On K_4^(3) of curves over number fields., by Rob de Jeu

In this paper we consider the group K_4^(3)(F) of a function field of a smooth geometrically irreducible curve over a number field. We do this by defining cohomological complexes for a field of characteristic zero, together with a natural map from the cohomology of the complex to the K-theory of the field. On the image in K_4^(3)(F) of those complexes, we derive a formula for the Beilinson regulator, and compute an approximation of the boundary map at the closed points of the curve in the localization sequence. We give a way of finding examples of elliptic curves E with elements in K_4^(3)(E), and in some cases use computer calculations to check numerically the relation between the regulator and the L-function, as conjectured by Beilinson.

This paper, slightly expanded, has been published in Inventiones Mathematicae 125, pp. 523-556 (1996).

Rob de Jeu <rob.de-jeu@durham.ac.uk>