DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

Joachim König,François Legrand

doi:10.1017/s1474748019000537

Abstract

AbstractWe 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.

Highlights

Given a finite Galois extension E of the rational function field Q(T ), and a point t0 ∈ P1(Q), there is a well-known notion of specialization Et0 /Q
If E is the splitting field of a monic separable polynomial P(T, Y ) ∈ Q[T ][Y ] and t0 ∈ Q is such that P(t0, Y ) is separable, the field Et0 is the splitting field over Q of P(t0, Y )
The specialization process has been much studied toward the inverse Galois problem, which asks whether every finite group G occurs as the Galois group of a finite Galois extension F/Q

Summary

Introduction

Given a finite Galois extension E of the rational function field Q(T ), and a point t0 ∈ P1(Q), there is a well-known notion of specialization Et0 /Q (see § 2.2.1 for more details). If E is the splitting field of a monic separable polynomial P(T, Y ) ∈ Q[T ][Y ] and t0 ∈ Q is such that P(t0, Y ) is separable, the field Et0 is the splitting field over Q of P(t0, Y ). The specialization process has been much studied toward the inverse Galois problem, which asks whether every finite group G occurs as the Galois group of a finite Galois extension F/Q. We shall say that such an extension F/Q is a G-extension

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Conclusion