Cohen–Lenstra Moments for Some Nonabelian Groups
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 631-664.

Cohen and Lenstra gave a heuristic for the distribution of odd p-class groups for imaginary (respectively real) quadratic fields. One such formulation of this distribution is that the expected number of surjections from the class group of an imaginary quadratic field k to a fixed abelian group of odd order is 1. Class field theory tells us that the class group of k is also the Galois group of the Hilbert class field, the maximal unramified abelian extension of k, so we could equivalently say that for a fixed abelian group G of odd order the expected number of unramified G-extensions of k is 1/#Aut(G). We generalize this to asking for the expected number of unramified G-extensions of k, Galois over , for a fixed finite group G, with no restrictions placed on G. We review cases where the answer is known or conjectured by Bhargava, Boston–Bush–Hajir, and Boston–Wood, and then answer this question in several new cases. In particular, we consider when the expected number is zero and give a nontrivial family of groups realizing this. Additionally, we prove the expected number for the quaternion group Q 8 and dihedral group D 4 of order 8 is infinite. Lastly, we give evidence for the special case of groups generated by elements of order 2 for which Malle’s conjecture predicts an infinite expected number.

Cohen et Lenstra ont proposé des heuristiques sur la répartition des p-parties impaires des groupes de classes des corps quadratiques imaginaires (respectivement réels). L’un des énoncés possibles de cette répartition prédit que le nombre de surjections du groupe de classes d’un corps quadratique imaginaire k vers un groupe abélien fixé d’ordre impair est un. Comme la théorie des corps de classes nous dit que le groupe de classes de k est aussi le groupe de Galois du corps de Hilbert, l’extension abélienne non ramifiée maximale de k, nous pouvons dire, de façon équivalente, que pour un groupe abélien fixé G d’ordre impair, le nombre attendu de G-extensions non ramifiées de k est 1/#Aut(G). Nous plaçons cette question dans un cadre plus général, en nous intéressant au nombre attendu de G-extensions galoisiennes non ramifiées de k pour un groupe fini fixé G, sans restrictions sur G. Nous donnons un aperçu des cas connus et des conjectures dans cette direction dus à Bhargava, Boston–Bush–Hajir et Boston–Wood, et donnons ensuite la réponse dans plusieurs nouveaux cas. En particulier, nous donnons une famille non triviale de groupes pour lesquels le nombre attendu est zéro. En outre, nous prouvons que pour le groupe des quaternions Q 8 et pour le groupe diédral D 4 d’ordre 8 ce nombre est infini. Pour conclure, nous considérons le cas spécial des groupes engendrés par des éléments d’ordre 2, dans lequel la conjecture de Malle prédit que le nombre attendu est infini.

Published online:
DOI: 10.5802/jtnb.1137
Classification: 11N56,  11R45,  20F28
Keywords: Cohen–Lenstra, class group, arithmetic statistics
Brandon Alberts 1

1 9500 Gilman Dr. La Jolla CA 92093, United States
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     title = {Cohen{\textendash}Lenstra {Moments} for {Some} {Nonabelian} {Groups}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
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     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Brandon Alberts. Cohen–Lenstra Moments for Some Nonabelian Groups. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 3, pp. 631-664. doi : 10.5802/jtnb.1137.

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