## Systèmes d’Euler $$p$$-adiques et théorie d’Iwasawa. ($$p$$-adic Euler systems and Iwasawa theory.).(French)Zbl 0930.11078

In number theory, the importance of the notion of Euler systems introduced by 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.

### MSC:

 11R23 Iwasawa theory 12G05 Galois cohomology 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R42 Zeta functions and $$L$$-functions of number fields

Zbl 0681.14016
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