Corps de nombres cubiques cycliques  ayant une capitulation harmonieusement équilibrée

Bill Allombert; Daniel C. Mayer

doi:10.5802/pmb.60

Corps de nombres cubiques cycliques ayant une capitulation harmonieusement équilibrée

Bill Allombert ¹ ; Daniel C. Mayer ²

¹ IMB/CNRS Université de Bordeaux 351 cours de la Libération 33405 Bordeaux France
² Naglergasse 53 8010 Graz Austria

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2025), pp. 21-46.

Abstract (VO)
Résumé

It is proved that $c=689\,347=31\cdot 37\cdot 601$ is the smallest conductor of a cyclic cubic number field $K$ whose maximal unramified pro-$3$-extension $E=\mathrm{F}_3^\infty (K)$ possesses an automorphism group $G=\mathrm{Gal}(E/K)$ of order $6561$ with coinciding relation and generator rank $d_2(G)=d_1(G)=3$ and harmonically balanced transfer kernels $\varkappa (G)\in S_{13}$. The result depends on computations done under the assumption of the GRH.

Nous établissons que $c=689\,347=31\cdot 37\cdot 601$ est le plus petit conducteur d’un corps cubique cyclique $K$ dont la pro-$3$-extension maximale non-ramifiée $E=\mathrm{F}_3^\infty (K)$ admet un groupe d’automorphismes $G=\mathrm{Gal}(E/K)$ d’ordre $6561$, avec égalité du rang des relations et des générateurs $d_2(G)=d_1(G)=3$, et des noyaux de transfert harmonieusement équilibrés $\varkappa (G)\in S_{13}$. Le résultat dépend de calculs fait sous l’hypothèse de Riemann généralisée.

Publié le : 2025-08-28

DOI : 10.5802/pmb.60

Classification : 20D15, 20E22, 20F05, 11R16, 11R29, 11R32, 11R37
Keywords: Finite $3$-groups, elementary tricyclic commutator quotient, relation rank, closed groups, Schur groups, Andozhskii–Tsvetkov groups, maximal and second maximal subgroups, kernels of Artin transfers, abelian quotient invariants, rank distribution, $p$-group generation algorithm, descendant tree, cyclic cubic number fields, harmonically balanced capitulation, Galois groups, $3$-class field tower

Affiliations des auteurs :

Bill Allombert ¹ ; Daniel C. Mayer ²

¹ IMB/CNRS Université de Bordeaux 351 cours de la Libération 33405 Bordeaux France
² Naglergasse 53 8010 Graz Austria

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Corps de nombres cubiques cycliques  ayant une capitulation harmonieusement \'equilibr\'ee},
     journal = {Publications math\'ematiques de Besan\c{c}on. Alg\`ebre et th\'eorie des nombres},
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Bill Allombert; Daniel C. Mayer. Corps de nombres cubiques cycliques  ayant une capitulation harmonieusement équilibrée. Publications mathématiques de Besançon. Algèbre et théorie des nombres (2025), pp. 21-46. doi : 10.5802/pmb.60. https://pmb.centre-mersenne.org/articles/10.5802/pmb.60/

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