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A p-adic analogue of the Borel regulator and the Bloch–Kato exponential map

Part of: Algebraic number theory: local and $p$-adic fields $K$-theory in number theory

Published online by Cambridge University Press:  05 August 2010

Annette Huber  and
Guido Kings
Annette Huber
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstrasse 1, 79104 Freiburg im Breisgau, Germany ([email protected])
Guido Kings
Affiliation:
Fakultät für Mathematik, Universität Regensburg, Universitätsstrasse 31, 93053 Regensburg, Germany ([email protected])
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Abstract

In this paper we define a p-adic analogue of the Borel regulator for the K-theory of p-adic fields. The van Est isomorphism in the construction of the classical Borel regulator is replaced by the Lazard isomorphism. The main result relates this p-adic regulator to the Bloch–Kato exponential and the Soulé regulator. On the way we give a new description of the Lazard isomorphism for certain formal groups. We also show that the Soulé regulator is induced by continuous and even analytic classes.

Keywords

p-adic regulatorBorel regulatorsyntomic cohomologycontinuous group cohomologyLie algebra cohomologyLazard isomorphism

MSC classification

Secondary: 11S70: $K$-theory of local fields 19F27: Étale cohomology, higher regulators, zeta and $L$-functions
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 10 , Issue 1 , January 2011 , pp. 149 - 190
Copyright
Copyright © Cambridge University Press 2010

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References

1Amice, Y., Duals, in Proc. Conf. p-adic Analysis, Nijmegen, 1978, pp. 1–15, Report 7806, Katholieke University, Nijmegen.Google Scholar
2.Beilinson, A., Higher regulators and values of L-functions, J. Sov. Math. 30 (1985), 2036–2070.CrossRefGoogle Scholar
3.Berthelot, P., Cohomologie rigide et cohomologie rigide à supports propres, prépublication 96-03, Université de Rennes (1996).Google Scholar
4.Besser, A., Syntomic regulators and p-adic integration, I, Rigid syntomic regulators, Israel J. Math. 120 (2000), 291–334.CrossRefGoogle Scholar
5.Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Volume I, pp. 333–400, Progress in Mathematics, Volume 86 (Birkhäuser, Boston, MA, 1990).Google Scholar
6.Borel, A., Stable real cohomology of arithmetic groups, Annales Scient. Éc. Norm. Sup. 7 (1974), 235–272.CrossRefGoogle Scholar
7Borel, A., Cohomologie de SLn et valeurs de fonctions zeta aux points entiers, Annali Scuola Norm. Sup. Pisa 4(4) (1977), 613–636.Google Scholar
8.Burgos Gil, J. I., The regulators of Beilinson and Borel, Centre de Recherches Mathématiques Monograph Series, Volume 15 (American Mathematical Society, Providence, RI, 2002).Google Scholar
9.Cartan, H., Notions d'algèbre différentielle, in Œuvres (ed. Remmert, R. and Serre, J.-P.), Volume III, pp. 1268–1282 (Springer, 1979).Google Scholar
10.Cartan, H. and Eilenberg, S., Homological algebra (Princeton University Press, 1956).Google Scholar
11.Casselman, W. and Wigner, D., Continuous cohomology and a conjecture of Serre's, Invent. Math. 24 (1974), 199–211.CrossRefGoogle Scholar
12.Colmez, P., Fonctions d'une variable p-adique, Astérisque 330 (2010), 13–59.Google Scholar
13.Greub, W., Halperin, S. and Vanstone, R., Connections, curvature and cohomology, Volume III (Academic Press, 1976).Google Scholar
14.Gros, M., Régulateurs syntomiques et valeurs de fonctions Lp-adiques, II, Invent. Math. 115 (1994), 61–79.CrossRefGoogle Scholar
15.Grosse-Klönne, E., De Rham-Kohomologie in der rigiden Analysis, Dissertation, Universität Münster (1999).Google Scholar
16.Grosse-Klönne, E., Rigid analytic spaces with overconvergent structure sheaf, J. Reine Angew. Math. 519 (2000), 73–95.Google Scholar
17.Grosse-Klönne, E., Finiteness of de Rham cohomology in rigid analysis, Duke Math. J. 113(1) (2002), 57–91.CrossRefGoogle Scholar
18.Hamida, N., Les régulateurs en K-théorie algébrique, Thesis, Institut de Mathématiques de Jussieu (2002).Google Scholar
19.Hochschild, G., Cohomology of algebraic linear groups, Illinois J. Math. 5 (1961), 492–519.CrossRefGoogle Scholar
20.Huber, A. and Kings, G., A cohomological Tamagawa number formula, Nagoya Math. J., in press.Google Scholar
21.Huber, A., Kings, G. and Naumann, N., Some complements to the Lazard isomorphism, Compositio Math., in press.Google Scholar
22.Karoubi, M., Homologie cyclique et régulateurs en K-théorie algébrique. C. R. Acad. Sci. Paris Sér. I 297 (1983), 557.Google Scholar
23.Karoubi, M., Sur la K-théorie multiplicative, in Cyclic cohomology and noncommutative geometry (ed. Cuntz, J. and Khalkali, M.), Fields Institute Communications Series (American Mathematical Society, Providence, RI, 1997).Google Scholar
24.Lazard, M., Groupes analytiques p-adiques, Publications Mathématiques de l'IHES, Volume 26 (Institut des Hautes Études Scientifiques, 1965).Google Scholar
25.Loday, J.-L., Cyclic homology, Grundlehren der Mathematischen Wissenschaften, Volume 301 (Springer, 1992).CrossRefGoogle Scholar
26.Meredith, D., Weak formal schemes, Nagoya Math. J. 45 (1972), 1–38.CrossRefGoogle Scholar
27.Niziol, W., On the image of p-adic regulators, Invent. Math. 127 (1997), 375–400.CrossRefGoogle Scholar
28.Rapoport, M., Comparison of the regulators of Beilinson and Borel, in Beĭlnson's conjectures on special values of L-functions, Perspectives in Mathematics, Volume 4, pp. 169–192 (Academic Press, 1988).CrossRefGoogle Scholar
29.Soulé, C., K-théorie des anneaux d'entiers de corps de nombres et cohomologie étale, Invent. Math. 55 (1979), 251–295.CrossRefGoogle Scholar
30.Soulé, C., On higher p-adic regulators, in Proc. Conf. Algebraic K-Theory, Evanston, IL, 1980, Lecture Notes in Mathematics, Volume 854, pp. 372–401 (Springer, 1981).CrossRefGoogle Scholar
31.Tamme, G., Comparison of the Karoubi regulator and the p-adic Borel regulator, preprint (2007; available at www.mathematik.uni-regensburg.de/FGAlgZyk/index.html), superseded by [132].Google Scholar
32.Tamme, G., The relative Chern character and regulators, Thesis, Universitat Regensburg (2010; arXive:1007.1385).Google Scholar