On an elliptic analogue of Zagier's conjecture
type: Item
instance of: scholarly article
title: On an elliptic analogue of Zagier's conjecture
zbMATH Open document ID: 0898.14001
DOI: 10.1215/S0012-7094-97-08714-7
author: Jörg Wildeshaus
published in: Duke Mathematical Journal
publication date: 1997-08-18T00:00:00Z
review text: Let \({\mathcal E}\to B\) be a smooth family of elliptic curves over a smooth quasiprojective connected scheme \(B\) over a number field \(K\) and \(i:B\to{\mathcal E}\) be its zero section. It is well-known that the group of sections \({\mathcal E}(B)\) is isomorphic to \(\text{Pic}_{{\mathcal E}/B}^0\) and the latter is isomorphic to the \((-1)\)-eigensubspace \(H_{\mathcal M}^2({\mathcal E},\mathbb{Q}(1))_-\) of the second motivic cohomology \(H_{\mathcal M}^2({\mathcal E},\mathbb{Q}(1))\) with respect to the action of the inversion automorphism. The author states a conjecture generalizing this fact which concerns higher motivic cohomology \(H_{\mathcal M}^{k-1}({\mathcal E}^{(k-2)},\mathbb{Q}(k-1))\cong K_{k-1}({\mathcal E}^{(k-1)}\otimes\mathbb{Q})^{k-1}\), \(k\geq 2\), where \({\mathcal E}^{(k-2)}= \text{Ker} (\sum:{\mathcal E}^{k-1}\to {\mathcal E})\). He also proves some partial results towards the conjecture. Let \({\mathcal L}_k\) be a vector space over \(\mathbb{Q}\) with basis \(\{s\}_k\), \(s\in {\mathcal E}(B)\setminus i(B)\) and let \[ D_k:{\mathcal L}_k\to{\mathcal L}_{k-1}\otimes_\mathbb{Q}({\mathcal E}(B)\otimes\mathbb{Q}) \] be a linear map defined by \(d_k(\{s\}_k)=\{s\}_{k-1}\otimes (s\otimes 1)\). The conjecture claims the existence of a homomorphism \(\text{Ker}(d_k)\to H_{\mathcal M}^{k-1}({\mathcal E}^{(k-2)},\mathbb{Q}(k-1))_{\text{sgn}}\) satisfying some functorial and naturality properties which we omit. Here the subscript sign denotes the eigensubspace of the sign character with respect to the natural action of the permutation group \(\Sigma_{k-1}\) on \({\mathcal E}^{(k-1)}\). When \(B=\text{Spec}(K)\), the author proves the torsion part of this conjecture by constructing the homomorphism defined on the subspace generated by the symbols of torsion sections. He also proves the conjecture for \(k=2\).
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 19E20
Mathematics Subject Classification ID: 14F99
zbMATH DE Number: 1026544
zbMATH Keywords: family of elliptic curves
zbMATH Keywords: motivic cohomology
Wikidata QID: Q122942946
reviewed by: Igor V. Dolgachev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1026544