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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

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)