Local Gouv\^ea-Mazur conjecture
Gouv\^ea-Mazur [GM] made a conjecture on the local constancy of slopes of modular forms when the weight varies $p$-adically. Since one may decompose the space of modular forms according to associated residual Galois representations, the Gouv\^ea-Mazur conjecture makes sense for each such component. We prove the localized Gouv\^ea-Mazur conjecture when the residual Galois representation is irreducible and its restriction to $\textrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ is reducible and very generic.