On the syntomic regulator for \(K_1\) of a surface
Let \(K\) be a finite extension field of \({\mathbb Q}_p\). If \(S\) is a proper variety over \(K\) we have the motivic cohomology group \(H_{\mathcal M}^{2i-j}(S,{\mathbb Q}(i))\) as an eigenspace of the Adams operations on \(K_j(S)\otimes{\mathbb Q}\). Soulé defined étale regulator maps \[ \text{reg}_{\text{{e}t}}:H_{\mathcal M}^{2i-j}(S,{\mathbb Q}(i))\longrightarrow H^1(K,H_{\text{{e}t}}^{2i-j-1}(\overline{S},{\mathbb Q}_p(i)). \] \textit{S. Bloch} and \textit{K. Kato} [in: The presentation functor and the compactified Jacobian. The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 333--400 (1990; Zbl 0768.14001)] constructed an exponential map \[ \exp:H_{\text{dR}}^{2i-j-1}({S}/K)/F^i\longrightarrow H^1(K,H_{\text{{e}t}}^{2i-j-1}(\overline{S},{\mathbb Q}_p(i)). \] The aim of the present paper is to compute the regulator for elements \(\theta\) in the higher Chow group \(CH^2(S,1)\), for \(S\) a surface. More precisely, using the isomorphism \(CH^2(S,1)\otimes{\mathbb Q}=H_{\mathcal M}^{3}(S,{\mathbb Q}(2))\) one may look at \[ \text{reg}_{\text{{e}t}}(\theta)\in H^1(K,H_{\text{{e}t}}^{2}(\overline{S},{\mathbb Q}_p(2)) \] and try to find an element \(\text{reg}_{\text{syn}}(\theta)\in H_{\mathrm{dR}}^2(S/K)/F^2\) such that \[ \text{exp}(\text{reg}_{\text{syn}}(\theta))=\text{reg}_{\text{{e}t}}(\theta). \] In the present paper this is achieved for \(S\) having reduction and under several integrality assumptions on \(\theta\). The formula describing \(\text{reg}_{\text{syn}}(\theta)\) involves a functional on (certain elements of) \(F^1H_{\mathrm{dR}}^2(S/K)\) defined in terms of Coleman integration.
scientific article; zbMATH DE number 6105500