Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



The theory of Coleman power series for $K_2$

Author: Takako Fukaya
Journal: J. Algebraic Geom. 12 (2003), 1-80
Published electronically: August 5, 2002
MathSciNet review: 1948685
Full-text PDF

Abstract | References | Additional Information

Abstract: The purpose of this paper is to define ``Coleman power series'' associated to norm compatible systems in $K_2$ groups of complete discrete valuation fields of mixed characteristic $(0,p)$ with imperfect residue fields ${\mathsf{k}}$ such that $[{\mathsf {k}}:{\mathsf {k}}^p]=p$. These ``Coleman power series'' are elements of $K_2$groups of certain power series rings. We use our ``Coleman power series'' to obtain some results on modular forms, and we also study properties of our ``Coleman power series''.

References [Enhancements On Off] (What's this?)

  • [Be] Berthelot, P., Cohomologie cristalline des schémas de caractéristique $p > 0$, Lecture Notes in Math. 407, Springer (1974).
  • [Co] Coleman, R., Division values in local fields, Invent. Math. 53 (1979) 91-116.
  • [CW] Coates, J. and Wiles, A., On $p$-adic $L$-functions and Elliptic Units, J. Austral. Math. Soc (Series A) 26 (1978) 1-25.
  • [Fa1] Faltings, G., Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory, Johns Hopkins Univ. Press (1988) 25-88.
  • [Fa2] Faltings, G., Almost étale extensions, preprint MPI Bonn (1998).
  • [Fe1] Fesenko, I., Explicit constructions in local fields, Thesis, St. Petersburg Univ. (1987).
  • [Fe2] Fesenko, I., Class field theory of multidimensional local fields of characteristic $0$, with the residue field of positive characteristic, Algebra i Analiz (1991); English translation in St. Petersburg Math. J. 3 (1992) 649-678.
  • [Fo1] Fontaine, J. -M., Sur certains types de répresentations $p$-adiques du groupe de Galois d'un corps local: construction d'un anneau de Barsotti-Tate, Ann. of Math. 115 (1982) 529-577.
  • [Fo2] Fontaine, J.-M., Représentations $p$-adiques des corps locaux, Grothendieck festschrift, vol. 2, Birkhäuser (1990) 249-309.
  • [Fo3] Fontaine, J. -M., Sur un théorème de Bloch et Kato $($lettre à B. Perrin-Riou$)$ appendice to PERRIN-RIOU, B.Théorie d'Iwasawa des représentations $p$-adiques, Invent. Math. 115 (1994) 151-161.
  • [FM] Fontaine, J. -M., and Messing, W.,$p$-adic periods and $p$-adic étale cohomology, Contemporary Math. 67 (1987) 179-207.
  • [Fu1] Fukaya, T., Explicit reciprocity laws for $p$-divisible groups over higher dimensional local fields, Journal für die reine und ang. Math. 531 61-119 (2001).
  • [Fu2] Fukaya, T., Coleman power series for $K_2$ and $p$-adic zeta functions of modular forms, in preparation.
  • [FW] Fontaine, J. -M. and Wintenberger, J.- P. , Le ``corps des normes'' de certaines extensions algébriques de corps locaux, C.R. Acad. Sci. 288 (1979) 367-370.
  • [Hi] Hida, H., Elementary theory of $L$-functions and Eisenstein series, London Math. Soc. Student Texts 26, Cambridge Univ. Press (1993).
  • [Hy] Hyodo, O., On the Hodge-Tate decomposition in the imperfect residue field case, Journal für die reine und ang. Math. 365 (1986) 97-113.
  • [Iw] Iwasawa, K., On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964) 42-82.
  • [Ka1] Kato, K., A generalization of local class field theory by using $K$-groups, ${\rom {I}}$, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 26 (1979) 303-376; ${\rom {II}}$, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 27 (1980) 603-683; ${\rom {III}}$, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 29 (1982) 31-43.
  • [Ka2] Kato, K., Residue Homomorphisms in Milnor $K$-theory, Advanced Studies in Pure Math. 2 (1983) 153-172.
  • [Ka3] Kato, K., Explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. Fr. 119 (1991) 397-441.
  • [Ka4] Kato, K., Lectures on the approach to Iwasawa theory for Hasse-Weil $L$-functions via $B_{\rom {dR}}$, Lecture Notes in Math. 1553, Springer (1993) 50-163.
  • [Ka5] Kato, K., Generalized explicit reciprocity laws, Advanced Studies in Contemporary Mathematics 1 (1999) 57-126.
  • [Ka6] Kato, K.,$p$-adic Hodge theory and values of zeta functions of modular forms, preprint, Univ. Tokyo (2000).
  • [Ku1] Kurihara, M., On two types of complete discrete valuation fields, Compositio Math. 63 (1987) 237-257.
  • [Ku2] Kurihara, M., The exponential homomorphism for the Milnor $K$-groups and explicit reciprocity law, Journal für die reine und ang. Math. 498 (1998) 201-221.
  • [La] Laubie, F., Extensions de Lie et groupes d'automorphismes de corps locaux, Compos. Math. 67 (1988) 165-189.
  • [Mi] Milnor, J., Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1970) 318-344.
  • [Na] Nakamura, J., On the Milnor $K$-groups of complete discrete valuation fields, the doctoral thesis, Univ. of Tokyo (2000).
  • [Qu] Quillen, D., Higher algebraic $K$-theory I, Lecture Notes in Math. 341, Springer (1973) 85-147.
  • [Sc] Scholl, J., An introduction to Kato's Euler systems, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press (1998) 379-460.
  • [Sh] Shimura, G., The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976) 783-804.
  • [Ta] Tate, J., Relations between $K_2$ and Galois cohomology, Invent. Math. 36 (1976) 257-274.
  • [Ts] Tsuji, T.,$p$-adic etale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999) 233-411.
  • [Vo1] Vostkov, V., An explicit form of the reciprocity law, English transl. in Math. USSR Izv. 13 (1979) 557-588.
  • [Vo2] Vostkov, V., Explicit construction of the theory of class fields of a multidimensional local field, English transl. in Math. USSR Izv. 26 (1986) 263-287.
  • [Wil] Wiles, A., Higher explicit reciprocity laws, Annals of Math. 107 (1978) 235-254.
  • [Win] Wintenberger, J.- P., Le corps des normes de certaines extensions infinies de corps locaux, Ann. Sc. ENS. 16 (1983) 59-89.
  • [Wit] Witt, E., Zyklische Körper und Algebren der Charakteristik $p$ vom Grad $p^n$, Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteritik $p$, Journal für die reine und ang. Math. 176 126-140 (1937).

Additional Information

Takako Fukaya
Affiliation: Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

Received by editor(s): August 3, 2000
Published electronically: August 5, 2002

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
AMS Website