UC San Diego

In this dissertation, we discuss mainly the corresponding geometric and representation theoretic aspects of relative $p$-adic Hodge theory and $p$-adic motives. To be more precise, we study the corresponding analytic geometry of the corresponding spaces over and attached to period rings in the relative $p$-adic Hodge theory, including derived topological de Rham complexes and derived topological logarithmic de Rham complexes after Bhatt, Gabber, Guo and Illusie which is in some sense equivalent to the derived prismatic cohomology of Bhatt-Scholze as shown in the work of Li-Liu, $\mathcal{O}\mathbb{B}_\mathrm{dR}$-sheaves after Scholze, $\varphi$-$\widetilde{C}_X$-sheaves and relative-$B$-pairs after Kedlaya-Liu, multidimensional rings after Carter-Kedlaya-Z abr adi and Pal-Z abr adi and many other possible general universal motivic rings or sheaves. Many contexts are expected to be sheafified, such as over Scholze's pro- etale sites of the considered analytic spaces by using perfectoids or the quasisyntomic sites by using quasiregular semiperfectoids as in the work of Bhatt-Morrow-Scholze and Bhatt-Scholze. The main motivation comes from the corresponding noncommutative Tamagawa Number conjectures after Burns-Flach-Fukaya-Kato, relative version of the generalized version of the period rings as in the work of Carter-Kedlaya-Z abr adi and Pal-Z abr adi, arithmetic families of the representations of fundamental groups in analytic geometry such as for analytification of the moduli stacks of algebraic curves after Reinecke, arithmetic families of general motivic structures in analytic geometry such as in the work of Andreatta-Brinon, Andreatta-Iovita, Berger, Bhatt-Morrow-Scholze, Bhatt-Scholze, Fargues-Fontaine, Fargues-Scholze, Fontaine and Kedlaya-Liu, noncommutative analytic geometry and noncommutative deformation, derived noncommutative analytic geometry and derived noncommutative deformation, Langlands programs, analytic approach to algebraic topology and so on. Due to the natural though not functorial correspondence between the linear topology and the one induced by a Banach norm, we do not restrict ourselves to the functional analytic point of view when we take completion after Bambozzi-Ben-Bassat-Kremnizer, Clausen-Scholze, Gabber-Ramero, Huber, Kedlaya-Liu and Scholze.