Families of Paraboline ($\phi ,{\mathrm{\Gamma }}_K$)-modules

Let $p$ be a prime and $K$ be a $p$-adic local field. We study the stack of quasi-deRham $(\phi ,{\mathrm{\Gamma }}_K)$-modules, i.e. $(\phi ,{\mathrm{\Gamma }}_K)$-modules that are deRham up to twist by characters. These objects are used to construct and then study the so called the paraboline varieties, which parametrize successive extensions of quasi-deRham $(\phi ,{\mathrm{\Gamma }}_K)$-modules of a certain type, generalizing the trianguline varieties. On the automorphic side, We construct relative eigenvarieties, and prove the existence of some local-global compatible morphism between them via showing the density of “classical points”.