Le résultat central est une nouvelle version du théorème d’irréductibilité de Hilbert qui fournit des bornes explicites pour le nombre de spécialisations de hauteur bornée d’un polynôme à deux variables sur un corps de nombres . Comme application, en démarrant d’une extension galoisienne finie régulière de groupe , nous pouvons compter le nombre d’extensions spécialisées conservant le groupe et dont la norme du discriminant de l’extension est majorée. En conséquence, nous contribuons à la conjecture de Malle sur le nombre d’extensions finies galoisiennes sur un corps de nombres , de groupe et de norme de discriminant . Pour chaque corps de nombres contenant un certain corps de nombres (dépendant de ), nous établissons cette minoration : pour et pour un exposant spécifique dépendant de . Nous pouvons aussi décrire le comportement local des extensions spécialisées en certains premiers. Nous déduisons ainsi de nouveaux résultats sur le problème local-global de Grunwald, en particulier pour certains groupes non résolubles.
The central result is a new version of the Hilbert Irreducibility Theorem which provides explicit bounds for the number of specializations of bounded height of two-variable polynomials over number fields . As an application, starting from a regular finite Galois extension of group , we can count the number of specialized extensions retaining the full Galois group and that are of bounded discriminant norm . Consequently, we contribute to the Malle conjecture on the number of finite Galois extensions of some number field , of group and of discriminant norm . For every number field containing a certain number field (depending on ), we establish this lower bound: for and some specific exponent depending on . We can also prescribe the local behaviour of the specialized extensions at some primes. We deduce new results on the local-global Grunwald problem, in particular for some non-solvable groups.
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Keywords: Galois extensions, Inverse Galois theory, Malle conjecture, Grunwald problem, Algebraic covers, Specialization, Diophantine geometry
Mot clés : Extensions galoisiennes, Théorie inverse de Galois, Conjecture de Malle, Problème de Grunwald, Revêtements algébriques, Spécialisation, Géométrie diophantienne.
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@article{AIF_2023__73_5_2099_0, author = {Motte, Fran\c{c}ois}, title = {Hilbert irreducibility, the {Malle} conjecture and the {Grunwald} problem}, journal = {Annales de l'Institut Fourier}, pages = {2099--2134}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {5}, year = {2023}, doi = {10.5802/aif.3567}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3567/} }
TY - JOUR AU - Motte, François TI - Hilbert irreducibility, the Malle conjecture and the Grunwald problem JO - Annales de l'Institut Fourier PY - 2023 SP - 2099 EP - 2134 VL - 73 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3567/ DO - 10.5802/aif.3567 LA - en ID - AIF_2023__73_5_2099_0 ER -
%0 Journal Article %A Motte, François %T Hilbert irreducibility, the Malle conjecture and the Grunwald problem %J Annales de l'Institut Fourier %D 2023 %P 2099-2134 %V 73 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3567/ %R 10.5802/aif.3567 %G en %F AIF_2023__73_5_2099_0
Motte, François. Hilbert irreducibility, the Malle conjecture and the Grunwald problem. Annales de l'Institut Fourier, Tome 73 (2023) no. 5, pp. 2099-2134. doi : 10.5802/aif.3567. https://aif.centre-mersenne.org/articles/10.5802/aif.3567/
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