DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS. (25th September 2021)

• Record Type: Journal Article
• Authors: König, Joachim; Legrand, François
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 20:Number 5(2021)
• Page Start: 1455
• Page End: 1496
• Publication Date: 2021-09-25
• Subjects: Galois theory -- specializations -- the abc-conjecture -- the Malle conjecture -- the uniformity conjecture -- hyperelliptic and superelliptic curves -- rational points -- twisted covers -- the Hasse principle; 11R32 -- 11G30 -- 14E20 -- 14G05; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748019000537
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract: 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.