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Analytic p-adic Banach spaces and the fundamental lemma of Colmez and Fontaine

This article gives a new proof of the fundamental lemma of the "weakly admissible implies admissible" theorem of Colmez-Fontaine that describes the semi-stable p-adic representations. To this end, we introduce the category of spectral Banach spaces, which are p-adic Banach spaces with a C_p-algebra of analytic functions, and the subcategory of effective Banach-Colmez spaces. The fundamental lemma states the surjectivity of certain analytic maps and describes their kernel. It is proven by an explicit count of solutions of the equations defining these maps. It is equivalent to the existence of functions of dimension and height of effective Banach-Colmez spaces.