The localized Gouvêa–Mazur conjecture

Gouvêa and Mazur (Math Comp 58:793-805, 1992) 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 Gouvêa–Mazur conjecture makes sense for each such component. We prove the localized Gouvêa–Mazur conjecture when the residual Galois representation is irreducible and its restriction to \({{\,\textrm{Gal}\,}}\left( \bar{\mathbb {Q}}_p/\mathbb {Q}_p \right) \) $\text{Gal}\left({\overline{\mathbb{Q}}}_p,/,{\mathbb{Q}}_p\right)$ is reducible and generic.