and Legrand recover the hyperelliptic result. They then extend the result to Galois covers of P 1 / Q , that is, when the Galois group of Q(C)/Q(P 1 ) contains in its center Z/nZ.…”

On the mod-$p$ distribution of discriminants of $G$-extensions

“…Already Hilbert's irreducibility theorem gave a partial answer to this question, ensuring the existence of infinitely many specializations with the same Galois group. Recent work by the author and others investigates the question whether the specialization set of a single regular Galois extension can answer (the lower bound of) Malle's conjecture in full ( [13]), or even contain all extensions of a prescribed Galois group ( [12], [14]), giving negative answers under weak assumptions in both cases. The above theorem complements these negative global results with a positive result in the direction of the local part of Malle's conjecture.…”