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

Bondarko, M.

doi:10.1090/S1061-0022-07-00972-7

Leopoldt’s problem for Abelian totally ramified extensions of complete discrete valuation fields
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by M. V. Bondarko
Translated by: the author

St. Petersburg Math. J. 18 (2007), 757-778

DOI: https://doi.org/10.1090/S1061-0022-07-00972-7

Published electronically: August 9, 2007

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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

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
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, DOI 10.1090/conm/300/05142
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
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, DOI 10.2969/aspm/01210287