$p$-adic Local Langlands Correspondence (2412.12055v4)

Abstract: We discuss symmetrical monoidal $\infty$-categoricalizations in relevant $p$-adic functional analysis and $p$-adic analytic geometry. Our motivation has three sources relevant in $p$-adic local Langlands correspondence: one is the corresponding foundation from Bambozzi-Ben-Bassat-Kremnizer on derived functional analysis and Clausen-Scholze on derived topologicalization, and the second one is representations of derived $E_1$-rings which is relevant in integral deformed version of the $p$-adic local Langlands correspondence with eventually Banach coefficients, and the third one is the corresponding comparison of the solid quasicoherent sheaves over two kinds of generalized prismatization stackifications over small arc stacks and small $v$-stacks: one from the de Rham Robba stackification, and the other one from the de Rham prismatization stackification. Small arc stacks imperfectize the prismatization stackification, which will then perfectize the prismatization stackification when we regard them as small $v$-stacks. After Scholze's philosophy, one can in fact imperfectize those significant $v$-stacks in Fargues-Scholze to still have the geometrization at least by using Berkovich motives. Both small arc-stacks and small $v$-stacks can be studied using local totally disconnectedness which is the key observation for condensed mathematics, theory of diamonds and perfectoid rings. Following Scholze, Richarz-Scholbach and Ayoub we then study the $p$-adic local Langlands correspondence by using $p$-adic motivic cohomology theories. We study in some uniform way many significant $p$-adic motivic cohomology theories in families, after the general framework and formalism in the recent work of Ayoub. We extend in some sense Ayoub's formalism after Scholze's recent theory of Berkovich motives, Scholze's theory of small $v$-stacks and Clausen-Scholze's analytic stacks.