Systèmes d'Euler \(p\)-adiques et théorie d'Iwasawa. (\(p\)-adic Euler systems and Iwasawa theory.)
In number theory, the importance of the notion of Euler systems introduced by \textit{V. A. Kolyvagin} in 1988 [Math. USSR, Izv. 33, 473-499 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, 1154-1180 (1988; Zbl 0681.14016)] can hardly be underestimated. In this paper, the author concentrates on the study of \(p\)-adic Euler systems, that is Euler systems attached to \(p\)-adic representations \(V\) of the absolute Galois group \(G_\mathbb{Q}\). The traditional congruences in cyclotomic Euler systems (namely \(1-\xi_{m\ell}\equiv 1-\xi_m\) modulo the places dividing the prime number \(\ell)\) are replaced by trace relations in cyclotomic extensions \(\mathbb{Q}(\xi_{m\ell})/\mathbb{Q}(\xi_m)\). These can be translated into relations between \(p\)-adic ``functions'' Gal\((\mathbb{Q}(\xi_{mp^\infty})/\mathbb{Q})\to{\mathbf D}_p(V)\), relations which in turn reflect the factorization of classical \(L\)-functions by means of Euler products. Modulo certain technical conditions on the representation \(V\), the author proves the ``Leopoldt conjecture for \(V\)'' (which asserts that a certain Iwasawa module attached to \(V\) is \(\Lambda\)-torsion), as well as some results on the divisibility of certain characteristic series in Iwasawa theory. Actually, the paper contains a host of informations on \(p\)-adic Euler systems, such as local properties of Euler systems, Kolyvagin's derivation, applications of Chebotarev's theorem \(\dots\), but the technicality inherent to the subject makes it difficult to give a fair summary.
trace relations in cyclotomic extensions
scientific article; zbMATH DE number 1227211
scientific article