The Grunwald problem and homogeneous spaces with nonsolvable stabilisers
[Problème de Grunwald et espaces homogènes à stabilisateurs non résolubles]
Boughattas, Elyes1 ; Neftin, Danny2
1Department of Mathematical Sciences, University of Bath – Claverton Down, Bath, BA2 7AY (United Kingdom)
2Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000 (Israel)
Annales de l'Institut Fourier, Online first, 45 p.

We give an affirmative answer to the Grunwald problem for new families of nonsolvable finite groups $G$, away from the set of primes dividing $|G|$. Furthermore, we show that such $G$ verify the condition (BM), that is, the Brauer–Manin obstruction to weak approximation is the only one for quotients of $\operatorname{SL}_n$ by $G$. These new families include extensions of groups satisfying (BM) by kernels which are products of symmetric groups $\mathfrak{S}_m$, with $m\ne 2,6$, and alternating groups $\mathfrak{A}_5$. We also investigate (BM) for small groups by giving an explicit list of small order groups for which (BM) is unknown and we show that for many of them (BM) holds under Schinzel’s hypothesis.

Nous apportons une réponse positive au problème de Grunwald pour de nouvelles familles de groupes finis non résolubles $G$, en dehors de l’ensemble des places divisant $|G|$. De plus, nous montrons que ces groupes $G$ vérifient la propriété (BM), c’est-à-dire que l’obstruction de Brauer–Manin à l’approximation faible est la seule pour les quotients de $\operatorname{SL}_n$ par $G$. Ces nouvelles familles sont formées d’extensions de groupes vérifiant (BM) par des produits de groupes symétriques $\mathfrak{S}_m$, pour $m\ne 2,6$, et de copies du groupe alterné $\mathfrak{A}_5$. Nous étudions également (BM) pour des groupes de petit cardinal, en donnant une liste explicite de tels groupes pour lesquels (BM) est inconnue, et nous démontrons que pour plusieurs d’entre eux, (BM) est vérifiée si l’hypothèse de Schinzel est vraie.

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DOI : 10.5802/aif.3784
Classification : 12F12, 14G12, 14E08
Keywords: Grunwald problem, inverse Galois problem, homogeneous spaces, Galois cohomology, rationality, weak approximation, Brauer–Manin obstruction.
Mots-clés : Problème de Grunwald, problème de Galois inverse, espaces homogènes, cohomologie galoisienne, rationalité, approximation faible, obstruction de Brauer–Manin.

Boughattas, Elyes  1   ; Neftin, Danny  2

1 Department of Mathematical Sciences, University of Bath – Claverton Down, Bath, BA2 7AY (United Kingdom)
2 Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000 (Israel) 
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Boughattas, Elyes; Neftin, Danny. The Grunwald problem and homogeneous spaces with nonsolvable stabilisers. Annales de l'Institut Fourier, Online first, 45 p.

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