Acta Mathematica Vietnamica

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

 

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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