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Towards regulator formulae for curves over number fields

In this paper we study the group K (n+1) 2n (F ) where F is the function field of a complete, smooth, geometrically irreducible curve C over a number field, assuming the Beilinson–Soulé conjecture on weights. In particular, we compute the Beilinson regulator on a subgroup of K (n+1) 2n (F ), using the complexes constructed in [dJ1]. We study the boundary map in the localization sequence for n = 3 (the case n = 2 was studied in [dJ2]). We combine our results with the results of Goncharov ([G1] and [G3], see also [G2]) in order to obtain a complete description of the image of the regulator map on K (3) 4 (C) and K (4) 6 (C), independent of any conjectures.