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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.

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
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