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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].

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