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
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References |
Additional Information
Abstract: The purpose of this paper is to define âColeman power seriesâ 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 âColeman power seriesâ are elements of $K_2$ groups of certain power series rings. We use our âColeman power seriesâ to obtain some results on modular forms, and we also study properties of our âColeman power seriesâ.
[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) 91â116.
[CW]CW Coates, J. and Wiles, A., On $p$-adic $L$-functions and Elliptic Units, J. Austral. Math. Soc (Series A) 26 (1978) 1â25.
[Fa1]Fa1 Faltings, G., Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory, Johns Hopkins Univ. Press (1988) 25â88.
[Fa2]Fa2 Faltings, G., Almost Ă©tale 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) 649â678.
[Fo1]Fo0 Fontaine, J. -M., Sur certains types de rĂ©presentations $p$-adiques du groupe de Galois dâun corps local: construction dâun anneau de Barsotti-Tate, Ann. of Math. 115 (1982) 529â577.
[Fo2]Fo1 Fontaine, J.-M., ReprĂ©sentations $p$-adiques des corps locaux, Grothendieck festschrift, vol. 2, BirkhĂ€user (1990) 249â309.
[Fo3]Fo2 Fontaine, J. -M., Sur un thĂ©orĂšme de Bloch et Kato $($lettre Ă B. Perrin-Riou$)$ appendice to Perrin-Riou, B.ThĂ©orie dâIwasawa des reprĂ©sentations $p$-adiques, Invent. Math. 115 (1994) 151-161.
[FM]FM Fontaine, J. -M., and Messing, W., $p$-adic periods and $p$-adic Ă©tale cohomology, Contemporary Math. 67 (1987) 179â207.
[Fu1]Fu1 Fukaya, T., Explicit reciprocity laws for $p$-divisible groups over higher dimensional local fields, Journal fĂŒr die reine und ang. Math. 531 61â119 (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 âcorps des normesâ de certaines extensions algĂ©briques de corps locaux, C.R. Acad. Sci. 288 (1979) 367â370.
[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 fĂŒr die reine und ang. Math. 365 (1986) 97â113.
[Iw]Iw Iwasawa, K., On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964) 42â82.
[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) 303â376; II, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 27 (1980) 603â683; III, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 29 (1982) 31â43.
[Ka2]Ka6 Kato, K., Residue Homomorphisms in Milnor $K$-theory, Advanced Studies in Pure Math. 2 (1983) 153â172.
[Ka3]Ka3 Kato, K., Explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. Fr. 119 (1991) 397â441.
[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) 50â163.
[Ka5]Ka5 Kato, K., Generalized explicit reciprocity laws, Advanced Studies in Contemporary Mathematics 1 (1999) 57â126.
[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) 237â257.
[Ku2]Ku2 Kurihara, M., The exponential homomorphism for the Milnor $K$-groups and explicit reciprocity law, Journal fĂŒr die reine und ang. Math. 498 (1998) 201â221.
[La]La Laubie, F., Extensions de Lie et groupes dâautomorphismes de corps locaux, Compos. Math. 67 (1988) 165â189.
[Mi]Mi Milnor, J., Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1970) 318â344.
[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) 85â147.
[Sc]Sc Scholl, J., An introduction to Katoâs Euler systems, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press (1998) 379â460.
[Sh]Sh Shimura, G., The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976) 783â804.
[Ta]Ta Tate, J., Relations between $K_2$ and Galois cohomology, Invent. Math. 36 (1976) 257â274.
[Ts]Ts Tsuji, T., $p$-adic etÌale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999) 233â411.
[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) 263â287.
[Wil]Wil Wiles, A., Higher explicit reciprocity laws, Annals of Math. 107 (1978) 235â254.
[Win]Win Wintenberger, J.- P., Le corps des normes de certaines extensions infinies de corps locaux, Ann. Sc. ENS. 16 (1983) 59â89.
[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 fĂŒr die reine und ang. Math. 176 126â140 (1937).
[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) 91â116.
