Abstract

Associated to each overconvergent p-adic modular form, there is a Galois representation, whose local component at $p$ is known to be triangulline. Conversely, a fixed residual Galois representation, the space of all its possible triangulline lifts typically is typically densely occupied by representations from overconvergent eigenforms (of various levels). So understanding the $U_p$-eigenvalues of modular forms is essentially equivalent to understanding the triangulline representations. Previously, we have worked mostly on the "automorphic side" by studying the $p$-adic valuations of $U_p$-operators, known as $p$-adic slopes of modular forms. In this talk, I will report on a joint work in progress with John Bergdall, Brandon Levin, and Yong-Suk Moon, in which we hope to recover the spectral halo conjecture of Coleman--Mazur using tools from the Galois side by studying the reduction of triangulline $(\phi, \Gamma)$-modules, and at the same time relate this to the reduced fiber of Emerton--Gee stack. The technical novelty is that, we hope to find a way to bypass Kedlaya's slope filtration theory to prove the etaleness of this family of $(\phi, \Gamma)$-modules.