Cyclic extensions and the local lifting problem

Abstract

The local Oort conjecture states that, if $\Gamma$ is cyclic and $k$ is an algebraically closed field of characteristic $p$, then all $\Gamma$-extensions of $k[[t]]$ should lift to characteristic zero. We prove a critical case of this conjecture. In particular, we show that the conjecture is always true when $v_p(|\Gamma|) \leq 3$ and is true for arbitrarily highly $p$-divisible cyclic groups $\Gamma$ when a certain condition on the higher ramification filtration is satisfied.

Note: To view the article, click on the URL link for the DOI number.

  • [ArzdorfWewers] K. Arzdorf and S. Wewers, A local proof of the semistable reduction theorem, 2011.
    @misc{ArzdorfWewers,
      author = {Arzdorf, Kai and Wewers, Stefan},
      TITLE = {A local proof of the semistable reduction theorem},
      YEAR = {2011},
      ARXIV={1211.4624} , }
  • [Bertin] Go to document J. Bertin, "Obstructions locales au relèvement de revêtements galoisiens de courbes lisses," C. R. Acad. Sci. Paris Sér. I Math., vol. 326, iss. 1, pp. 55-58, 1998.
    @article {Bertin, MRKEY = {1649485},
      AUTHOR = {Bertin, Jos{é}},
      TITLE = {Obstructions locales au relèvement de revêtements galoisiens de courbes lisses},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. I Math.},
      FJOURNAL = {Comptes Rendus de l'Académie des Sciences. Série I. Mathématique},
      VOLUME = {326},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {55--58},
      ISSN = {0764-4442},
      CODEN = {CASMEI},
      MRCLASS = {14H30},
      MRNUMBER = {1649485},
      ZBLNUMBER = {0952.14018},
      DOI = {10.1016/S0764-4442(97)82712-7},
      }
  • [BM] Go to document J. Bertin and A. Mézard, "Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques," Invent. Math., vol. 141, iss. 1, pp. 195-238, 2000.
    @article {BM, MRKEY = {1767273},
      AUTHOR = {Bertin, Jos{é} and M{é}zard, Ariane},
      TITLE = {Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {141},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {195--238},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14D15 (14D20 14H10 14H30 14H37)},
      MRNUMBER = {1767273},
      MRREVIEWER = {Akio Tamagawa},
      ZBLNUMBER = {0993.14014},
      DOI = {10.1007/s002220000071},
      }
  • [BGR] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Berlin: Springer-Verlag, 1984, vol. 261.
    @book {BGR, MRKEY={0746961},
      AUTHOR = {Bosch, Siegfried and G{ü}ntzer, Ulrich and Remmert, Reinhold},
      TITLE = {Non-{A}rchimedean analysis},
      SERIES = {Grundlehren Math. Wiss.},
      VOLUME = {261},
      NOTE = {A systematic approach to rigid analytic geometry},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {Berlin},
      YEAR = {1984},
      PAGES = {xii+436},
      ISBN = {3-540-12546-9},
      MRCLASS = {32K10 (30G05 46P05)},
      MRNUMBER = {0746961},
      MRREVIEWER = {W. Bartenwerfer},
      ZBLNUMBER = {0539.14017},
      }
  • [BLstable] Go to document S. Bosch and W. Lütkebohmert, "Stable reduction and uniformization of abelian varieties. II," Invent. Math., vol. 78, iss. 2, pp. 257-297, 1984.
    @article {BLstable, MRKEY = {0767194},
      AUTHOR = {Bosch, Siegfried and L{ü}tkebohmert, Werner},
      TITLE = {Stable reduction and uniformization of abelian varieties. {II}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {78},
      YEAR = {1984},
      NUMBER = {2},
      PAGES = {257--297},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {14K15 (11G10)},
      MRNUMBER = {0767194},
      MRREVIEWER = {Frank Herrlich},
      ZBLNUMBER = {0554.14015},
      DOI = {10.1007/BF01388596},
      }
  • [BLrigid] Go to document S. Bosch and W. Lütkebohmert, "Formal and rigid geometry. I. Rigid spaces," Math. Ann., vol. 295, iss. 2, pp. 291-317, 1993.
