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A cohomological Tamagawa number formula

Published online by Cambridge University Press:  11 January 2016

Annette Huber
Affiliation:
Mathematisches Institut, Universität Freiburg, 9102 Freiburg, [email protected]
Guido Kings
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, [email protected]
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Abstract

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For smooth linear group schemes over ℤ, we give a cohomological interpretation of the local Tamagawa measures as cohomological periods. This is in the spirit of the Tamagawa measures for motives defined by Bloch and Kato. We show that in the case of tori, the cohomological and the motivic Tamagawa measures coincide, which proves again the Bloch-Kato conjecture for motives associated to tori.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2011

References

[B] Bloch, S., A note on height pairings, Tamagawa numbers, and the Birch-Swinnerton-Dyer conjecture, Invent. Math. 58 (1980), 6576.CrossRefGoogle Scholar
[BK] Bloch, S. and Kato, K., “ L-functions and Tamagawa numbers of motives” in The Grothendieck Festschrift, Vol. I, Progr. Math. 86, Birkhäuser, Boston, 1990, 333400.Google Scholar
[Bo] Borel, A., Cohomologie de SLn et valeurs de fonctions zeta aux points entiers, Ann. Sc. Norm. Supér. Pisa Cl. Sci. (5) 4 (1977), 613636.Google Scholar
[BLR] Bosch, S., Lütkebohmert, W., and Raynaud, M., Néron Models, Ergeb. Math. Grenzgeb. (3) 21, Springer, Berlin, 1990.Google Scholar
[CR] Curtis, C. W. and Reiner, I., Methods of Representation Theory, Vol. II: With Applications to Finite Groups and Orders, Pure Appl. Math., Wiley, New York, 1987.Google Scholar
[F] Fontaine, J.-M., Valeurs spéciales des fonctions L de motifs, Astérisque 206 (1992), Séminaire Bourbaki, no. 751.Google Scholar
[G] Gross, B. H., On the motive of a reductive group, Invent. Math. 130 (1997), 287313.CrossRefGoogle Scholar
[H] Huber, A., Poincaré duality for p-adic Lie groups, Arch. Math. 95 (2010), 509517.CrossRefGoogle Scholar
[HK1] Huber, A. and Kings, G., Bloch-Kato conjecture and main conjecture of Iwasawa theory for Dirichlet characters, Duke Math. J. 119 (2003), 393464.CrossRefGoogle Scholar
[HK2] Huber, A. and Kings, G., A p-adic analogue of the Borel regulator and the Bloch-Kato exponential map, J. Inst. Math. Jussieu 10 (2011), 149190.CrossRefGoogle Scholar
[HKN] Huber, A., Kings, G., and Naumann, N., Some complements to the Lazard isomorphism, Compos. Math. 147 (2011), 235262.CrossRefGoogle Scholar
[L] Lazard, M., Groupes analytiques p-adiques, Publ. Math. Inst. Hautes Etudes Sci. 26 (1965).Google Scholar
[M] Milne, J. S., Arithmetic Duality Theorems, Perspect. Math. 1, Academic Press, Boston, 1986.Google Scholar
[N] Naumann, N., Arithmetically defined dense subgroups of Morava stabilizer groups, Compos. Math. 144 (2008), 247270.CrossRefGoogle Scholar
[O] Ono, T., On the Tamagawa number of algebraic tori, Ann. of Math. (2) 78 (1963), 4773.CrossRefGoogle Scholar
[PR] Platonov, V. and Rapinchuk, A., Algebraic Groups and Number Theory, Pure Appl. Math. 139, Academic Press, Boston, 1994.Google Scholar
[S] Serre, J.-P., Lie Algebras and Lie Groups, Harvard University Lectures, W. A. Benjamin, New York, 1965.Google Scholar
[W] Weil, A., Adeles and Algebraic Groups, with appendices by Demazure, M. and Ono, T., Progr. Math. 23, Birkhäuser, Boston, 1982.Google Scholar