Hodge-Tate and de Rham representations in the imperfect residue field case

Kazuma Morita

Annales scientifiques de l'École Normale Supérieure (2010)

  • Volume: 43, Issue: 2, page 341-355
  • ISSN: 0012-9593

Abstract

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Let K be a p -adic local field with residue field k such that [ k : k p ] = p e < + ∞ and V be a p -adic representation of Gal ( K ¯ / K ) . Then, by using the theory of p -adic differential modules, we show that V is a Hodge-Tate (resp. de Rham) representation of Gal ( K ¯ / K ) if and only if V is a Hodge-Tate (resp. de Rham) representation of Gal ( K pf ¯ / K pf ) where K pf / K is a certain p -adic local field with residue field the smallest perfect field k pf containing k .

How to cite

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Morita, Kazuma. "Hodge-Tate and de Rham representations in the imperfect residue field case." Annales scientifiques de l'École Normale Supérieure 43.2 (2010): 341-355. <http://eudml.org/doc/272211>.

@article{Morita2010,
abstract = {Let $K$ be a $p$-adic local field with residue field $k$ such that $[k : k^p] = p^e &lt; +\infty $ and $V$ be a $p$-adic representation of $\text\{\rm Gal\}(\overline\{K\}/K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a Hodge-Tate (resp. de Rham) representation of $\text\{\rm Gal\}(\overline\{K\}/K)$ if and only if $V$ is a Hodge-Tate (resp. de Rham) representation of $\text\{\rm Gal\}(\overline\{K^\{\text\{\rm pf\}\}\}/K^\{\text\{\rm pf\}\})$ where $K^\{\text\{\rm pf\}\}/K$ is a certain $p$-adic local field with residue field the smallest perfect field $k^\{\text\{\rm pf\}\}$ containing $k$.},
author = {Morita, Kazuma},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {$p$-adic Galois representation; $p$-adic cohomology; $p$-adic differential equation},
language = {eng},
number = {2},
pages = {341-355},
publisher = {Société mathématique de France},
title = {Hodge-Tate and de Rham representations in the imperfect residue field case},
url = {http://eudml.org/doc/272211},
volume = {43},
year = {2010},
}

TY - JOUR
AU - Morita, Kazuma
TI - Hodge-Tate and de Rham representations in the imperfect residue field case
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2010
PB - Société mathématique de France
VL - 43
IS - 2
SP - 341
EP - 355
AB - Let $K$ be a $p$-adic local field with residue field $k$ such that $[k : k^p] = p^e &lt; +\infty $ and $V$ be a $p$-adic representation of $\text{\rm Gal}(\overline{K}/K)$. Then, by using the theory of $p$-adic differential modules, we show that $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{\rm Gal}(\overline{K}/K)$ if and only if $V$ is a Hodge-Tate (resp. de Rham) representation of $\text{\rm Gal}(\overline{K^{\text{\rm pf}}}/K^{\text{\rm pf}})$ where $K^{\text{\rm pf}}/K$ is a certain $p$-adic local field with residue field the smallest perfect field $k^{\text{\rm pf}}$ containing $k$.
LA - eng
KW - $p$-adic Galois representation; $p$-adic cohomology; $p$-adic differential equation
UR - http://eudml.org/doc/272211
ER -

References

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  8. [8] K. Kato, Generalized explicit reciprocity laws, Adv. Stud. Contemp. Math. (Pusan) 1 (1999), 57–126. Zbl1024.11029MR1701912
  9. [9] K. Kato, p -adic Hodge theory and values of zeta functions of modular forms, Astérisque295 (2004), 117–290. Zbl1142.11336MR2104361
  10. [10] S. Sen, Continuous cohomology and p -adic Galois representations, Invent. Math. 62 (1980/81), 89–116. Zbl0463.12005MR595584
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