Localized Gouv\^ea-Mazur conjecture

TL;DR

This paper proves a localized version of the Gouv extsuperscript{e}a-Mazur conjecture on the p-adic stability of slopes of modular forms for certain residual Galois representations, advancing understanding in p-adic modular forms.

Contribution

It establishes the conjecture for irreducible residual Galois representations with reducible and very generic restrictions to the local Galois group.

Findings

Proves the localized Gouv extsuperscript{e}a-Mazur conjecture under specified conditions.

Advances the understanding of p-adic variation of modular form slopes.

Provides new insights into the structure of residual Galois representations.

Abstract

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.