    @article {BLrigid, MRKEY = {1202394},
      AUTHOR = {Bosch, Siegfried and L{ü}tkebohmert, Werner},
      TITLE = {Formal and rigid geometry. {I}. {R}igid spaces},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {295},
      YEAR = {1993},
      NUMBER = {2},
      PAGES = {291--317},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {11G25 (14D15 14G20)},
      MRNUMBER = {1202394},
      MRREVIEWER = {Qing Liu},
      ZBLNUMBER = {0808.14017},
      DOI = {10.1007/BF01444889},
      }
  • [arizona] Go to document I. Bouw and S. Wewers, Group actions on curves and the lifting problem.
    @misc{arizona,
      author = {Bouw, Irene and Wewers, Stefan},
      title = {Group actions on curves and the lifting problem},
      note = {Course notes for the Arizona Winter School, 2012},
      URL={http://swc.math.arizona.edu/aws/2012},
      }
  • [Brewisthesis] Go to document Brewis, Louis Hugo, Ramification theory of the $p$-adic open disc and the lifting problem.
    @Unpublished{Brewisthesis,
      author = {Brewis, Louis Hugo},
      title = {Ramification theory of the $p$-adic open disc and the lifting problem},
      note = {Ph.D. Thesis},
      URL={http://webdoc.sub.gwdg.de/ebook/dissts/Ulm/Brewis2009.pdf},
      }
  • [Brewis] Go to document L. H. Brewis and S. Wewers, "Artin characters, Hurwitz trees and the lifting problem," Math. Ann., vol. 345, iss. 3, pp. 711-730, 2009.
    @article {Brewis, MRKEY = {2534115},
      AUTHOR = {Brewis, Louis Hugo and Wewers, Stefan},
      TITLE = {Artin characters, {H}urwitz trees and the lifting problem},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {345},
      YEAR = {2009},
      NUMBER = {3},
      PAGES = {711--730},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {14G20 (11S31 13F25 14H30 14H37 14L30)},
      MRNUMBER = {2534115},
      MRREVIEWER = {Daniel Barsky},
      ZBLNUMBER = {1222.14045},
      DOI = {10.1007/s00208-009-0374-0},
      }
  • [CGH1] Go to document T. Chinburg, R. Guralnick, and D. Harbater, "Oort groups and lifting problems," Compos. Math., vol. 144, iss. 4, pp. 849-866, 2008.
    @article {CGH1, MRKEY = {2441248},
      AUTHOR = {Chinburg, T. and Guralnick, R. and Harbater, D.},
      TITLE = {Oort groups and lifting problems},
      JOURNAL = {Compos. Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {144},
      YEAR = {2008},
      NUMBER = {4},
      PAGES = {849--866},
      ISSN = {0010-437X},
      MRCLASS = {14L30 (14H37)},
      MRNUMBER = {2441248},
      MRREVIEWER = {Vasily A. Chernecky},
      ZBLNUMBER = {1158.12003},
      DOI = {10.1112/S0010437X08003515},
      }
  • [Epp73] Go to document H. P. Epp, "Eliminating wild ramification," Invent. Math., vol. 19, pp. 235-249, 1973.
    @article {Epp73, MRKEY = {0321929},
      AUTHOR = {Epp, Helmut P.},
      TITLE = {Eliminating wild ramification},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {19},
      YEAR = {1973},
      PAGES = {235--249},
      ISSN = {0020-9910},
      MRCLASS = {14E20 (13B15)},
      MRNUMBER = {0321929},
      MRREVIEWER = {Manfred Herrmann},
      ZBLNUMBER = {0254.13008},
      DOI = {10.1007/BF01390208},
      }
  • [Garuti96] Go to document M. A. Garuti, "Prolongement de revêtements galoisiens en géométrie rigide," Compositio Math., vol. 104, iss. 3, pp. 305-331, 1996.
    @article {Garuti96, MRKEY = {1424559},
      AUTHOR = {Garuti, Marco A.},
      TITLE = {Prolongement de revêtements galoisiens en géométrie rigide},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {104},
      YEAR = {1996},
      NUMBER = {3},
      PAGES = {305--331},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14G20 (14E20)},
      MRNUMBER = {1424559},
      ZBLNUMBER = {0885.14011},
      URL = {http://www.numdam.org/item?id=CM_1996__104_3_305_0},
      }
  • [Garuti99] Go to document M. A. Garuti, "Linear systems attached to cyclic inertia," in Arithmetic Fundamental Groups and Noncommutative Algebra, Providence, RI: Amer. Math. Soc., 2002, vol. 70, pp. 377-386.
