On \(K_ 4^{(3)}\) of curves over number fields
type: Item
instance of: scholarly article
title: On \(K_ 4^{(3)}\) of curves over number fields
zbMATH Open document ID: 0864.11059
author: Rob de Jeu
published in: Inventiones Mathematicae
publication date: 1996-11-24T00:00:00Z
review text: Let \(F\) be the function field of a smooth geometrically irreducible curve \(C\) over a number field \(k\). The main object of interest in this paper is the group \(K_4^{(3)} (F)\) -- where the superscript denotes an eigenspace for Adams operations --, which contains \(K_4^{(3)} (C)\) and goes to \(\coprod_{x\in C^1} K_3^{(2)} (k(x))\) by a boundary map. The author introduces a complex \(\widetilde {\mathcal M}^\bullet_{(3)}(F)\) previously constructed by him in his study of the Zagier conjecture [Compos. Math. 96, 197-247 (1995)], and he computes the boundary map on the image of \(H^2 (\widetilde {\mathcal M}^\bullet_{(3)} (F))\) in \(K_4^{(3)} (F)\), as well as the Beilinson regulator on the intersection of \(K_4^{(3)} (C)\) and this image. He also introduces an auxiliary complex \({\mathcal C}^\bullet\), originally due to Bloch, which makes calculations easier to handle (for example, both \(H^2 (\widetilde {\mathcal M}^\bullet_{(3)})\) and \(K_4^{(3)} (F)\) go to \(H^1({\mathcal C}^\bullet))\). Finally, he gives methods for constructing explicit examples in \(H^1({\mathcal C}^\bullet)\) when the curves \(C\) are certain elliptic curves; in certain cases, computer calculations can be used to check numerically the relation between regulators and \(L\)-functions conjectured by Beilinson.
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 19F27
zbMATH DE Number: 931980
zbMATH Keywords: curves over number fields
zbMATH Keywords: Beilinson conjecture
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: function field
zbMATH Keywords: Adams operations
zbMATH Keywords: boundary map
zbMATH Keywords: Beilinson regulator
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: s002220050085
OpenAlex ID: W1979918078
DOI: 10.1007/S002220050085
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description: scientific article; zbMATH DE number 931980