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$$p$$-adic representations, $$p$$-adic periods and $$p$$-adic $$L$$-functions. (Représentations $$p$$-adiques, périodes et fonctions $$L$$ $$p$$-adiques.) (French) Zbl 0783.11044
Sémin. Théor. Nombres, Paris/Fr. 1987-88, Prog. Math. 81, 213-258 (1990).
[For the entire collection see Zbl 0686.00006.]
This paper completes in some sense a previous paper by J. Coates and the author in [Adv. Stud. Pure Math. 17, 23-54 (1989; Zbl 0783.11050)], about the existence of $$p$$-adic $$L$$-functions attached to motives over $$\mathbb{Q}$$.
The paper is divided into two parts. The first one is devoted to recall the main fact about $$p$$-adic representations of the absolute Galois group of a $$p$$-adic field, Fontaine’s theory on $$\varphi$$-filtered modules, as well as the relationship between these two concepts, in the ordinary case.
In the second part of the paper the author, by means of $$p$$-adic comparison isomorphisms between de Rham cohomology and Fontaine-Messing cohomology, elaborates a theory of $$p$$-adic periods in the spirit of Deligne’s theory of complex periods. The behaviour of $$p$$-adic periods under isogeny is discussed.
A conjecture predicts the role of the $$p$$-adic periods in the study of special values of $$p$$-adic $$L$$-functions of motives which are ordinary at $$p$$. Some calculations are pursued in the case of elliptic curves defined over $$\mathbb{Q}$$, provided that they have ordinary good reduction at $$p$$.

##### MSC:
 11S40 Zeta functions and $$L$$-functions 11F85 $$p$$-adic theory, local fields 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols