Localized Gouvêa-Mazur conjecture

Abstract

Gouvêa-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 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 $\text{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$ is reducible and very generic.