Cohen–Lenstra Moments for Some Nonabelian Groups

Alberts, Brandon

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 3, pp. 631-664.

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 $1/\#\mathrm{Aut}\left(G\right)$. 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.

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 $1/\#\mathrm{Aut}\left(G\right)$. We generalize this to asking for the expected number of unramified $G$-extensions of $k$, Galois over $\mathbb{Q}$, 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.

Classification : 11N56, 11R45, 20F28
Mots clés : Cohen–Lenstra, class group, arithmetic statistics

[1] Bhargava, Manjul The geometric sieve and the density of squarefree values of invariant polynomials (2014) (https://arxiv.org/abs/1402.0031)

[2] Boston, Nigel Embedding 2-groups in groups generated by involutions, J. Algebra, Volume 300 (2006) no. 1, pp. 73-76 | DOI | MR | Zbl

[3] Boston, Nigel; Bush, Michael; Hajir, Farshid Heuristics for p-class towers of imaginary quadratic fields, Math. Ann., Volume 368 (2017) no. 1-2, pp. 633-669 | DOI | MR | Zbl

[4] Boston, Nigel; Wood, Melanie M. Non-abelian Cohen–Lenstra Heuristics over Function Fields, Compos. Math., Volume 153 (2017) no. 7, pp. 1372-1390 | DOI | MR | Zbl

[5] Cohen, Henri; Lenstra, Hendrik W. Jr Heuristics on class groups of number fields, Number theory, Noordwijkerhout 1983 (Lecture Notes in Mathematics), Volume 1068, Springer, 1984, pp. 33-62 | DOI | MR

[6] Davenport, Harold; Heilbronn, Hans On the Density of Cubic Fields. II, Proc. R. Soc. Lond., Ser. A, Volume 322 (1971), pp. 405-420 | MR | Zbl

[7] Delange, Hubert Généralisation du théorème de Ikehara, Ann. Sci. Éc. Norm. Supér., Volume 71 (1954), pp. 213-242 | DOI | Numdam | MR | Zbl

[8] Ellenberg, Jordan; Venkatesh, Akshay Statistics of Number Fields and Function Fields, Proceedings of the international congress of mathematicians (ICM 2010), World Scientific, 2011 | Zbl

[9] Goldfeld, Dorian; Hoffstein, Jeffrey Eisenstein series of $\frac{1}{2}$-integral weight and the mean value of real Dirichlet $L$-series, Invent. Math., Volume 80 (1985), pp. 187-208 | MR | Zbl

[10] Grönwall, Thomas H. Some asymptotic expressions in the theory of numbers, Trans. Am. Math. Soc., Volume 14 (1913), pp. 113-122 | DOI | MR | Zbl

[11] Helleloid, Geir T.; Martin, Ursula The automorphism group of a finite $p$-group is almost always a $p$-group, J. Algebra, Volume 312 (2007) no. 1, pp. 294-329 | DOI | MR | Zbl

[12] Klüners, Jürgen A counter example to Malle’s conjecture on the asymptotics of discriminants, C. R. Math. Acad. Sci. Paris, Volume 340 (2005) no. 6, pp. 411-414 | DOI | Zbl

[13] Lemmermeyer, Franz Unramified quaternion extensions of quadratic number fields, J. Théor. Nombres Bordeaux, Volume 9 (1997) no. 1, pp. 51-68 | DOI | Numdam | MR | Zbl

[14] Malle, Gunter On the Distribution of Galois Groups, J. Number Theory, Volume 92 (2002) no. 2, pp. 315-329 | DOI | MR | Zbl

[15] Malle, Gunter On the Distribution of Galois Groups, II, Exp. Math., Volume 13 (2004) no. 2, pp. 129-135 | DOI | MR | Zbl

[16] Narkiewicz, Władysław Elementary and Analytic Theory of Algebraic Numbers, PWN-Polish Scientific Publishers, 1974 | Zbl

[17] Rotman, Joseph J. An introduction to the theory of groups, Graduate Texts in Mathematics, 148, Springer, 1995 | Zbl

[18] Wang, Lei On the Automorphism Groups of Frobenius Groups, Commun. Algebra, Volume 48 (2020) no. 12, pp. 5330-5342 | DOI | MR | Zbl

[19] Wood, Melanie M. Nonabelian Cohen–Lenstra Moments, Duke Math. J., Volume 168 (2019) no. 3, pp. 377-427 | DOI | MR | Zbl