Let be a complete discrete valuation field of mixed characteristic and the absolute Galois group of . In this paper, we will prove the -adic monodromy theorem for -adic representations of without any assumption on the residue field of , for example the finiteness of a -basis of the residue field of . The main point of the proof is a construction of -module for a de Rham representation , which is a generalization of Pierre Colmez’s . In particular, our proof is essentially different from Kazuma Morita’s proof in the case when the residue field admits a finite -basis.
We also give a few applications of the -adic monodromy theorem, which are not mentioned in the literature. First, we prove a horizontal analogue of the -adic monodromy theorem. Secondly, we prove an equivalence of categories between the category of horizontal de Rham representations of and the category of de Rham representations of an absolute Galois group of the canonical subfield of . Finally, we compute of some -adic representations of , which is a generalization of Osamu Hyodo’s results.
"The $p$-adic monodromy theorem in the imperfect residue field case." Algebra Number Theory 7 (8) 1977 - 2037, 2013. https://doi.org/10.2140/ant.2013.7.1977