    @incollection {Garuti99, MRKEY = {1935414},
      AUTHOR = {Garuti, Marco A.},
      TITLE = {Linear systems attached to cyclic inertia},
      BOOKTITLE = {Arithmetic Fundamental Groups and Noncommutative Algebra},
      VENUE={{B}erkeley, {CA},
      1999},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {70},
      PAGES = {377--386},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2002},
      MRCLASS = {14E20 (11S15 13K05)},
      MRNUMBER = {1935414},
      MRREVIEWER = {Henri Gaudier},
      ZBLNUMBER = {1072.14017},
      DOI = {10.1090/pspum/070/1935414},
      }
  • [Gre04] Go to document B. Green, "Realizing deformations of curves using Lubin-Tate formal groups," Israel J. Math., vol. 139, pp. 139-148, 2004.
    @article {Gre04, MRKEY = {2041226},
      AUTHOR = {Green, Barry},
      TITLE = {Realizing deformations of curves using {L}ubin-{T}ate formal groups},
      JOURNAL = {Israel J. Math.},
      FJOURNAL = {Israel Journal of Mathematics},
      VOLUME = {139},
      YEAR = {2004},
      PAGES = {139--148},
      ISSN = {0021-2172},
      CODEN = {ISJMAP},
      MRCLASS = {11S31 (14L05)},
      MRNUMBER = {2041226},
      MRREVIEWER = {Antoine Chambert-Loir},
      ZBLNUMBER = {1147.11344},
      DOI = {10.1007/BF02787544},
      }
  • [GM98] Go to document B. Green and M. Matignon, "Liftings of Galois covers of smooth curves," Compositio Math., vol. 113, iss. 3, pp. 237-272, 1998.
    @article {GM98, MRKEY = {1645000},
      AUTHOR = {Green, Barry and Matignon, Michel},
      TITLE = {Liftings of {G}alois covers of smooth curves},
      JOURNAL = {Compositio Math.},
      FJOURNAL = {Compositio Mathematica},
      VOLUME = {113},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {237--272},
      ISSN = {0010-437X},
      CODEN = {CMPMAF},
      MRCLASS = {14H30 (11G20 14D15 14E22)},
      MRNUMBER = {1645000},
      MRREVIEWER = {Akio Tamagawa},
      ZBLNUMBER = {0923.14006},
      DOI = {10.1023/A:1000455506835},
      }
  • [GM99] Go to document B. Green and M. Matignon, "Order $p$ automorphisms of the open disc of a $p$-adic field," J. Amer. Math. Soc., vol. 12, iss. 1, pp. 269-303, 1999.
    @article {GM99, MRKEY = {1630112},
      AUTHOR = {Green, Barry and Matignon, Michel},
      TITLE = {Order {$p$} automorphisms of the open disc of a {$p$}-adic field},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {12},
      YEAR = {1999},
      NUMBER = {1},
      PAGES = {269--303},
      ISSN = {0894-0347},
      MRCLASS = {13F25 (11S31 14L05)},
      MRNUMBER = {1630112},
      MRREVIEWER = {Noriko Yui},
      ZBLNUMBER = {0923.14007},
      DOI = {10.1090/S0894-0347-99-00284-2},
      }
  • [Ha80] Go to document D. Harbater, "Moduli of $p$-covers of curves," Comm. Algebra, vol. 8, iss. 12, pp. 1095-1122, 1980.
    @article {Ha80, MRKEY = {0579791},
      AUTHOR = {Harbater, David},
      TITLE = {Moduli of {$p$}-covers of curves},
      JOURNAL = {Comm. Algebra},
      FJOURNAL = {Communications in Algebra},
      VOLUME = {8},
      YEAR = {1980},
      NUMBER = {12},
      PAGES = {1095--1122},
      ISSN = {0092-7872},
      CODEN = {COALDM},
      MRCLASS = {14D15 (14H30)},
      MRNUMBER = {0579791},
      MRREVIEWER = {David Cox},
      ZBLNUMBER = {0471.14011},
      DOI = {10.1080/00927878008822511},
      }
  • [KatoVC] Go to document K. Kato, "Vanishing cycles, ramification of valuations, and class field theory," Duke Math. J., vol. 55, iss. 3, pp. 629-659, 1987.
