## On $$K_ 4^{(3)}$$ of curves over number fields.(English)Zbl 0864.11059

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.

### MSC:

 11R70 $$K$$-theory of global fields 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects)
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