On the average size of 3-torsion in class groups of $C_2 \wr H$-extensions
Jonas Iskander1 ; Hari R. Iyer2
1 Department of Mathematics, Harvard University, Cambridge, MA 02138, United States
2 Department of Mathematics, Princeton University, Princeton, NJ 08540, United States
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 1031-1039

The Cohen–Lenstra–Martinet heuristics lead one to conjecture that the average size of the $p$-torsion in class groups of $G$-extensions of a number field is finite. In a 2021 paper, Lemke Oliver, Wang, and Wood proved this conjecture in the case of $p = 3$ for permutation groups $G$ of the form $C_2 \wr H$ for a broad family of permutation groups $H$, including most nilpotent groups. However, their theorem does not apply for some nilpotent groups of interest, such as $H = C_5$. We extend their results to prove that the average size of $3$-torsion in class groups of $C_2 \wr H$-extensions is finite for any nilpotent group $H$.

Les heuristiques de Cohen, Lenstra, et Martinet conduisent à conjecturer que la taille moyenne de la $p$-torsion dans les groupes de classes des $G$-extensions d’un corps de nombres est finie. Dans un article de 2021, Lemke Oliver, Wang et Wood ont démontré cette conjecture dans le cas $p = 3$ pour les groupes de permutations $G$ de la forme $C_2 \wr H$, pour une large famille de groupes de permutations $H$, incluant la plupart des groupes nilpotents. Cependant, leur théorème ne s’applique pas à certains groupes nilpotents d’intérêt, tels que $H = C_5$. Nous étendons leurs résultats afin de montrer que la taille moyenne de la $3$-torsion dans les groupes de classes des $C_2 \wr H$-extensions est finie pour tout groupe nilpotent $H$.

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DOI : 10.5802/jtnb.1350
Classification : 11R29
Keywords: Class groups, arithmetic statistics

Jonas Iskander 1 ; Hari R. Iyer 2

1 Department of Mathematics, Harvard University, Cambridge, MA 02138, United States
2 Department of Mathematics, Princeton University, Princeton, NJ 08540, United States
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Jonas Iskander; Hari R. Iyer. On the average size of 3-torsion in class groups of $C_2 \wr H$-extensions. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 3, pp. 1031-1039. doi: 10.5802/jtnb.1350

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