[CW]CW Coates, J. and Wiles, A., On $p$-adic $L$-functions and Elliptic Units, J. Austral. Math. Soc (Series A) 26 (1978) 1â25.
[Fa1]Fa1 Faltings, G., Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory, Johns Hopkins Univ. Press (1988) 25â88.
[Fa2]Fa2 Faltings, G., Almost Ă©tale 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) 649â678.
[Fo1]Fo0 Fontaine, J. -M., Sur certains types de rĂ©presentations $p$-adiques du groupe de Galois dâun corps local: construction dâun anneau de Barsotti-Tate, Ann. of Math. 115 (1982) 529â577.
[Fo2]Fo1 Fontaine, J.-M., ReprĂ©sentations $p$-adiques des corps locaux, Grothendieck festschrift, vol. 2, BirkhĂ€user (1990) 249â309.
[Fo3]Fo2 Fontaine, J. -M., Sur un thĂ©orĂšme de Bloch et Kato $($lettre Ă B. Perrin-Riou$)$ appendice to Perrin-Riou, B.ThĂ©orie dâIwasawa des reprĂ©sentations $p$-adiques, Invent. Math. 115 (1994) 151-161.
[FM]FM Fontaine, J. -M., and Messing, W., $p$-adic periods and $p$-adic Ă©tale cohomology, Contemporary Math. 67 (1987) 179â207.
[Fu1]Fu1 Fukaya, T., Explicit reciprocity laws for $p$-divisible groups over higher dimensional local fields, Journal fĂŒr die reine und ang. Math. 531 61â119 (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 âcorps des normesâ de certaines extensions algĂ©briques de corps locaux, C.R. Acad. Sci. 288 (1979) 367â370.
[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 fĂŒr die reine und ang. Math. 365 (1986) 97â113.
[Iw]Iw Iwasawa, K., On some modules in the theory of cyclotomic fields, J. Math. Soc. Japan 16 (1964) 42â82.
[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) 303â376; II, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 27 (1980) 603â683; III, J. Fac. Sci. Univ. of Tokyo, Sec. IA, 29 (1982) 31â43.
[Ka2]Ka6 Kato, K., Residue Homomorphisms in Milnor $K$-theory, Advanced Studies in Pure Math. 2 (1983) 153â172.
[Ka3]Ka3 Kato, K., Explicit reciprocity law and the cohomology of Fontaine-Messing, Bull. Soc. Math. Fr. 119 (1991) 397â441.
[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) 50â163.
[Ka5]Ka5 Kato, K., Generalized explicit reciprocity laws, Advanced Studies in Contemporary Mathematics 1 (1999) 57â126.
[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) 237â257.
[Ku2]Ku2 Kurihara, M., The exponential homomorphism for the Milnor $K$-groups and explicit reciprocity law, Journal fĂŒr die reine und ang. Math. 498 (1998) 201â221.
[La]La Laubie, F., Extensions de Lie et groupes dâautomorphismes de corps locaux, Compos. Math. 67 (1988) 165â189.
[Mi]Mi Milnor, J., Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1970) 318â344.
[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) 85â147.
[Sc]Sc Scholl, J., An introduction to Katoâs Euler systems, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press (1998) 379â460.
[Sh]Sh Shimura, G., The special values of the zeta functions associated with cusp forms, Comm. Pure Appl. Math. 29 (1976) 783â804.
[Ta]Ta Tate, J., Relations between $K_2$ and Galois cohomology, Invent. Math. 36 (1976) 257â274.
[Ts]Ts Tsuji, T., $p$-adic etÌale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999) 233â411.
[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) 263â287.
[Wil]Wil Wiles, A., Higher explicit reciprocity laws, Annals of Math. 107 (1978) 235â254.
[Win]Win Wintenberger, J.- P., Le corps des normes de certaines extensions infinies de corps locaux, Ann. Sc. ENS. 16 (1983) 59â89.
[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 fĂŒr die reine und ang. Math. 176 126â140 (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
