The boundary of the Eisenstein symbol
type: Item
instance of: scholarly article
title: The boundary of the Eisenstein symbol
zbMATH Open document ID: 0729.11027
DOI: 10.1007/BF01459247
author: Norbert Schappacher
author: Anthony J. Scholl
published in: Mathematische Annalen
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164820
review text: For an elliptic curve \(E\) over a field \(F\) (supposed to have a nontrivial discrete valuation \(v\), valuation ring \(\mathcal O\) and perfect residue field \(k\)) with a finite subgroup scheme \(P\subset E\) defined over \(F\), and for any integer \(n\geq 1\), one has the Eisenstein symbol map \[ \mathcal E^ n_ P: \mathbb Q[P]^ 0\to H_{\mathcal M}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}}, \] where \(H^ i_{\mathcal M}(-,\mathbb Q(j))\) is motivic cohomology, \(\mathbb Q[P]^ 0\) is the \(\mathbb Q\)-vector space of \(\text{Gal}(\bar F/F)\)-invariant functions \(\beta: P(\bar F)\to \mathbb Q\) satisfying \(\sum_{x\in P(\bar F)}\beta (x)=0\), \(E^ n\) is identified with the kernel of the sum map \(E^{n+1}\to E\) (thus giving an action of the symmetric group \(\mathcal S_{n+1}\) on \(E^ n)\), and where the subscript `sgn' denotes the image under the projector \[ \prod_{\text{sgn}}=\frac{1}{(n+1)!}\sum_{\sigma \in\mathcal S_{n+1}}\text{sgn}(\sigma)\cdot \sigma. \] Write \(E/k\) for the special fibre of the minimal regular model \(E/{\mathcal O}\) of \(E\) and suppose that \(E/k\) is a Néron \(N\)-gon for some \(N\geq 1\). Furthermore suppose that \(P\) extends to a finite flat subgroup scheme \(P/\mathcal O\) of the Néron model of \(E\) over \(\mathcal O\). Also, let \(\overset \circ E\) denote the connected component of the Néron model of \(E\) over \(\mathcal O\). An isomorphism \(\overset \circ E/k\overset \sim \rightarrow\mathbb G_ m\) induces a bijection between \(\mathbb Z/N\mathbb Z\) and the set of components \(C_{\nu}\) of \(E/k\). Thus \(E/k=\cup_{\nu \in\mathbb Z/N\mathbb Z}C_{\nu}\). For \(\beta\in\mathbb Q[P]^ 0\) let \(d_{\beta}(\nu)\) be the degree of the restriction of the flat extension of \(\beta\) to \(C_{\nu}\). The localization sequence for the pair (\(\overset \circ E^ n/{\mathcal O},\overset \circ E^ n/k)\) gives a boundary map \[ \partial^ n: H_{{\mathcal M}}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}}\to H^ n_{\mathcal M}(\overset \circ E^ n/k,\mat…
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 19F15
Mathematics Subject Classification ID: 19D45
zbMATH DE Number: 4204445
zbMATH Keywords: Eisenstein symbol map
zbMATH Keywords: motivic cohomology
zbMATH Keywords: Néron model
zbMATH Keywords: Bernoulli polynomials
zbMATH Keywords: boundary maps
zbMATH Keywords: K-theory
zbMATH Keywords: place of bad reduction
zbMATH Keywords: elliptic curve
zbMATH Keywords: modular curve
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4204445