no. 4
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, Farshid1 ; Maire, Christian2 ; Ramakrishna, Ravi3
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, Tome 75 (2025) no. 4, pp. 1415-1462.

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

\[0\le \mathrm{Def}(\mathrm{G}_\emptyset ) \le \dim (Ø_\mathbb{K}^\times /(Ø_\mathbb{K}^{\times })^p).\]

We explore connections between the relations of $\mathrm{G}_\emptyset $ and the Galois module structure of the units in the tower $\mathbb{K}_\emptyset /\mathbb{K}$. If $\mu _p \lnot \subset \mathbb{K}$, we give an exact formula for $\mathrm{Def}(\mathrm{G}_\emptyset )$ in terms of the number of independent Minkowski units in the tower. We also study the depth of the relations of $\mathrm{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 $\mathrm{G}_\emptyset $ with $\mathrm{Def}(\mathrm{G}_\emptyset )=0$ and $\dim (Ø_\mathbb{K}^\times /(Ø_\mathbb{K}^{\times })^p)$ large, so that the upper bound is not sharp.

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

\[0\le \mathrm{Def}(\mathrm{G}_\emptyset ) \le \dim (Ø_\mathbb{K}^\times /(Ø_\mathbb{K}^{\times })^p).\]

Nous explorons les connexions entre les relations de $\mathrm{G}_\emptyset $ et la structure galoisienne des unités dans la tour $\mathbb{K}_\emptyset /\mathbb{K}$. Si $\mu _p \lnot \subset \mathbb{K}$, nous donnons une formule exacte pour $\mathrm{Def}(\mathrm{G}_\emptyset )$ en termes du nombre d’unités de Minkowski indépendantes dans la tour. Nous étudions également la profondeur des relations de $\mathrm{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 $\mathrm{G}_\emptyset $ est infini avec $\mathrm{Def}(\mathrm{G}_\emptyset )=0$ et $\dim (Ø_\mathbb{K}^\times /(Ø_\mathbb{K}^{\times })^p)$ grand, de sorte que la borne supérieure n’est pas optimale.

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Révisé le :
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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

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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Deficiency of $p$-class tower groups and {Minkowski} units},
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Hajir, Farshid; Maire, Christian; Ramakrishna, Ravi. Deficiency of $p$-class tower groups and Minkowski units. Annales de l'Institut Fourier, Tome 75 (2025) no. 4, pp. 1415-1462. doi : 10.5802/aif.3677. https://aif.centre-mersenne.org/articles/10.5802/aif.3677/

[1] Andožski, I. V. On some classes of closed pro-p-groups, Math. USSR, Izv., Volume 9 (1965) no. 4, pp. 663-691 | DOI | Zbl

[2] Benjamin, Elliot; Lemmermeyer, Franz; Snyder, Chip Imaginary quadratic fields k with cyclic Cl 2 (k 1 ), J. Number Theory, Volume 67 (1997) no. 2, pp. 229-245 | DOI | MR | Zbl

[3] Bosma, Wieb; Cannon, John; Playoust, Catherine The MAGMA algebra system. I. The user language, J. Symb. Comput., Volume 24 (1997) no. 3-4, pp. 235-265 | DOI | MR | Zbl

[4] Boston, Nigel; Wang, Jiuya The 2-class tower of ℚ(-5460), Geometry, algebra, number theory, and their information technology applications (Springer Proceedings in Mathematics & Statistics), Volume 251, Springer (2018), pp. 71-80 | Zbl

[5] Curtis, Charles W.; Reiner, Irving Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, 11, John Wiley & Sons, 1988 | MR | Zbl

[6] Dixon, John D.; Du Sautoy, Marcus P. F.; Mann, Avinoam; Segal, Dan Analytic pro-p groups, London Mathematical Society Lecture Note Series, 157, Cambridge University Press, 1999 | DOI | MR | Zbl

[7] Fontaine, Jean-Marc; Mazur, Barry Geometric Galois representations, Elliptic curves, modular forms, and Fermat’s last theorem (Series in Number Theory), Volume 1, International Press, 1995, pp. 41-78 | Zbl

[8] Fouvry, Étienne; Klüners, Jürgen On the 4-rank of the class groups of quadratic number fields, Invent. Math., Volume 167 (2007) no. 3, pp. 455-513 | DOI | MR | Zbl

[9] Golod, E. S.; Shafarevich, Igor R. On the class field tower, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 28 (1964), pp. 261-272 | MR | Zbl

[10] Gras, Georges Class Field Theory, From theory to practice, Springer Monographs in Mathematics, Springer, 2005 | MR

[11] Gras, Georges On the T-ramified, S-split p-class field towers over an extension of degree prime to p, J. Number Theory, Volume 129 (2009) no. 11, pp. 2843-2852 | DOI | MR | Zbl

[12] Gras, Georges; Munnier, Adeline Extensions cycliques T-totalement ramifiées, Publ. Math. Besançon, Volume 1998 (1998), 6, 17 pages | Numdam

[13] Hajir, Farshid; Maire, Christian; Ramakrishna, Ravi On the Shafarevich group of restricted ramification extensions of number fields in the tame case, Indiana Univ. Math. J., Volume 70 (2021) no. 6, pp. 2693-2710 | DOI | MR | Zbl