    @article {KatoVC, MRKEY = {0904945},
      AUTHOR = {Kato, Kazuya},
      TITLE = {Vanishing cycles, ramification of valuations, and class field theory},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {55},
      YEAR = {1987},
      NUMBER = {3},
      PAGES = {629--659},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {11S15 (11S31 11S70 19F27)},
      MRNUMBER = {0904945},
      MRREVIEWER = {Jerzy Browkin},
      ZBLNUMBER = {0665.14005},
      DOI = {10.1215/S0012-7094-87-05532-3},
      }
  • [Ka86] Go to document N. M. Katz, "Local-to-global extensions of representations of fundamental groups," Ann. Inst. Fourier $($Grenoble$)$, vol. 36, iss. 4, pp. 69-106, 1986.
    @article {Ka86, MRKEY = {0867916},
      AUTHOR = {Katz, Nicholas M.},
      TITLE = {Local-to-global extensions of representations of fundamental groups},
      JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
      FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
      VOLUME = {36},
      YEAR = {1986},
      NUMBER = {4},
      PAGES = {69--106},
      ISSN = {0373-0956},
      CODEN = {AIFUA7},
      MRCLASS = {14H30 (14B20 14E20)},
      MRNUMBER = {0867916},
      MRREVIEWER = {Noriko Yui},
      ZBLNUMBER = {0564.14013},
      DOI = {10.5802/aif.1069},
      }
  • [Lang] Go to document S. Lang, Algebra, third ed., New York: Springer-Verlag, 2002, vol. 211.
    @book {Lang, MRKEY = {1878556},
      AUTHOR = {Lang, Serge},
      TITLE = {Algebra},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {211},
      EDITION = {third},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2002},
      PAGES = {xvi+914},
      ISBN = {0-387-95385-X},
      MRCLASS = {00A05 (15-02)},
      MRNUMBER = {1878556},
      ZBLNUMBER = {0984.00001},
      DOI = {10.1007/978-1-4613-0041-0},
      }
  • [LiuAG] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford: Oxford University Press, 2002, vol. 6.
    @book {LiuAG, MRKEY = {1917232},
      AUTHOR = {Liu, Qing},
      TITLE = {Algebraic Geometry and Arithmetic Curves},
      SERIES = {Oxf. Grad. Texts Math.},
      VOLUME = {6},
      NOTE = {translated from the French by Reinie Ern{é},
      Oxford Science Publications},
      PUBLISHER = {Oxford University Press},
      ADDRESS = {Oxford},
      YEAR = {2002},
      PAGES = {xvi+576},
      ISBN = {0-19-850284-2},
      MRCLASS = {14-01 (11G30 14A05 14A15 14Gxx 14Hxx)},
      MRNUMBER = {1917232},
      ZBLNUMBER = {0996.14005},
      MRREVIEWER = {C{\'ı}cero Fernandes de Carvalho},
      }
  • [Nagata] M. Nagata, Local Rings, Huntington, N.Y.: Robert E. Krieger Publishing Co., 1975.
    @book {Nagata, MRKEY = {0460307},
      AUTHOR = {Nagata, Masayoshi},
      TITLE = {Local Rings},
      NOTE = {corrected reprint},
      PUBLISHER = {Robert E. Krieger Publishing Co.},
      ADDRESS={ Huntington, N.Y.},
      YEAR = {1975},
      PAGES = {xiii+234},
      ISBN = {0-88275-228-6},
      MRCLASS = {13-01 (16-01)},
      MRNUMBER = {0460307},
      ZBLNUMBER = {0386.13010},
      }
  • [Ob11] A. Obus, "The (local) lifting problem for curves," in Galois-Teichmüller Theory and Arithmetic Geometry, Tokyo: Math. Soc. Japan, 2012, vol. 63, pp. 359-412.
