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Galois descent and capitulation between Bertrandias-Payan modules. (Descente Galoisienne et capitulation entre modules de Bertrandias-Payan.) (French. English summary) Zbl 1380.11093
Algèbre et théorie des nombres 2016. Besançon: Presses Universitaires de Franche-Comté. Publications Mathématiques de Besançon. Algèbre et Théorie des Nombres 2016, 59-79 (2017).
Let \(K\) be a number field and \(L/K\) a finite Galois extension with Galois group \(G\). Let \(p\) be an odd prime: denote by \(\widetilde{K}\) the compositum of all the \(\mathbb{Z}_p\)-extensions of \(K\) and by \(BP_K\) the compositum of all the cyclic \(p\)-extensions of \(K\) which can be embedded in cyclic \(p\)-extensions of arbitrary high degree. The Bertrandias-Payan module of \(K\) is \(\mathcal{BP}_K:=\mathrm{Gal}(BP_K/\widetilde{K})\), i.e., the \(\mathbb{Z}_p\)-torsion of \(\mathrm{Gal}(BP_K/K)\), and the paper deals with the elements of the exact sequence \[ \mathrm{Ker}\,j_{L/K} \hookrightarrow \mathcal{BP}_K \rightarrow \mathcal{BP}_L^G \twoheadrightarrow \mathrm{Coker}\,j_{L/K} \] (where the map \(j_{L/K}\) is the natural capitulation map arising from \(G\)-cohomology). Together with the companion papers by G. Gras [Publ. Math. Besançon, Algèbre et Théorie des Nombres 2016, 25–44 (2017; Zbl 1406.11112)] and F. Jaulent [Publ. Math. Besançon, Algèbre et Théorie des Nombres 2016, 45–58 (2017; Zbl 1406.11113)], the article under review provides several results on (and different approaches to the study of) \(\mathcal{BP}_K\) and, in particular on the quotient \(\mathcal{BP}_L^G/\mathcal{BP}_K\) (whose order coincides with \(|\mathrm{Coker}\,j_{L/K}|/|\mathrm{Ker}\,j_{L/K}|\)) assuming Lepoldt’s conjecture for \(L\) in \(p\).
After a few natural reduction steps one can study cyclic \(p\)-extensions \(L/K\) (even Kummer extensions \(L=K(\sqrt[p]{\alpha})\) assuming \(\mu_p\in K\)): the author uses local and global Galois cohomology (and \(S\)-ramified Galois cohomology, where \(S\) is the set of primes ramified in \(L/K\)) to show that the injectivity of \(j_{L/K}\) depends on the relation between local and global roots of unity in \(L\) and \(K\). In particular, when \(\mu_p\in K\), \(j_{L/K}\) is not injective if \(L/K\) is not (globally) cyclotomic but it is locally cyclotomic everywhere, i.e., \(L_w/K_v\) is cyclotomic for any pair of primes \(w|v\). A similar cohomological approach is carried out for \(\mathrm{Coker}\,j_{L/K}\) distinguishing between the exceptional case (when \(\mathrm{Ker}\,j_{L/K}\neq 1\)) and the trivial one. In particular, for the exceptional case, \(\mathrm{Coker}\,j_{L/K}\) has order 1 or \(p\), can be related (via Galois descent/co-descent) to \(H^2(G,\mu(L))\) and is linked to the possibility of embedding \(L\) in one (or several) \(\mathbb{Z}_p\)-extensions of \(K\).
For the entire collection see [Zbl 1365.11004].

11R37 Class field theory
11R34 Galois cohomology
11R23 Iwasawa theory
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