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
DOI: https://doi.org/10.1090/S1056-3911-02-00324-7
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]Be Berthelot, P., Cohomologie cristalline des schémas de caractéristique $p > 0$, Lecture Notes in Math. 407, Springer (1974). [Co]Co Coleman, R., Division values in local fields, Invent. Math. 53 (1979) 91–116. [CW]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]Fa1 Faltings, G., Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory, Johns Hopkins Univ. Press (1988) 25–88. [Fa2]Fa2 Faltings, G., Almost étale extensions, preprint MPI Bonn (1998). [Fe1]Fe1 Fesenko, I., Explicit constructions in local fields, Thesis, St. Petersburg Univ. (1987). [Fe2]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]Fo0 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]Fo1 Fontaine, J.-M., Représentations $p$-adiques des corps locaux, Grothendieck festschrift, vol. 2, Birkhäuser (1990) 249–309. [Fo3]Fo2 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]FM Fontaine, J. -M., and Messing, W., $p$-adic periods and $p$-adic étale cohomology, Contemporary Math. 67 (1987) 179–207. [Fu1]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]Fu2 Fukaya, T., Coleman power series for $K_2$ and $p$-adic zeta functions of modular forms, in preparation. [FW]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]Hi Hida, H., Elementary theory of $L$-functions and Eisenstein series, London Math. Soc. Student Texts 26, Cambridge Univ. Press (1993). [Hy]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]Iw Iwasawa, K., On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964) 42–82. [Ka1]Ka1 Kato, K., A generalization of local class field theory by using $K$-groups, I, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 26 (1979) 303–376; II, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 27 (1980) 603–683; III, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 29 (1982) 31–43. [Ka2]Ka6 Kato, K., Residue Homomorphisms in Milnor $K$-theory, Advanced Studies in Pure Math. 2 (1983) 153–172. [Ka3]Ka3 Kato, K., Explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. Fr. 119 (1991) 397–441. [Ka4]Ka8 Kato, K., Lectures on the approach to Iwasawa theory for Hasse-Weil $L$-functions via $B_{\mathrm {dR}}$, Lecture Notes in Math. 1553, Springer (1993) 50–163. [Ka5]Ka5 Kato, K., Generalized explicit reciprocity laws, Advanced Studies in Contemporary Mathematics 1 (1999) 57–126. [Ka6]Ka7 Kato, K., $p$-adic Hodge theory and values of zeta functions of modular forms, preprint, Univ. Tokyo (2000). [Ku1]Ku1 Kurihara, M., On two types of complete discrete valuation fields, Compositio Math. 63 (1987) 237–257. [Ku2]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]La Laubie, F., Extensions de Lie et groupes d’automorphismes de corps locaux, Compos. Math. 67 (1988) 165–189. [Mi]Mi Milnor, J., Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1970) 318–344. [Na]Na Nakamura, J., On the Milnor $K$-groups of complete discrete valuation fields, the doctoral thesis, Univ. of Tokyo (2000). [Qu]Qu Quillen, D., Higher algebraic $K$-theory I, Lecture Notes in Math. 341, Springer (1973) 85–147. [Sc]Sc Scholl, J., An introduction to Kato’s Euler systems, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press (1998) 379–460. [Sh]Sh Shimura, G., The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976) 783–804. [Ta]Ta Tate, J., Relations between $K_2$ and Galois cohomology, Invent. Math. 36 (1976) 257–274. [Ts]Ts Tsuji, T., $p$-adic et́ale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999) 233–411. [Vo1]Vo1 Vostkov, V., An explicit form of the reciprocity law, English transl. in Math. USSR Izv. 13 (1979) 557-588. [Vo2]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]Wil Wiles, A., Higher explicit reciprocity laws, Annals of Math. 107 (1978) 235–254. [Win]Win Wintenberger, J.- P., Le corps des normes de certaines extensions infinies de corps locaux, Ann. Sc. ENS. 16 (1983) 59–89. [Wit]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
Email: takako@ms357.ms.u-tokyo.ac.jp

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