Towards Regulator Formulae for the K-Theory of Curves over Number Fields

Abstract

In this paper we study the group K_2n⁽ⁿ⁺¹⁾(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_2n⁽ⁿ⁺¹⁾(F), using the complexes constructed in Compositio Math. 96 (1995), pp. 197–247. We study the boundary map in the localization sequence for n=2 and n=3. We combine our results with results of Goncharov in order to obtain a complete description of the image of the regulator map on K₄⁽³⁾(C) and K₆⁽⁴⁾(C) (which have the same images as K₄(C)$⊗ _{\Bbb Z}\, {\Bbb Q}$, and K₆(C)$⊗ _{\Bbb Z}\, {\Bbb Q}$, respectively), independent of any conjectures.