Galois covers of \mathbb{P}^1 over \mathbb{Q} with prescribed local or global behavior by specialization

Plans, Bernat; Vila, Núria Sala i

doi:10.5802/jtnb.490

Cited by 16 publications

(10 citation statements)

References 18 publications

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“…Similar Hilbert-Grunwald-RIGP questions are addressed in a paper of Plans and Vila [35], for a few groups and for specific G-covers X → P 1 , generally derived from the rigidity method. Here, G can be any group in Theorem 1.3 and Theorem 1.2 a priori applies to all K-G-covers X → P 1 .…”

Section: Earlier Workmentioning

confidence: 88%

“…Here, G can be any group in Theorem 1.3 and Theorem 1.2 a priori applies to all K-G-covers X → P 1 . We have however a big enough condition on q v and p v for v ∈ S. This condition can in fact not be removed: as Wang's counter-example to Grunwald's theorem or other examples in [35] show, there are situations where some local unramified behaviours cannot occur. These counter-examples however all involve the prime p = 2 and it seems unknown whether counter-examples exist with other primes.…”

Section: Earlier Workmentioning

confidence: 99%

See 1 more Smart Citation

Galois Covers and the Hilbert-Grunwald Property

Dèbes¹,

Ghazi²

2012

Ann. inst. Fourier

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Tome 62, n o 3 (2012), p. 989-1013. © Association des Annales de l'institut Fourier, 2012, tous droits réservés. L'accès aux articles de la revue « Annales de l'institut Fourier » (http://aif.cedram.org/), implique l'accord avec les conditions générales d'utilisation (http://aif.cedram.org/legal/). Toute reproduction en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l'utilisation à fin strictement personnelle du copiste est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/

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Section: Earlier Workmentioning

confidence: 88%

Section: Earlier Workmentioning

confidence: 99%

Galois Covers and the Hilbert-Grunwald Property

Dèbes¹,

Ghazi²

2012

Ann. inst. Fourier

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“…Related conclusions can be found in an earlier paper of Plans and Vila [PV05], for specific finite Galois extensions E/Q(T ) such that E/Q is regular, generally derived from the rigidity method. Here there are no restriction on the extension E/Q(T ) and the inertia groups may be specified.…”

Section: Introductionmentioning

confidence: 93%

Specialization results and ramification conditions

Legrand

2016

Isr. J. Math.

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Abstract. Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T ) with group G such that E/k is regular, we produce some specializations of E/k(T ) at points t 0 ∈ P 1 (k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior -ramified or unramified -at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T ) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T ).

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“…Since the sets of specialization values s 1 ∈ Q admissible for the above form an S-adically open set for S the finite set of bad primes of E/Q(s), the well-known compatibility of Hilbert's irreducibility theorem with weak approximation (e.g., [19,Proposition 2.1]) instantly yields infinitely many linearly disjoint extensions E s1 /Q with full Galois group M 22 and embedding into 4.M 22 -extensions. Proof.…”

Section: 1mentioning

confidence: 99%