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Towards regulator formulae for the $$K$$-theory of curves over number fields. (English) Zbl 0985.19002
The author computes the Beilinson regulator on a subgroup of the motivic cohomology group $$H^2_{\mathcal M}(F,{\mathbb Q}(n+1)) = K_{2n}^{(n+1)}(F)$$, where $$F$$ is the function field of a complete smooth geometrically irreducible curve $$\mathcal C$$ over a number field, under the Beilinson-Soulé conjecture on weights, by using the motivic complexes $$\widetilde{\mathcal M}_{(m)} ^{\bullet}$$ that he constructed in a previous article [R. De Jeu, Compos. Math. 96, No. 2, 197-247 (1995; Zbl 0868.19002)].
More precisely, let $$\mathcal C$$ be a smooth geometrically irreducible projective curve over a number field $$k$$, with function field $$F = k(\mathcal C)$$, and let $${\mathbb Q} \otimes_{\mathbb Z} K_m(\mathcal C) = \bigoplus_{n=1}^{m+1} K_m^{(n)}(\mathcal C)$$ be the decomposition of the $$K$$-theory with rational coefficients, given by the eigenspaces under Adams operators. There are regulator maps to Deligne cohomology $$K_m^{n}(\mathcal C) \rightarrow H^{2n-m}_\mathcal D({\mathcal C}_{\text{an}}; {\mathbb R}(n))$$, where $${\mathcal C}_{\text{an}}$$ is the analytic manifold associated to $$\mathbb C \otimes_\mathbb Q \mathcal C$$ and $${\mathbb R}(n) = (2\pi i)^n {\mathbb R} \subset \mathbb C$$. For $$n \geq 2$$ and $$m=2n-2$$, this gives a map with values in $$H^2_\mathcal D({\mathcal C}_{\text{an}};{\mathbb R}(n)) \simeq H_{\text{dR}}^1({\mathcal C}_{\text{an}}; {\mathbb R}(n-1))$$; and in fact in the subspace $$H^1_{\text{dR}}({\mathcal C}_{\text{an}}; {\mathbb R}(n-1))^+$$ of the forms $$\psi$$ satisfying the identity $$\psi \circ \sigma = \overline\psi$$, where $$\sigma$$ denotes the canonical involution on $${\mathcal C}_{\text{an}}$$. So, by wedging with an holomorphic 1-form on $${\mathcal C}_{\text{an}}$$ and integrating, one lands in $${\mathbb R}(1)$$.
In the paper under review the author uses the quotient complexes $$\widetilde {\mathcal M}_{(m)}^{\bullet}$$ that he introduced in a previous work [op. cit.] and the Beilinson-Soulé conjecture for fields of characteristic 0 in order to construct symbols $$[f]_n \otimes g$$ in $$K_{2n}^{(n+1)}(F)$$; then he proves that their images under the map $$H^2(\widetilde{\mathcal M}^\bullet_{(n+1)}(F)) \rightarrow K_{2n}^{(n+1)}(F) \overset{\text{reg}}\longrightarrow H^1_{\text{dR}}(F; {\mathbb R}(n))^+ \rightarrow {\mathbb R}(1)$$ has the form conjectured by A. B. Goncharov [“Polylogarithms in arithmetic and geometry”, Proc. Internat. Congr. Math. (Zürich), 374-387 (1994; Zbl 0849.11087)]. In addition, for $$n=2$$ and $$n=3$$, he observes that the map above exists without assumption.
The author also studies compatibility of Gersten complexes for $$\widetilde {\mathcal M}_{(m)}$$ and $$K$$-theory on $$\mathcal C$$. Most proofs in the paper are general, but unfortunately the combinatorics become more and more complicated as $$n$$ increases. So the author restricts his final computations to $$K_4^{(3)}$$ and $$K_6^{(4)}$$. Finally he shows that $$K_4^{(3)}(\mathcal C)$$ (resp. $$K_6^{(4)}(\mathcal C)$$), $$H^2(\widetilde{\mathcal M}^\bullet _{(3)}(\mathcal C))$$ (resp. $$H^2(\widetilde{\mathcal M}^\bullet _{(4)}(\mathcal C))$$) and Goncharov’s version $$H^2(\Gamma'({\mathcal C},3))$$ (resp. $$H^2(\Gamma'({\mathcal C},4))$$) of the latter group all have the same image in $$H^1_{\text{dR}}({\mathcal C}_{\text{an}}; {\mathbb R}(2))^+$$ (resp. in $$H^1_{\text{dR}}({\mathcal C}_{\text{an}}; {\mathbb R}(3))^+$$) under the regulator map.

##### MSC:
 19F27 Étale cohomology, higher regulators, zeta and $$L$$-functions ($$K$$-theoretic aspects) 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 19D45 Higher symbols, Milnor $$K$$-theory 19E08 $$K$$-theory of schemes
##### Citations:
Zbl 0868.19002; Zbl 0849.11087
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