Iwasawa-theory of abelian varieties at primes of non-ordinary reduction
type: Item
instance of: scholarly article
title: Iwasawa-theory of abelian varieties at primes of non-ordinary reduction
zbMATH Open document ID: 0847.14026
DOI: 10.1007/BF02570472
author: Heiko Knospe
published in: Manuscripta Mathematica
publication date: 1995-08-29T00:00:00Z
full work available at URL: https://eudml.org/doc/156085
review text: The well known Birch and Swinnerton-Dyer conjectures for abelian varieties can be reformulated for \(p\)-adic \(L\)-functions. There are two approaches to a definition of \(p\)-adic \(L\)-function of an abelian variety \(A\), the analytical one and the algebraic one. The last one working with the Selmer groups of infinite cyclotomic extensions is more directly related with the Birch-Swinnerton-Dyer formulas for value at \(s = 1\). The algebraic definition was given by \textit{B. Mazur} [Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] and the formulas for \(L(1)\) were established by \textit{P. Schneider} [Invent. Math. 71, 251-293 (1983; Zbl 0511.14010)]. The severe restriction used in these papers was that the abelian variety must have an ordinary reduction at the primes \(p\). This restriction was overcome by \textit{B. Perrin-Riou} in Invent. Math. 99, No. 2, 247-292 (1990; Zbl 0715.11030) for supersingular elliptic curves. The author gives here an extension to the case of general abelian varieties \(A\). It is defined some module \(L_0\) over Iwasawa algebra of \(A\) and the \(L\)-function is a characteristic function related to \(L_0\). The part of the Birch-Swinnerton-Dyer formulas is checked.
Mathematics Subject Classification ID: 14K05
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11S40
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11G25
Mathematics Subject Classification ID: 11G10
zbMATH DE Number: 784072
zbMATH Keywords: \(p\)-adic \(L\)-function of an abelian variety
zbMATH Keywords: Birch and Swinnerton-Dyer conjectures
reviewed by: A. N. Parshin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 784072