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Mathematische Annalen

, Volume 371, Issue 1–2, pp 795–881 | Cite as

Notes on the local p-adic Simpson correspondence

  • Takeshi Tsuji
Article

Abstract

We complete the theory of the local p-adic Simpson correspondence developed by Faltings for rational coefficients. It asserts that there is an equivalence between the category of small \(\mathbb {Q}_p\)-generalized representations of the geometric fundamental group and that of small \(\mathbb {Q}_p\)-Higgs bundles for a certain kind of log smooth affine scheme over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. The difficulty lies in the construction of the latter from the former, and we give it via a generalized Sen’s theory for the log smooth affine scheme, which depends on Faltings’ almost purity theorem. Inspired by a recent work by R. Liu and X. Zhu, we also give a formulation of local p-adic Simpson correspondence for \(\mathbb {Q}_p\)-generalized representations of the arithmetic fundamental group, and a characterization of Hodge-Tate generalized representations in terms of the correspondence.

Mathematics Subject Classification

14F30 14F35 

Notes

Acknowledgements

The author would like to thank Ahmed Abbes for telling him the problem solved as the first main theorem in this paper, encouraging him to write down the proof, and suggesting, in the process of revision, to work with log smooth schemes and not only with semi-stable ones as he did in [1, IV.5.3] and [31]. Some of this work was done during his stay at IHES (Institut des Hautes Études Scientifiques) in March 2014 and in March-April, 2016. He would like to thank the institute for its support and hospitality. Finally the author is most grateful to the referee for reading the paper carefully and giving many helpful comments. This work was financially supported by JSPS Grants-in-Aid for Scientific Research, Grant Numbers 24540009 and 15H02050.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

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