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REFINED SWAN CONDUCTORS $\text{mod}~p$ OF ONE-DIMENSIONAL GALOIS REPRESENTATIONS

Part of: Algebraic number theory: local and $p$-adic fields (Co)homology theory

Published online by Cambridge University Press:  03 June 2019

KAZUYA KATO ,
ISABEL LEAL  and
TAKESHI SAITO
KAZUYA KATO
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA email kkato@math.uchicago.edu
ISABEL LEAL
Affiliation:
Courant Institute of Mathematical Sciences, New York, NY 10012-1185, USA email leal@courant.nyu.edu
TAKESHI SAITO
Affiliation:
Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan email t-saito@ms.u-tokyo.ac.jp
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Abstract

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

MSC classification

Primary: 14F30: $p$-adic cohomology, crystalline cohomology 11S31: Class field theory; $p$-adic formal groups
Type
Article
Information
Nagoya Mathematical Journal , Volume 236: Celebrating the 60th Birthday of Shuji Saito , December 2019 , pp. 134 - 182
Copyright
© 2019 Foundation Nagoya Mathematical Journal  

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Footnotes

One of the authors (K.K.) is partially supported by NSF Award 1601861 and (T.S.) is partially supported by JSPS Grant-in-Aid for Scientific Research (A) 26247002.

References

Abbes, A. and Saito, T., Ramification of local fields with imperfect residue fields , Amer. J. Math. 124 (2002), 879–920.10.1353/ajm.2002.0026Google Scholar
Abbes, A. and Saito, T., Analyse Micro-locale l-adique en caractéristique p > 0. Le cas d’un trait , RIMS, Kyoto Univ. 45 (2009), 25–74.10.2977/prims/1234361154+0.+Le+cas+d’un+trait+,+RIMS,+Kyoto+Univ.+45+(2009),+25–74.10.2977/prims/1234361154>Google Scholar
Barrientos, I., Log ramification via curves in rank 1 , Int. Math. Res. Not. IMRN 19 (2017), 5769–5799.Google Scholar
Bhatt, B., Morrow, M. and Scholze, P., Topological Hochschild homology and integral $p$ -adic Hodge theory, preprint, 2018, arXiv:1802.03261.Google Scholar
Borger, J. M., Conductors and the moduli of residual perfection , Math. Ann. 329 (2004), 1–30.10.1007/s00208-003-0490-1Google Scholar
Borger, J. M., Kato’s conductor and generic residual perfection, preprint, 2011, arXiv:0112306v2.Google Scholar
Brylinski, J.-L., Théorie du corps de classes de Kato et revêtements abéliens de surfaces , Ann. Inst. Fourier (Grenoble) 33 (1983), 23–38.Google Scholar
Esnault, H. and Kerz, M., A finiteness theorem for Galois representations of function fields over finite fields (after Deligne) , Acta Math. Vietnam. 37 (2012), 531–562.Google Scholar
Fujiwara, K. and Kato, F., Foundations of Rigid Geometry. I, EMS Monographs in Mathematics, European Mathematical Society, Zurich, 2018.10.4171/135Google Scholar
Garel, E., An extension of the trace map , J. Pure Appl. Algebra 32 (1984), 301–313.10.1016/0022-4049(84)90094-XGoogle Scholar
Geisser, T. and Hesselholt, L., The de Rham–Witt complex and p-adic vanishing cycles , J. Amer. Math. Soc. 19 (2006), 1–36.10.1090/S0894-0347-05-00505-9Google Scholar
Grothendieck, A., Élements de géométrie algébrique IV (première partie, quatrième partie) , Publ. Math. Inst. Hautes Études Sci. 20 (1964), 32 (1967).10.1007/BF02684747Google Scholar
Hartshorne, R., Residue and Duality, Lecture Notes in Mathematics, 20 , Springer, Berlin, Heidelberg, New York, 1966.10.1007/BFb0080482Google Scholar
Hazewinkel, M., Corps de classes local, Appendix to Demazure M. and Gabriel M., Groupes algébriques, Tome I: Géométrie algébrique, généralités, groupes commutatifs (1970).Google Scholar
Hu, H., Logarithmic ramifications of étale sheaves by restricting to curve, preprint, 2017, arXiv:1704.04734.10.1093/imrn/rnx228Google Scholar
Illusie, L., Complexe de de Rham–Witt et cohomologie cristalline , Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), 501–661.Google Scholar
Kato, K., A generalization of local class field theory by using K-groups. II , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 603–683.Google Scholar
Kato, K., A generalization of local class field theory by using K-groups. III , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), 31–34.Google Scholar
Kato, K., Residue homomorphisms in Milnor K-theory , Adv. Stud. Pure Math. 2 (1983), 153–172.10.2969/aspm/00210153Google Scholar
Kato, K., Swan conductors for characters of degree one in the imperfect residue field case , Contemp. Math. 83 (1989), 101–131.10.1090/conm/083/991978Google Scholar
Kato, K., “ Logarithmic structures of Fontaine–Illusie ”, in Algebraic Analysis, Geometry, and Number Theory, Johns Hopkins University Press, Baltimore, 1989, 191–224.Google Scholar
Kato, K., “ Existence theorem for higher local fields ”, in Invitation to Higher Local Fields, Geometry and Topology Monographs, 3 , Mathematical Sciences Publisher, Berkeley, 2000, 165–195.Google Scholar
Kato, K. and Saito, S., Global class field theory of arithmetic schemes , Contemp. Math. 55 (1986), 255–331.10.1090/conm/055.1/862639Google Scholar
Kato, K. and Saito, T., Coincidence of two Swan conductors of abelian characters, preprint, 2019, arXiv:1904.08604.Google Scholar
Kato, K. and Suzuki, T., Duality theories for $p$ -primary étale cohomology, III, J. Math. Sci. Univ. Tokyo (to appear).Google Scholar
Kerz, M. and Saito, S, Chow group of 0-cycles with modulus and higher-dimensional class field theory , Duke Math. J. 165 (2016), 2811–2897.10.1215/00127094-3644902Google Scholar
Kurihara, M., On two types of complete discrete valuation fields , Compositio Math. 63 (1987), 237–257.Google Scholar
Kurihara, M., The exponential homomorphisms for the Milnor K-groups and an explicit reciprocity law , J. Reine Angew. Math. 498 (1998), 201–221.10.1515/crll.1998.051Google Scholar
Leal, I., On ramification in transcendental extensions of local fields , J. Algebra 495 (2018), 15–50.10.1016/j.jalgebra.2017.11.004Google Scholar
Matsuda, S., On the Swan conductor in positive characteristic , Amer. J. Math. 119 (1997), 705–739.10.1353/ajm.1997.0026Google Scholar
Parshin, A. N., Local class field theory , Trudy Mat. Inst. Steklov. 165 (1984), 143–170.Google Scholar
Raskind, W., Abelian Class Field Theory of Arithmetic Schemes, Proc. Sympos. Pure Math., 58, Part 1, American Mathematical Society, Providence, 1995, 85–187.Google Scholar
Serre, J.-P., Sur les corps locaux à corps résiduel algébriquement clos , Bull. Soc. Math. France 89 (1961), 105–154.Google Scholar
Wiesend, G., Class field theory for arithmetic schemes , Math. Z. 256 (2007), 717–729.10.1007/s00209-006-0095-yGoogle Scholar
Yatagawa, Y., Equality of two non-logarithmic ramification filtrations of abelianized Galois group in positive characteristic , Documenta Math. 22 (2017), 917–952.Google Scholar