The distribution of second $p$-class groups on coclass graphs

Mayer, Daniel C.

Journal de théorie des nombres de Bordeaux, Tome 25 (2013) no. 2, pp. 401-456.

Des concepts et des stratégies généraux sont développés pour identifier le type d’isomorphisme du deuxième $p$-groupe de classes $G=\mathrm{Gal}\left({\mathrm{F}}_p^2\left(K\right)\right|K)$, c’est donc le groupe de Galois du deuxième $p$-corps de classes de Hilbert ${\mathrm{F}}_p^2\left(K\right)$, d’un corps de nombres $K$, pour un nombre premier $p$. Le type d’isomorphisme détermine la position de $G$ sur un des graphes de coclasses $𝒢(p,r)$, $r\ge 0$, dans le sens de Eick, Leedham-Green et Newman. Il est démontré que, pour des types spéciaux du corps de base $K$ et de son $p$-groupe de classes ${\mathrm{Cl}}_p\left(K\right)$, la position de $G$ est limitée à certaines branches d’arbres de coclasses admissibles par des règles de sélection. Une compréhension plus profonde, en particulier la densité de population de sommets particuliers au niveau des graphes de coclasses, est acquise en calculant la distribution effective des deuxièmes $p$-groupes de classes $G$ pour diverses séries de corps de nombres $K$ ayant des $p$-groupes de classes ${\mathrm{Cl}}_p\left(K\right)$ de type fixe et $p\in \{2,3,5,7\}$.

General concepts and strategies are developed for identifying the isomorphism type of the second $p$-class group $G=\mathrm{Gal}\left({\mathrm{F}}_p^2\left(K\right)\right|K)$, that is the Galois group of the second Hilbert $p$-class field ${\mathrm{F}}_p^2\left(K\right)$, of a number field $K$, for a prime $p$. The isomorphism type determines the position of $G$ on one of the coclass graphs $𝒢(p,r)$, $r\ge 0$, in the sense of Eick, Leedham-Green, and Newman. It is shown that, for special types of the base field $K$ and of its $p$-class group ${\mathrm{Cl}}_p\left(K\right)$, the position of $G$ is restricted to certain admissible branches of coclass trees by selection rules. Deeper insight, in particular, the density of population of individual vertices on coclass graphs, is gained by computing the actual distribution of second $p$-class groups $G$ for various series of number fields $K$ having $p$-class groups ${\mathrm{Cl}}_p\left(K\right)$ of fixed type and $p\in \{2,3,5,7\}$.

Classification : 11R29, 11R37, 11R11, 11R16, 11R20, 20D15
Mots clés : $p$-class groups, $p$-class field tower, principalization of $p$-classes, quadratic fields, cubic fields, dihedral fields, metabelian $p$-groups, coclass graphs

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