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Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze

Abstract

The monodromy weight conjecture is one of the main remaining open problems on Galois representations. It implies that the local Galois action on the -adic cohomology of a proper smooth variety is almost completely determined by the traces. Peter Scholze proved the conjecture in many cases including smooth complete intersections in a projective space, using a new powerful tool in rigid geometry called perfectoid spaces. The main arguments of the proof as well as basic ingredients in the theory of perfectoid spaces are presented.

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Acknowledgements

The author thanks A. Abbes, T. Mihara, K. Miyatani, K. Tokimoto and N. Umezaki for useful comments on a preliminary version.

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Correspondence to Takeshi Saito.

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Saito, T. Perfectoid spaces and the weight-monodromy conjecture, after Peter Scholze. Acta Math Vietnam. 39, 55–68 (2014). https://doi.org/10.1007/s40306-013-0044-x

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Keywords

  • Perfectoid spaces
  • The weight-monodromy conjecture
  • Galois representations
  • -adic cohomology

Mathematics Subject Classification (2000)

  • 11G25
  • 14G20
  • 14G22