    @incollection {Ob11, MRKEY = {3051249},
      AUTHOR = {Obus, Andrew},
      TITLE = {The (local) lifting problem for curves},
      BOOKTITLE = {Galois-{T}eichmüller Theory and Arithmetic Geometry},
      SERIES = {Adv. Stud. Pure Math.},
      VOLUME = {63},
      PAGES = {359--412},
      PUBLISHER = {Math. Soc. Japan},
      ADDRESS = {Tokyo},
      YEAR = {2012},
      MRCLASS = {14H37 (11G20 12Fxx 14G17)},
      MRNUMBER = {3051249},
      }
  • [Oort95] Go to document F. Oort, Some questions in algebraic geometry.
    @misc{Oort95,
      author = {Oort, Frans},
      title = {Some questions in algebraic geometry},
      URL = {http://www.staff.science.uu.nl/~oort0109/},
      SORTYEAR={2012},
      }
  • [Oort87] Go to document F. Oort, "Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero," in Algebraic Geometry, Bowdoin, 1985, Providence, RI: Amer. Math. Soc., 1987, vol. 46, pp. 165-195.
    @incollection {Oort87, MRKEY = {0927980},
      AUTHOR = {Oort, Frans},
      TITLE = {Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero},
      BOOKTITLE = {Algebraic Geometry, {B}owdoin, 1985},
      VENUE={{B}runswick, {M}aine, 1985},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {46},
      PAGES = {165--195},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1987},
      MRCLASS = {14K10 (11G10 14H10)},
      MRNUMBER = {0927980},
      MRREVIEWER = {Tsutomu Sekiguchi},
      DOI = {10.1090/pspum/046.2/927980},
      ZBLNUMBER={0645.14017},
      }
  • [Pop] Go to document F. Pop, "Lifting of curves: The Oort conjecture," Ann. of Math., vol. 180, iss. 1, pp. 285-322, 2014.
    @article{Pop,
      author = {Pop, Florian},
      title = {Lifting of curves: The {O}ort conjecture},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {180},
      YEAR = {2014},
      NUMBER = {1},
      PAGES = {285--322},
      DOI = {10.4007/annals.2014.180.1.6},
      }
  • [OSS] Go to document T. Sekiguchi, F. Oort, and N. Suwa, "On the deformation of Artin-Schreier to Kummer," Ann. Sci. École Norm. Sup., vol. 22, iss. 3, pp. 345-375, 1989.
    @article {OSS, MRKEY = {1011987},
      AUTHOR = {Sekiguchi, T. and Oort, F. and Suwa, N.},
      TITLE = {On the deformation of {A}rtin-{S}chreier to {K}ummer},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {22},
      YEAR = {1989},
      NUMBER = {3},
      PAGES = {345--375},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {14L15 (11G20)},
      MRNUMBER = {1011987},
      ZBLNUMBER = {0714.14024},
      URL = {http://www.numdam.org/item?id=ASENS_1989_4_22_3_345_0},
      }
  • [SS94] T. Sekiguchi and N. Suwa, On the unified Kummer-Artin-Schreier-Witt theory, 1994.
    @misc{SS94,
      author = {Sekiguchi, Tsutomi and Suwa, Noriyuki},
      title = {On the unified {K}ummer-{A}rtin-{S}chreier-{W}itt theory},
      NOTE={Chuo Math. preprint series},
      YEAR={1994},
      }
  • [SS99] T. Sekiguchi and N. Suwa, On the unified Kummer-Artin-Schreier-Witt theory, 1999.
    @misc{SS99,
      author = {Sekiguchi, Tsutomi and Suwa, Noriyuki},
      title = {On the unified {K}ummer-{A}rtin-{S}chreier-{W}itt theory},
      note = {Laboratoire de Mathématiques Pures de Bordeaux preprint series},
      year = {1999},
      }
  • [SS01] Go to document T. Sekiguchi and N. Suwa, "A note on extensions of algebraic and formal groups. IV. Kummer-Artin-Schreier-Witt theory of degree $p^2$," Tohoku Math. J., vol. 53, iss. 2, pp. 203-240, 2001.
    @article {SS01, MRKEY = {1829979},
      AUTHOR = {Sekiguchi, Tsutomu and Suwa, Noriyuki},
      TITLE = {A note on extensions of algebraic and formal groups. {IV}. {K}ummer-{A}rtin-{S}chreier-{W}itt theory of degree {$p\sp 2$}},
      JOURNAL = {Tohoku Math. J.},
      FJOURNAL = {The Tohoku Mathematical Journal. Second Series},
      VOLUME = {53},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {203--240},
      ISSN = {0040-8735},
      CODEN = {TOMJAM},
      MRCLASS = {14L05 (13K05 14L15 20G10)},
      MRNUMBER = {1829979},
      MRREVIEWER = {Henri Gaudier},
      ZBLNUMBER = {1073.14546},
      DOI = {10.2748/tmj/1178207479},
      }
  • [SerreCL] . J-P. Serre, Corps Locaux, Paris: Hermann, 1968.
