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On the \(p\)-adic realization of elliptic polylogarithms for CM-elliptic curves. (English) Zbl 1019.11018
We cite from the introduction: “The aim of this paper is to study the \(p\)-adic analogue of the elliptic polylogarithm. Let \(E\) be a CM-elliptic curve over \(\mathbb{Q}\) with good reduction at a prime \(p \geq 5\). Using the tools developed in the author’s paper [J. Reine Angew. Math. 529, 205-237 (2000; Zbl 1006.19002)], we construct the \(p\)-adic elliptic polylogarithm in rigid syntomic cohomology following the method of Beilinson and Levin. Our main result is that this element, when specialized to the torsion points of \(E\), gives the special values of the one-variable \(p\)-adic \(L\)-function of the Grössencharakter associated to \(E\).”
The method of Beilinson and Levin is explained in the paper [A. Beilinson and A. Levin, The elliptic polylogarithm. In: Motives (Seattle 1991), Proc. Symp. Pure Math. 55, Part 2, 123-190 (1994; Zbl 0817.14014)].
The publication under review is a revised version of the author’s Ph.D. thesis, finished in the year 2000 at the University of Tokyo. The main result is a generalization to the case of elliptic curves of his paper cited above. The construction of the elliptic polylogarithm is based on the paper [A. Huber and G. Kings, Invent. Math. 135, 545-594 (1999; Zbl 0955.11027)].
The publication is well-written, long (so that the table of contents at the beginning is very useful) and rather sophisticated. Therefore, it does not make too much sense to go into any more details.

MSC:
11G55 Polylogarithms and relations with \(K\)-theory
11G05 Elliptic curves over global fields
11G15 Complex multiplication and moduli of abelian varieties
14F30 \(p\)-adic cohomology, crystalline cohomology
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R23 Iwasawa theory
11G07 Elliptic curves over local fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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