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DENSITY RESULTS FOR SPECIALIZATION SETS OF GALOIS COVERS

Part of: Birational geometry Algebraic number theory: global fields Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  25 October 2019

Joachim König Open the ORCID record for Joachim König [Opens in a new window]  and
François Legrand
Joachim König
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu Daejeon34141, South Korea (jkoenig@kaist.ac.kr)
François Legrand
Affiliation:
Institut für Algebra, Fachrichtung Mathematik, TU Dresden, 01062Dresden, Germany (francois.legrand@tu-dresden.de)
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Abstract

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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.

Keywords

Galois theoryspecializationsthe abc-conjecturethe Malle conjecturethe uniformity conjecturehyperelliptic and superelliptic curvesrational pointstwisted coversthe Hasse principle

MSC classification

Primary: 11R32: Galois theory
Secondary: 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 14E20: Coverings 14G05: Rational points
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 5 , September 2021 , pp. 1455 - 1496
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2019. Published by Cambridge University Press

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