Deformations of Trianguline B-Pairs and Zariski Density of Two Dimensional Crystalline Representations

摘要

Theaimsofthisarticlearetostudythedeformationtheoryoftrianguline$B$-pairsandtoconstructa$p$-adicfamilyoftwodimensionaltriangulinerepresentationsforany$p$-adicfield.ThedeformationtheoryisthegeneralizationofBella"iche-Chenevier'sandChenevier'sworksinthe$mathbb{Q}_p$-case,wheretheyused($varphi,Gamma$)-modulesovertheRobbaringinsteadofusing$B$-pairs.GeneralizingandmodifyingKisin'stheoryof$X_{fs}$forany$p$-adicfield,weconstructa$p$-adicfamilyoftwodimensionaltriangulinerepresentations.Asanapplicationofthesetheories,weproveatheoremconcerningtheZariskidensityoftwodimensionalcrystallinerepresentationsforany$p$-adicfield,whichisageneralizationofColmez'sandKisin'stheoremforthe$mathbb{Q}_p$-case.