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Izv. RAN. Ser. Mat., 1999, Volume 63, Issue 1, Pages 3–40 (Mi izv226)  

This article is cited in 2 scientific papers (total in 2 papers)

The image of the Galois group for some crystalline representations

V. A. Abrashkin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $K$ be the field of fractions of the ring $W=W(k)$ of Witt vectors, where $k$ is an algebraically closed field of characteristic $p>0$, and let $\Gamma=\operatorname{Gal}(\overline K/K)$. If $U$ is a $\Gamma$-invariant lattice in a continuous $\mathbb Q_p[\Gamma]$-module $V$ of finite dimension over $\mathbb Q_p$ and if the set of characters $S$ of the semisimple envelope of $U\otimes\mathbb F_p$ satisfies some additional assumptions, then one can associate to $U$ a function $n_U\colon S\times S\to\mathbb Z_{\geqslant 0}\cup\{\infty\}$ containing a considerable amount of information about the image $H_U$ of $\Gamma$ in $\operatorname{Aut}_{\mathbb Z_p}U$. In this paper we describe the set of functions arising from crystalline modules $V$ with Hodge–Tate weights in the interval $[0,p-2]$. Moreover, we explicitly express these functions in terms of the corresponding filtered modules. This is applied to the description of the image $H_{T(\mathcal G)}$, where $T(\mathcal G)$ is the Tate module of a 1-dimensional formal group $\mathcal G$ over $W(k)$ of finite height.

DOI: https://doi.org/10.4213/im226

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English version:
Izvestiya: Mathematics, 1999, 63:1, 1–36

Bibliographic databases:

MSC: 11S
Received: 14.10.1997

Citation: V. A. Abrashkin, “The image of the Galois group for some crystalline representations”, Izv. RAN. Ser. Mat., 63:1 (1999), 3–40; Izv. Math., 63:1 (1999), 1–36

Citation in format AMSBIB
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\paper The image of the Galois group for some crystalline representations
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\yr 1999
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\pages 1--36
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Gardeyn F., “Openness of the Galois image for $\tau$-modules of dimension 1”, J. Number Theory, 102:2 (2003), 306–338  crossref  mathscinet  zmath  isi  elib  scopus
    2. Pink R., Rütsche E., “Adelic openness for Drinfeld modules in generic characteristic”, J. Number Theory, 129:4 (2009), 882–907  crossref  mathscinet  zmath  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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