## $$p$$-adic $$L$$-functions. (Fonctions $$L$$ $$p$$-adiques.)(French)Zbl 0964.11055

Séminaire Bourbaki. Volume 1998/99. Exposés 850-864. Paris: Société Mathématique de France, Astérisque 266, 21-58, Exp. No. 851 (2000).
This article is based on work of B. Perrin-Riou [Invent. Math. 115, 81–149 (1994; Zbl 0838.11071); Fonctions $$L$$ $$p$$-adiques des représentations $$p$$-adiques, Astérisque. 229. Paris: Société Mathématique de France (1995; Zbl 0845.11040)], which aims to construct $$p$$-adic $$L$$-functions for motives. The machine of Perrin-Riou starts with a motive having good reduction at the prime $$p$$ and having a compatible system of special elements, whose existence is not known in general, and constructs a $$p$$-adic $$L$$-function. In the classical case of Tate motives, this yields the construction of the Kubota-Leopoldt $$p$$-adic $$L$$-functions via Coleman power series, where the special elements are cyclotomic units. This method of R. Coleman is explained in the book of the reviewer [Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics, 83, New York, NY: Springer (1997; Zbl 0966.11047)]. This construction in the general case allows one to formulate $$p$$-adic analogues of the conjectures of Deligne-Beilinson for the values of $$L$$-functions at integers. The work of Perrin-Riou also attaches Iwasawa functions to motives with good reduction at $$p$$. This permits the formulation of Main Conjectures, generalizing the one proved by B. Mazur and A. Wiles [Invent. Math. 76, 179–330 (1984; Zbl 0545.12005)] in the cyclotomic case.
The present exposition starts by reviewing the construction of Kubota-Leopoldt $$p$$-adic $$L$$-functions and Manin’s work on attaching $$p$$-adic $$L$$-functions to modular forms. It then gives a detailed account of the work of Perrin-Riou mentioned above and of the author [Ann. Math. (2) 148, No. 2, 485–571 (1998; Zbl 0928.11045)].
The present article is a good introduction to some important modern work on $$p$$-adic $$L$$-functions and related areas.
For the entire collection see [Zbl 0939.00019].

### MSC:

 11S40 Zeta functions and $$L$$-functions 11R42 Zeta functions and $$L$$-functions of number fields 14F42 Motivic cohomology; motivic homotopy theory 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F85 $$p$$-adic theory, local fields 11G55 Polylogarithms and relations with $$K$$-theory 11R23 Iwasawa theory
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