Iwasawa theory and motivic \(L\)-functions
type: Item
instance of: scholarly article
title: Iwasawa theory and motivic \(L\)-functions
zbMATH Open document ID: 1179.11036
author: Matthias Flach
published in: Pure and Applied Mathematics Quarterly
publication date: 2009-04-14T00:00:00Z
review text: This paper is a continuation of the survey article by the author in [Contemp. Math. 358, 79--125 (2004; Zbl 1070.11025)], on the Equivariant Tamagawa Number Conjecture (ETNC). Without pretending to give a precise statement of the ETNC, let us just say that a pure motif over \({\mathbb Q}\) on which a semi-simple \({\mathbb Q}\)-algebra \(A\) acts, gives rise to a ``motivic structure'' consisting of the realizations and the motivic cohomology groups of \(M\) together with comparison isomorphisms and exact sequences relating these groups. There is then a whole corpus of conjectures (no less than eight are listed in this paper) which describe, on the one hand, the motivic cohomology with \({\mathbb R}\)-coefficients and \({\mathbb Q}_\ell\)-coefficients, on the other hand, the behaviour at \(s = 0\) of a hypothetical complex \(L\)-function \(L({}_\wedge M, s).\) Of particular importance are a rationality conjecture and an integrality conjecture for the leading term \(L^\ast({}_\wedge M).\) Note that all these conjectures have been completely (resp. essentially) proved for Tate motives (resp. motives associated to CM elliptic curves) over abelian extensions of \({\mathbb Q}\) (resp. of an imaginary quadratic field) thanks to the efforts of many people, culminating in \textit{D. Burns} and \textit{C. Greither}'s [Invent. Math. 153, No. 2, 303--359 (2003; Zbl 1142.11076), resp. \textit{W. Bley}, Doc. Math., J. DMV 11, 73--118 (2006; Zbl 1178.11070)]. The ``modern'' point of view due to Kato, Fontaine--Perrin-Riou, Burns--Flach et al., presents both the theory of complex \(L\)-values as well as of \(p\)-adic \(L\)-functions as the construction of ``trivializations'' of determinant line bundles. The integrality conjecture pertains mainly to Iwasawa theory via the Main Conjecture, but to address it, it is not sufficient to know the rationality conjecture as such (i.e. up to an unspecified factor in \(\wedge^\ast),\) a more precise description of \(L^\ast({}_\wedge M)\) is ne…
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11-02
zbMATH DE Number: 5543150
MaRDI profile type: Publication
full work available at URL: pamq.2009.v5.n1.a8
OpenAlex ID: W2063879018
DOI: 10.4310/PAMQ.2009.V5.N1.A8
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description: scientific article; zbMATH DE number 5543150