Infinitesimal deformations and the \(\ell\)-invariant
type: Item
instance of: scholarly article
title: Infinitesimal deformations and the \(\ell\)-invariant
zbMATH Open document ID: 1235.11098
author: Denis Benois
published in: Documenta Mathematica
publication date: 2010-12-16T00:00:00Z
full work available at URL: benois.html
review text: The main results of this paper are two formulas for Greenberg's generalized \(\ell\)-invariant in terms of derivatives of eigenvalues of Frobenius. Rather heavy preliminaries are required, which we shall try to summarize in a minimal way. Let \(M\) be a pure motive over \({\mathbb Q}\) with coefficients in a number field \(E\), and assume that the \(L\)-function \(L(M, s)\) is well-defined. Suppose moreover that \(M\) is critical and that \(L(M,0) \not= 0.\) Fix a \(p\)-place \(\lambda\) of \(E\) and assume that the \(\lambda\)-adic realization \(M_\lambda\) of \(M\) is semistable in the sense of Fontaine. The \((\varphi, N)\)-module \({\mathbb D}_{st} \;(M_\lambda)\) associated to \(M_\lambda\) is a finite dimensional \(E_\lambda\)-vector space equipped with an exhaustive decreasing filtration, an \(E_\lambda\)-linear bijective Frobenius map \(\varphi\) and a nilpotent monodromy operator \(N\) such that \(N \varphi = p \varphi N\). A \((\varphi, N)\)-submodule \(D\) of \({\mathbb D}_{st}\;(M_\lambda)\) is called regular if \( {\mathbb D}_{st}\;(M_\lambda) = D \oplus \;Fil^o\;{\mathbb D}_{st}\;(M_\lambda)\) as \(E_\lambda\)-vector spaces. The theory of [\textit{B. Perrin-Riou}, \(p\)-adic \(L\)-functions and \(p\)-adic representations. Astérisque. 229. Paris: Société Mathématique de France (1995; Zbl 0845.11040)] suggests that any regular \(D\) should give rise to a \(p\)-adic \(L\)-function \(L_p (M, D, s)\) interpolating rational parts of special values of \(L(M, s).\) In particular, the interpolation formula at \(s = 0\) should read: \(L_p (M, D, 0) = \text{Eul}(M, D)\;L(M, 0)/\Omega_\infty (M),\) where \(\Omega_\infty (M)\) is the Deligne period and \(\text{Eul}(M, D)\) is a certain product of Euler-like factors. One can naturally expect that \(L_p (M, D, 0) = 0\) if and only if \(\text{Eul}(M, D) = 0,\) in which case one says that \(L_p(M, D, s) = 0\) has a trivial zero at \(s = 0\). In his preprint [``A generalization of Greenberg's \(\mathcal{L}\)-invariant,'…
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 5827437
zbMATH Keywords: \((\varphi, \Gamma)\)-module
zbMATH Keywords: \(L\)-function
zbMATH Keywords: \(p\)-adic representation
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5827437