×

On the syntomic regulator for \(K_1\) of a surface. (English) Zbl 1285.19001

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 \[ \text{reg}_{\text{{e}t}}:H_{\mathcal M}^{2i-j}(S,{\mathbb Q}(i))\longrightarrow H^1(K,H_{\text{{e}t}}^{2i-j-1}(\overline{S},{\mathbb Q}_p(i)). \] S. Bloch and 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.

MSC:

19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
14F42 Motivic cohomology; motivic homotopy theory
11G07 Elliptic curves over local fields
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14C15 (Equivariant) Chow groups and rings; motives
14F30 \(p\)-adic cohomology, crystalline cohomology

Citations:

Zbl 0768.14001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. Baldassarri and P. Berthelot, On Dwork cohomology for singular hypersurfaces, in Geometric Aspects of Dwork Theory, Vols. I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 177–244. · Zbl 1117.14022
[2] F. Baldassarri, M. Cailotto and L. Fiorot, Poincaré duality for algebraic de Rham cohomology, Manuscripta Mathematica 114 (2004), 61–116. · Zbl 1071.14023
[3] A. A. Beilinson, Higher regulators and values of L-functions, Journal of Soviet Mathematics 30 (1985), 2036–2070. · Zbl 0588.14013
[4] A. A. Beilinson, Notes on absolute Hodge cohomology, in Applications of Algebraic K-Theory to Algebraic Geometry and Number Theory, Parts I, II (Boulder, Colo., 1983), Contemporary Mathematics, Vol. 55, American Mathematical Society, Providence, RI, 1986, pp. 35–68.
[5] P. Berthelot, Dualité de Poincaré et formule de Künneth en cohomologie rigide, Comptes Rendus de l’Académie des Sciences. Série I. Mathématique 325 (1997), 493–498.
[6] P. Berthelot, Finitude et pureté cohomologique en cohomologie rigide, Inventiones Mathematicae 128 (1997), 329–377, With an appendix in English by A. J. de Jong. · Zbl 0908.14005
[7] A. Besser, A generalization of Coleman’s p-adic integration theory, Inventiones Mathematicae 142 (2000), 397–434. · Zbl 1053.14020
[8] A. Besser, Syntomic regulators and p-adic integration I: rigid syntomic regulators, Israel Journal of Mathematics 120 (2000), 291–334. · Zbl 1001.19003
[9] A. Besser, Syntomic regulators and p-adic integration II: K 2 of curves, Israel Journal of Mathematics 120 (2000), 335–360. · Zbl 1001.19004
[10] A. Besser, Coleman integration using the Tannakian formalism, Mathematische Annalen 322 (2002), 19–48. · Zbl 1013.11028
[11] A. Besser and R. de Jeu, The syntomic regulator for the K-theory of fields, Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 36 (2003), 867–924. · Zbl 1106.11024
[12] A. Besser and R. de Jeu, The syntomic regulator for K 4 of curves, Algorithmic Number Theory (ANTS-IX), Lecture Notes in Computer Science 6197, Springer-Verlag, 2010, pp. 16–31. · Zbl 1261.14011
[13] S. Bloch and K. Kato, L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift I (Boston), Progress in Mathematics, Vol. 86, Birkhäuser, Basel, 1990, pp. 333–400. · Zbl 0768.14001
[14] R. Bradshaw, J. Balakrishnan and K. Kedlaya, Cycle classes and the syntomic regulator, 2010, http://arxiv.org/abs/1006.0132 .
[15] B. Chiarellotto, A. Ciccioni and N. Mazzari, On rigid syntomic cohomology with compact support, preprint, 2010. · Zbl 1330.14030
[16] R. Coleman, Dilogarithms, regulators, and p-adic L-functions, Inventiones Mathematicae 69 (1982), 171–208. · Zbl 0516.12017
[17] R. Coleman, Reciprocity laws on curves, Compositio Mathematica 72 (1989), 205–235. · Zbl 0706.14013
[18] R. Coleman and E. de Shalit, p-adic regulators on curves and special values of p-adic L-functions, Inventiones Mathematicae 93 (1988), 239–266. · Zbl 0655.14010
[19] M. Gros, Régulateurs syntomiques et valeurs de fonctions L p-adiques. II, Inventiones Mathematicae 115 (1994), 61–79. · Zbl 0799.14010
[20] E. Grosse-Klönne, Rigid analytic spaces with overconvergent structure sheaf, Journal für die Reine und Angewandte Mathematik 519 (2000), 73–95. · Zbl 0945.14013
[21] I. Gutnik, Kedlays’s algorithm and Coleman integration, M.Sc. Thesis, Ben-Gurion University, 2006.
[22] A. Langer, On the syntomic regulator for products of elliptic curves, Journal of the London Mathematical Society, to appear. · Zbl 1228.14016
[23] W. Nizioł, On the image of p-adic regulators, Inventiones Mathematicae 127 (1997), 375–400. · Zbl 0928.14014
[24] V. Vologodsky, Hodge structure on the fundamental group and its application to p-adic integration, Moscow Mathematical Journal 3 (2003), 205–247. · Zbl 1050.14013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.