\(p\)-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 \(p\).

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 \(p\).

Reviewer: P.Bayer (Barcelona)

##### 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 |