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