Groups with no parametric  Galois realizations

Dèbes, Pierre

doi:10.24033/asens.2353

Groups with no parametric Galois realizations
[Groupes sans réalisations galoisiennes paramétriques]

Dèbes, Pierre

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 143-179.

Résumé
Abstract

Nous réfutons une forme forte du problème inverse de Galois régulier: il existe des groupes finis $G$ qui n'ont pas de réalisation régulière $F / ℚ (T)$ induisant toutes les extensions galoisiennes $L / ℚ (U)$ de groupe $G$ par spécialisation de $T$ en $f (U) \in ℚ (U)$ . Une propriété de relèvement bien plus faible est même infirmée pour ces groupes: deux réalisations $L / ℚ (U)$ existent qui ne peuvent être induites par des réalisations ayant le même type de ramification. Nos exemples de tels groupes $G$ incluent les groupes symétriques $S_{n}$ , $n \geq 6$ , une infinité de ${PSL}_{2} (𝔽_{p})$ , le Monstre.

Deux variantes de la question, où $ℚ (U)$ est remplacé par $ℂ (U)$ et $ℚ$ , ont une réponse similaire, la seconde sous une « hypothèse de travail » liée à un problème de Fried-Schinzel.

Nous introduisons deux nouveaux outils: un théorème de comparaison entre les invariants d'une extension $F / ℂ (T)$ et ceux de celle obtenue en spécialisant $T$ en $f (U) \in ℂ (U)$ ; et, étant données deux extensions régulières galoisiennes de $k (T)$ , un ensemble fini de $k (U)$ -courbes qui disent si ces extensions ont une spécialisation commune $E / k$ .

We disprove a strong form of the Regular Inverse Galois Problem: there exist finite groups $G$ which do not have a $ℚ (U)$ -parametric extension, i.e., a regular realization $F / ℚ (T)$ that induces all Galois extensions $L / ℚ (U)$ of group $G$ by specializing $T$ to $f (U) \in ℚ (U)$ . A much weaker Lifting Property is even disproved for these groups: two realizations $L / ℚ (U)$ exist that cannot be induced by realizations with the same ramification type. Our examples of such groups $G$ include symmetric groups $S_{n}$ , $n \geq 6$ , infinitely many ${PSL}_{2} (𝔽_{p})$ , the Monster.

Two variants of the question with $ℚ (U)$ replaced by $ℂ (U)$ and $ℚ$ are answered similarly, the second one under a diophantine “working hypothesis” going back to a problem of Fried-Schinzel.

We introduce two new tools: a comparison theorem between the invariants of an extension $F / ℂ (T)$ and those obtained by specializing $T$ to $f (U) \in ℂ (U)$ ; and, given two regular Galois extensions of $k (T)$ , a finite set of $k (U)$ -curves that say whether these extensions have a common specialization $E / k$ .

MR Zbl

DOI : 10.24033/asens.2353

Classification : 12F12, 11R58, 14E20; 14E22, 12E30, 11Gxx.
Keywords: Galois extensions, inverse Galois theory, specialization, parametric extensions, twisting.
Mot clés : Extensions galoisiennes, théorie inverse de Galois, spécialisation, extensions paramétriques, twisting.

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     author = {D\`ebes, Pierre},
     title = {Groups with no parametric  {Galois} realizations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {143--179},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {1},
     year = {2018},
     doi = {10.24033/asens.2353},
     mrnumber = {3764040},
     zbl = {1387.12003},
     language = {en},
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}

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Dèbes, Pierre. Groups with no parametric  Galois realizations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 1, pp. 143-179. doi : 10.24033/asens.2353. http://www.numdam.org/articles/10.24033/asens.2353/

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