Analytic Geometry and Hodge-Frobenius Structure

Abstract

In this paper, we study Frobenius structures in higher dimensional $p$-adic analytic geometry and the corresponding $p$-adic functional analysis. This will build up foundations for further study on some generalized cohomology of Frobenius modules and the corresponding generalized Iwasawa theory and generalized noncommutative Tamagawa number conjectures in the spirit of Burns-Flach-Fukaya-Kato and Nakamura (as well as certainly the original noncommutative Tamagawa number conjectures as observed by Pal-Z\'abr\'adi). We will work in the program proposed by Carter-Kedlaya-Z\'abr\'adi and after Pal-Z\'abr\'adi, and we will follow closely the approach from Kedlaya-Pottharst-Xiao to investigate the corresponding deformation of the generalized $p$-adic Hodge structures.