On the \(p\)-adic realization of elliptic polylogarithms for CM-elliptic curves
type: Item
instance of: scholarly article
title: On the \(p\)-adic realization of elliptic polylogarithms for CM-elliptic curves
zbMATH Open document ID: 1019.11018
DOI: 10.1215/S0012-7094-02-11321-0
author: Kenichi Bannai
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: 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 [\textit{A. Beilinson} and \textit{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 [\textit{A. Huber} and \textit{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.
Mathematics Subject Classification ID: 11G55
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11G07
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 1820905
zbMATH Keywords: elliptic curve
zbMATH Keywords: polylogarithm
zbMATH Keywords: \(p\)-adic fields
zbMATH Keywords: syntomic cohomology
zbMATH Keywords: logarithmic sheaf
reviewed by: Horst G. Zimmer
MaRDI profile type: Publication
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full work available at URL: s0012-7094-02-11321-0
OpenAlex ID: W2089117223
description: scientific article; zbMATH DE number 1820905