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Conductors of wild extensions of local fields, especially in mixed characteristic $ (0,2)$


Author: Andrew Obus
Journal: Proc. Amer. Math. Soc. 142 (2014), 1485-1495
MSC (2010): Primary 11S15, 11S20; Secondary 11R18, 11R20, 12F05, 12F10
DOI: https://doi.org/10.1090/S0002-9939-2014-11881-8
Published electronically: February 4, 2014
MathSciNet review: 3168456
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Abstract | References | Similar Articles | Additional Information

Abstract: If $ K_0$ is the fraction field of the Witt vectors over an algebraically closed field $ k$ of characteristic $ p$, we calculate upper bounds on the conductor of higher ramification for (the Galois closure of) extensions of the form $ K_0(\zeta _{p^c}, \sqrt [p^c]{a})/K_0$, where $ a \in K_0(\zeta _{p^c})$. Here $ \zeta _{p^c}$ is a primitive $ p^c$th root of unity. In certain cases, including when $ a \in K_0$ and $ p=2$, we calculate the conductor exactly. These calculations can be used to determine the discriminants of various extensions of $ \mathbb{Q}$ obtained by adjoining roots of unity and radicals.


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  • [dOV82] Mariá Acosta de Orozco and William Yslas Vélez, The lattice of subfields of a radical extension, J. Number Theory 15 (1982), no. 3, 388-405. MR 680540 (84e:12027), https://doi.org/10.1016/0022-314X(82)90040-3
  • [Epp73] Helmut P. Epp, Eliminating wild ramification, Invent. Math. 19 (1973), 235-249. MR 0321929 (48 #294)
  • [JV90] Eliot T. Jacobson and William Y. Vélez, The Galois group of a radical extension of the rationals, Manuscripta Math. 67 (1990), no. 3, 271-284. MR 1046989 (91j:11096), https://doi.org/10.1007/BF02568433
  • [Obu09] Andrew Obus, Fields of moduli of three-point $ G$-covers with cyclic $ p$-Sylow. I, Algebra Number Theory 6 (2012), no. 5, 833-883. MR 2968628, https://doi.org/10.2140/ant.2012.6.833
  • [Obu10] Obus, Andrew. Fields of moduli of three-point $ G$-covers with cyclic $ p$-Sylow, II. To appear in J. Théor. Numbres Bordeaux.
  • [Ray99] Michel Raynaud, Spécialisation des revêtements en caractéristique $ p>0$, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 1, 87-126 (French, with English and French summaries). MR 1670532 (2000e:14016), https://doi.org/10.1016/S0012-9593(99)80010-X
  • [Ser79] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237 (82e:12016)
  • [Viv04] Filippo Viviani, Ramification groups and Artin conductors of radical extensions of $ \mathbb{Q}$, J. Théor. Nombres Bordeaux 16 (2004), no. 3, 779-816 (English, with English and French summaries). MR 2144967 (2006j:11148)
  • [Wew03b] Stefan Wewers, Three point covers with bad reduction, J. Amer. Math. Soc. 16 (2003), no. 4, 991-1032 (electronic). MR 1992833 (2005f:14065), https://doi.org/10.1090/S0894-0347-03-00435-1

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Additional Information

Andrew Obus
Affiliation: Department of Mathematics, MC4403, Columbia University, 2990 Broadway, New York, New York 10027
Address at time of publication: Department of Mathematics, University of Virginia, P. O. Box 400137, Charlottesville, Virginia 229904-4137
Email: obus@virginia.edu

DOI: https://doi.org/10.1090/S0002-9939-2014-11881-8
Keywords: Higher ramification groups, local fields, conductor, cyclotomic extensions
Received by editor(s): October 2, 2011
Received by editor(s) in revised form: May 29, 2012
Published electronically: February 4, 2014
Additional Notes: The author was supported by an NSF Postdoctoral Research Fellowship in the Mathematical Sciences. Final preparation of this paper took place at the Max-Planck-Institut für Mathematik in Bonn
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society

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