    @book {SerreCL, MRKEY = {0354618},
      AUTHOR = {Serre, {\relax J-P}},
      TITLE = {Corps Locaux},
      NOTE = {Deuxi{è}me {é}dition, Publications de l'Universit{é} de Nancago, No. VIII},
      PUBLISHER = {Hermann},
      ADDRESS = {Paris},
      YEAR = {1968},
      PAGES = {245},
      MRCLASS = {12BXX (14GXX)},
      MRNUMBER = {0354618},
      }
  • [Tossici10] Go to document D. Tossici, "Models of $\mu_{p^2,K}$ over a discrete valuation ring," J. Algebra, vol. 323, iss. 7, pp. 1908-1957, 2010.
    @article {Tossici10, MRKEY = {2594655},
      AUTHOR = {Tossici, Dajano},
      TITLE = {Models of {$\mu\sb {p\sp 2,K}$} over a discrete valuation ring},
      NOTE = {with an appendix by Xavier Caruso},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {323},
      YEAR = {2010},
      NUMBER = {7},
      PAGES = {1908--1957},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {14L15},
      MRNUMBER = {2594655},
      MRREVIEWER = {Alan Koch},
      ZBLNUMBER = {1193.14059},
      DOI = {10.1016/j.jalgebra.2010.01.012},
      }
  • [boundary] S. Wewers, Swan conductors on the boundary of Lubin-Tate spaces, 2005.
    @misc{boundary,
      author = {Wewers, Stefan},
      title = {Swan conductors on the boundary of {L}ubin-{T}ate spaces},
      year = {2005},
      ARXIV={math.NT/0511434},
      }
  • [cyclic] Go to document S. Wewers, "Fiercely ramified cyclic extensions of $p$-adic fields with imperfect residue field," Manuscripta Math., pp. 1-28, 2013.
    @article{cyclic, MRKEY={3167623},
      year={2013},
      issn={0025-2611},
      journal={Manuscripta Math.},
      doi={10.1007/s00229-013-0630-1},
      title={Fiercely ramified cyclic extensions of $p$-adic fields with imperfect residue field},
      publisher={Springer-Verlag},
      ADDRESS={New York},
      keywords={11S15; 11S31; 14F05; 19F05},
      author={Wewers, Stefan},
      MRNUMBER={3167623},
      pages={1-28},
     }
  • [Witt37] E. Witt, "Zyklische Körper und Algebren der Charakteristik $p$ vom Grad $p^n$," J. Reine Angew. Math., vol. 176, pp. 126-140, 1937.
    @Article{Witt37,
      author = {Witt, Ernst},
      title = {Zyklische {K}örper und {Al}gebren der {C}harakteristik $p$ vom {G}rad $p^n$},
      journal = {J. Reine Angew. Math.},
      year = {1937},
      volume = {176},
      pages = {126-140},
      ZBLNUMBER = {0016.05101},
      }
  • [ZariskiSamuelII] Go to document O. Zariski and P. Samuel, Commutative Algebra. Vol. II, New York: Springer-Verlag, 1975.
    @book {ZariskiSamuelII, MRKEY = {0389876},
      AUTHOR = {Zariski, Oscar and Samuel, Pierre},
      TITLE = {Commutative Algebra. {V}ol. {II}},
      NOTE = {preprint of the 1960 edition, \emph{Graduate Texts in Math.} {\bf 29}},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1975},
      PAGES = {x+414},
      MRCLASS = {13-02 (12JXX 14-02)},
      MRNUMBER = {0389876},
      ZBLNUMBER = {0322.13001},
      DOI = {10.1007/978-3-662-29244-0},
      }

Authors

Andrew Obus

Department of Mathematics, University of Virginia, Charlottesville VA 22904

Stefan Wewers

Institut für Reine Mathematik, Universität Ulm, 89081 Ulm, Germany