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\(p\)-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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