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Slope filtrations in families

Published online by Cambridge University Press:  17 May 2012

Ruochuan Liu
Ruochuan Liu*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (ruochuan@umich.edu)
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Abstract

This paper concerns arithmetic families of $\varphi $-modules over reduced affinoid spaces. For such a family, we first prove that the slope polygons are lower semicontinuous around any rigid point. We further prove that if the slope polygons are locally constant around a rigid point, then around this point, the family has a global slope filtration after base change to some extended Robba ring.

Keywords

slope filtrationϕ-modules
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 2 , April 2013 , pp. 249 - 296
Copyright
©Cambridge University Press 2012

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References

Berger, Laurent, Construction de $(\varphi , \Gamma )$-modules: représentations $p$-adiques et $B$-paires, Algebra Number Theory 2 (1) (2008), 91–120.CrossRefGoogle Scholar
Berger, Laurent and Colmez, Pierre, Familles de représentations de de Rham et monodromie p-adique, Astérisque 319 (2008), 303–337.Google Scholar
Berkovich, Vladimir, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, Volume 33. p. 169 (American Mathematical Society, Providence, RI, 1990).Google Scholar
Bosch, Siegfried, Güntzer, Ulrich and Remmert, Reinhold, Non-Archimedean analysis, Grundlehren der Math. Wiss., Volume 261 (Springer-Verlag, Berlin, 1984).Google Scholar
Bourbaki, Nicolas, Espaces Vectoriels Topologiques, reprint of the 1981 original (Springer, Berlin, 2007).Google Scholar
Colmez, Pierre, Espaces Vectoriels de dimension finie et représentations de de Rham, Astérisque 319 (2008), 117–186.Google Scholar
Colmez, Pierre, Représentations triangulines de dimension 2, Astérisque 319 (2008), 213–258.Google Scholar
Colmez, Pierre, Fonctions d’une Variable p-adique, Astérisque 330 (2010), 13–59.Google Scholar
de Jong, Aise Johan, Homomorphisms of Barsotti–Tate groups and crystals in positive characteristic, Invent. Math. 134 (1998), 301–333.Google Scholar
Fargues, Laurent and Fontaine, Jean-Marc, Courbes et fibrés vectoriels en théorie de Hodge p-adique, in preparation.Google Scholar
Hellmann, Eugen, On arithmetic families of filtered $\varphi $-modules and crystalline representations, preprint, Bonn, 2011.Google Scholar
Kedlaya, Kiran S., The algebraic closure of the power series field in positive characteristic, Proc. Amer. Math. Soc. 129 (2001), 3461–3470.Google Scholar
Kedlaya, Kiran S., A $p$-adic local monodromy theorem, Ann. of Math. (2) 160 (2004), 93–184.Google Scholar
Kedlaya, Kiran S., Slope filtrations revisited, Doc. Math. 10 (2005), 447–525.Google Scholar
Kedlaya, Kiran S., Slope filtrations for relative Frobenius, Astérisque 319 (2008), 259–301.Google Scholar
Kedlaya, Kiran S., p-adic differential equations, Cambridge Studies in Advanced Mathematics, Volume 125 (Cambridge University Press, 2010).Google Scholar
Kedlaya, Kiran S. and Liu, Ruochuan, On families of $(\varphi , \Gamma )$-modules, Algebra Number Theory 4 (2010), 943–967.CrossRefGoogle Scholar
Kedlaya, Kiran S. and Liu, Ruochuan, Relative $p$-adic Hodge theory, I: Foundations, preprint 2011.Google Scholar
Kedlaya, Kiran S. and Liu, Ruochuan, Relative $p$-adic Hodge theory, II: $(\varphi , \Gamma )$-modules, preprint 2011.Google Scholar
Liu, Ruochuan, Cohomology and duality for $(\varphi , \Gamma )$-modules over the Robba ring, Int. Math. Res. Not. 2008 3; Art ID. rnm150.Google Scholar
Lazard, Michael, Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Publ. Math. IHÉS 14 (1962), 47–75.Google Scholar
Schneider, Peter, Non-Archimedean functional analysis. (Springer-Verlag, Berlin, 2002).Google Scholar