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

de Jeu, Rob

Invent. Math. 125, No. 3, 523-556 (1996).

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.

Reviewer: T.Nguyen Quang Do (Besançon)

MSC:

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

Keywords:

curves over number fields; Beilinson conjecture; \(K\)-theory; function field; Adams operations; boundary map; Beilinson regulator