Structure galoisienne relative de la racine carrée de la codifférente d’extensions métacycliques non abéliennes
Volume 199 / 2021
Abstract
Let $k$ be a number field and $O_k$ its ring of integers. Let $\Gamma $ be a finite group of odd order. Let $\mathcal {M}$ be a maximal $O_k$-order in the semisimple algebra $k[\Gamma ]$ containing $O_k[\Gamma ]$, and ${\rm Cl}(\mathcal {M})$ its locally free classgroup. We define the set $\mathcal {R}(\mathcal {A}, \mathcal {M})$ of classes realizable by the square root of the inverse different to be the set of classes $c \in {\rm Cl}(\mathcal {M})$ such that there exists a Galois extension $N/k$ which is tame, with Galois group isomorphic to $\Gamma $, and for which the class of $\mathcal {M} \otimes _{O_k[\Gamma ]} \mathcal {A}_{N/k}$ is equal to $c$, where $\mathcal {A}_{N/k}$ is the square root of the inverse different of $N/k$. Let $l, q$ be odd prime numbers. Let $\xi _l$ (resp. $\xi _{q}$) be a primitive $l$th (resp. $q$th) root of unity. First, when $\Gamma $ is cyclic of order $l$, under the hypothesis that $k/\mathbb {Q}$ and $\mathbb {Q}(\xi _l)/\mathbb {Q}$ are linearly disjoint, we apply a recent theorem due to the second author to show that $ \mathcal {R} (\mathcal {A}, \mathcal {M}) $ is a subgroup of $ {\rm Cl} (\mathcal {M}) $ explicitly described by means of a Stickelberger ideal. Next, we apply that result to the case where $\Gamma $ is a nonabelian metacyclic group of order $lq$; under the hypothesis that $k/\mathbb {Q}$ and $\mathbb {Q}(\xi _l, \xi _q)/\mathbb {Q}$ are linearly disjoint, we define a subset of $\operatorname{Cl} (\mathcal {M})$ (which can be interpreted via the notion of domestic extensions) by means of two Stickelberger ideals, and prove that it is a subgroup of $\operatorname{Cl} (\mathcal {M})$ contained in $\mathcal {R}(\mathcal {A}, \mathcal {M})$. Finally, under only the hypothesis that $k/\mathbb {Q}$ and $\mathbb {Q}(\xi _l)/\mathbb {Q}$ are linearly disjoint, we show a nonexplicit generalisation of the preceding result to nonabelian metacyclic extensions of degree $lm$, where $m$ is an odd integer.
