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Leopoldt's problem for Abelian totally ramified extensions of complete discrete valuation fields


Author: M. V. Bondarko
Translated by: the author
Original publication: Algebra i Analiz, tom 18 (2006), nomer 5.
Journal: St. Petersburg Math. J. 18 (2007), 757-778
MSC (2000): Primary 12F99
DOI: https://doi.org/10.1090/S1061-0022-07-00972-7
Published electronically: August 9, 2007
MathSciNet review: 2301042
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Abstract | References | Similar Articles | Additional Information

Abstract: By using methods described in earlier papers of the author, it is proved that, in many cases, if an Abelian totally ramified $ p$-extension contains an ideal free over its associated order, then the extension is of the type described and completely classified in an earlier paper of the author (such extensions are said to be semistable). A counterexample to this statement is presented in the case where the conditions on the extension are not fulfilled. Several other properties of extensions in question are proved.


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

  • [B1] M. V. Bondarko, Local Leopoldt’s problem for rings of integers in abelian 𝑝-extensions of complete discrete valuation fields, Doc. Math. 5 (2000), 657–693. MR 1808921
  • [B2] M. V. Bondarko, Local Leopoldt’s problem for ideals in totally ramified 𝑝-extensions of complete discrete valuation fields, Algebraic number theory and algebraic geometry, Contemp. Math., vol. 300, Amer. Math. Soc., Providence, RI, 2002, pp. 27–57. MR 1936366, https://doi.org/10.1090/conm/300/05142
  • [BVZ] M. V. Bondarko, S. V. Vostokov, and I. B. Zhukov, Additive Galois modules in complete discrete valuation fields, Algebra i Analiz 9 (1997), no. 4, 28–46 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 4, 675–693. MR 1604032
  • [H] Osamu Hyodo, Wild ramification in the imperfect residue field case, Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986) Adv. Stud. Pure Math., vol. 12, North-Holland, Amsterdam, 1987, pp. 287–314. MR 948250

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Additional Information

M. V. Bondarko
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ Prospect 28, Staryĭ Peterhof, St. Petersburg 198504, Russia
Email: mbondarko@hotmail.com

DOI: https://doi.org/10.1090/S1061-0022-07-00972-7
Keywords: Semistable extension, Leopoldt ideal
Received by editor(s): December 1, 2005
Published electronically: August 9, 2007
Article copyright: © Copyright 2007 American Mathematical Society