[14] Kisilevsky, H. Number Fields with Class Number congruent to 4 mod 8 and Hilbert’s Theorem 94, J. Number Theory, Volume 8 (1976), pp. 271-279 | DOI | MR | Zbl

[15] Koch, Helmu Galois Theory of p-Extensions, Springer Monographs in Mathematics, Springer, 2002 | DOI | MR | Zbl

[16] Labute, John Mild pro-p-groups and Galois groups of p-extensions of ℚ, J. Reine Angew. Math., Volume 596 (2006), pp. 155-182 | MR | Zbl

[17] Labute, John; Mináč, Ján Mild pro-2-groups and 2-extensions of ℚ with restricted ramification, J. Algebra, Volume 332 (2011), pp. 136-158 | DOI | MR | Zbl

[18] Lazard, Michel Groupes analytiques p-adiques, Publ. Math., Inst. Hautes Étud. Sci., Volume 26 (1965), pp. 389-603 | MR | Zbl

[19] Liu, Yuan; Wood, Melanie Matchett; Zureick-Brown, David A predicted distribution for Galois groups of maximal unramified extensions, Invent. Math., Volume 237 (2024), pp. 49-116 | DOI | MR | Zbl

[20] Maire, Christian Cohomology of number fields and analytic pro-p-groups, Mosc. Math. J., Volume 10 (2010) no. 2, pp. 399-414 | DOI | MR | Zbl

[21] Maire, Christian On the quotients of the maximal unramified 2-extensions of a number field, Doc. Math., Volume 23 (2018), pp. 1263-1290 | DOI | MR | Zbl

[22] Martinet, Jacques Tours de corps de classes et estimations de discriminants, Invent. Math., Volume 44 (1978), pp. 65-73 | DOI | MR | Zbl

[23] Mazur, Barry Notes on étale cohomology of number fields, Ann. Sci. Éc. Norm. Supér., Volume 6 (1973), pp. 521-553 | DOI | Numdam | MR | Zbl

[24] Neukirch, Jürgen; Schmidt, Alexander; Wingberg, Kay Cohomology of Number Fields, Grundlehren der Mathematischen Wissenschaften, 323, Springer, 2013 (second edition, corrected second printing) | MR

[25] Ozaki, Manabu Construction of maximal unramified p-extensions with prescribed Galois groups, Invent. Math., Volume 183 (2011) no. 3, pp. 649-680 | DOI | MR | Zbl

[26] The PARI Group PARI/GP version 2.9.4, 2018 (available from http://pari.math.u-bordeaux.fr/)

[27] Ribes, Luis; Zalesskii, Pavel Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 40, Springer, 2010 | DOI | MR | Zbl

[28] Roquette, Peter On Class Field Towers, Algebraic Number Theory (Cassels, John W. S.; Fröhlich, Albrecht, eds.), Academic Press Inc. (1967), pp. 231-249 | Zbl

[29] Schmidt, Alexander Bounded defect in partial Euler characteristics, Bull. Lond. Math. Soc., Volume 28 (1996) no. 5, pp. 463-464 | DOI | MR | Zbl

[30] Schmidt, Alexander Rings of integers of type K(π,1), Doc. Math., Volume 12 (2007), pp. 441-471 | DOI | MR | Zbl

[31] Schmidt, Alexander Über Pro-p-Fundamentalgruppen markierter arithmetischer Kurven, J. Reine Angew. Math., Volume 640 (2010), pp. 203-235 | MR | Zbl

[32] Scholz, Arnold Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. Reine Angew. Math., Volume 166 (1932), pp. 201-203 | DOI | MR | Zbl

[33] Schoof, René Infinite class field towers of quadratic fields, J. Reine Angew. Math., Volume 372 (1986), pp. 209-220 | MR | Zbl

[34] Scott, William R. Group Theory, Dover Publications, 1987 | MR

[35] Serre, Jean-Pierre Cohomologie Galoisienne, Lecture Notes in Mathematics, 5, Springer, 1997 | DOI | Numdam | MR

[36] Shafarevich, Igor R. Algebraic number fields, Proceedings of the International Congress of Mathematicians 1962, Inst. Mittag-Leffler (1963), pp. 163-176 | MR | Zbl

[37] Shafarevich, Igor R. Extensions with prescribed ramification points, Publ. Math., Inst. Hautes Étud. Sci., Volume 18 (1964), pp. 71-95 (in Russian; English translation Amer. Math. Soc. Transl., 59, p. 128-149) | MR

[38] Vinberg, Èrnest Borisovich On a theorem concerning on infinite dimensionality of an associative algebra, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 29 (1965), pp. 208-214 in Russian; English translation Amer. Mat. Soc. Transl. (2) 82, p. 237-242 | MR | Zbl

[39] Wingberg, Kay On Demuškin groups with involution, Ann. Sci. Éc. Norm. Supér., Volume 22 (1989) no. 4, pp. 555-567 | DOI | Numdam | MR | Zbl

[40] Wingberg, Kay On the Fontaine–Mazur conjecture for CM-fields, Compos. Math., Volume 131 (2002) no. 3, pp. 341-354 | DOI | MR | Zbl

[41] Wingberg, Kay Free quotients of Demushkin groups with operators (2004) (preprint)

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