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On $$p$$-adic $$L$$-functions. (English) Zbl 0706.11064
Sémin. Bourbaki, Vol. 31, 41e année (1988/1989), Exp. No. 701, Astérisque 177-178, 33-59 (1989).
The principal conjecture of this article asserts that there is a two-variable $$p$$-adic $$L$$-function interpolating special values of a complex motivic $$L$$-function. To be more precise, let $$H$$ and $$D$$ denote, respectively, the Galois groups of the field generated by all $$p$$-power roots of unity and of its maximal real subfield, and let $$\psi: H\to {\mathbb Z}^*_ p$$ be the canonical isomorphism defined by the action of $$H$$ on the group of all $$p$$-power roots of unity. Then, according to the author’s conjecture, there is a unique “pseudo-measure” $$\mu$$ on $$D$$ such that $$L(\chi,n):=\int_{D}\psi \chi^ n \,d\mu$$ is equal to a properly normalised special value of a certain motivic $$L$$-function; here $$n\in {\mathbb Z}$$, and $$\chi$$ ranges over all the Dirichlet characters of $$p$$-power conductor, with $$\chi (-1)=(-1)^ n.$$
According to the author, this conjecture is basically taken from his joint work with B. Perrin-Riou [cf. Algebraic number theory - in honour of K. Iwasawa, Proc. Workshop Iwasawa Theory Spec. Values $$L$$-Funct., Berkeley/CA (USA) 1987, Adv. Stud. Pure Math. 17, 23–55 (1989; Zbl 0783.11039)] however, P. Deligne “pointed out that there is a much more elegant and succinct way of expressing our conjectures using the local $$\varepsilon$$-factors of the motive”.
The relevant definitions and results concerning $$p$$-adic analogues of Dirichlet $$L$$-functions and complex motivic $$L$$-functions are reviewed in some detail. In the introduction, the author states a theorem from his forthcoming joint work with M. Flach. This theorem relates the value at $$s=1$$ of the complex $$L$$-function attached to the motive, which is the symmetric square of an elliptic curve, with Kummer’s theory.
For the entire collection see [Zbl 0691.00001].
Reviewer: B. Z. Moroz

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11R23 Iwasawa theory 11S40 Zeta functions and $$L$$-functions 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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