Deficiency of $p$-class tower groups and Minkowski units

Hajir, Farshid; Maire, Christian; Ramakrishna, Ravi

doi:10.5802/aif.3677

Deficiency of

p

-class tower groups and Minkowski units
[Défaut du groupe de Galois des

p

-tours de Hilbert et les unités de Minkowski]

Hajir, Farshid ¹ ; Maire, Christian ² ; Ramakrishna, Ravi ³

¹ Department of Mathematics & Statistics University of Massachusetts Amherst, MA 01003, USA
² FEMTO-ST Institute, Université Bourgogne Franche-Comté, CNRS 15B avenue des Montboucons 25000 Besançon, France
³ Department of Mathematics, Cornell University Ithaca, USA

Annales de l'Institut Fourier, Online first, 48 p.

Résumé
Abstract

Le défaut $Def (G)$ d’un pro- $p$ groupe $G$ de type fini est la différence entre son nombre minimal de relations et son nombre minimal de générateurs. Pour un corps de nombres $𝕂$ de $p$ -extension non ramifiée maximale $𝕂_{\emptyset}$ , posons $G_{\emptyset} = Gal (𝕂_{\emptyset} / 𝕂)$ . Shafarevich (et indépendamment Koch) a montré que

0 \leq Def (G_{\emptyset}) \leq dim (Ø_{𝕂}^{\times} / {(Ø_{𝕂}^{\times})}^{p}) .

Nous explorons les connexions entre les relations de $G_{\emptyset}$ et la structure galoisienne des unités dans la tour $𝕂_{\emptyset} / 𝕂$ . Si $μ_{p} \neg \subset 𝕂$ , nous donnons une formule exacte pour $Def (G_{\emptyset})$ en termes du nombre d’unités de Minkowski indépendantes dans la tour. Nous étudions également la profondeur des relations de $G_{\emptyset}$ suivant la filtration de Zassenhaus et montrons que la borne supérieure de Shafarevich–Koch est « presque toujours » optimale. Mais dans l’autre sens, nous donnons les premiers exemples pour lesquels $G_{\emptyset}$ est infini avec $Def (G_{\emptyset}) = 0$ et $dim (Ø_{𝕂}^{\times} / {(Ø_{𝕂}^{\times})}^{p})$ grand, de sorte que la borne supérieure n’est pas optimale.

The deficiency $Def (G)$ of a finitely presented pro- $p$ group $G$ is the difference between its minimal numbers of relations and generators. For a number field $𝕂$ with maximal unramified $p$ -extension $𝕂_{\emptyset}$ , set $G_{\emptyset} = Gal (𝕂_{\emptyset} / 𝕂)$ . Shafarevich (and independently Koch) showed

0 \leq Def (G_{\emptyset}) \leq dim (Ø_{𝕂}^{\times} / {(Ø_{𝕂}^{\times})}^{p}) .

We explore connections between the relations of $G_{\emptyset}$ and the Galois module structure of the units in the tower $𝕂_{\emptyset} / 𝕂$ . If $μ_{p} \neg \subset 𝕂$ , we give an exact formula for $Def (G_{\emptyset})$ in terms of the number of independent Minkowski units in the tower. We also study the depth of the relations of $G_{\emptyset}$ in the Zassenhaus filtration and provide evidence that the Shafarevich–Koch upper bound is “almost always” sharp. In the other direction, we give the first examples of infinite $G_{\emptyset}$ with $Def (G_{\emptyset}) = 0$ and $dim (Ø_{𝕂}^{\times} / {(Ø_{𝕂}^{\times})}^{p})$ large, so that the upper bound is not sharp.

Reçu le : 2021-07-24
Révisé le : 2022-05-23
Accepté le : 2022-11-10
Première publication : 2025-02-10

DOI : 10.5802/aif.3677

Classification : 11R29, 11R37
Keywords: Deficiency, Golod–Shafarevich polynomial, $p$-class field tower, Zassenhaus filtration
Mots-clés : Défaut, polynôme de Golod–Shafarevich, $p$-tour de Hilbert, filtration de Zassenhaus

Affiliations des auteurs :

Hajir, Farshid ¹ ; Maire, Christian ² ; Ramakrishna, Ravi ³

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     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     note = {Online first},
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Hajir, Farshid; Maire, Christian; Ramakrishna, Ravi. Deficiency of $p$-class tower groups and Minkowski units. Annales de l'Institut Fourier, Online first, 48 p.

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