The slopes of modular forms are the $p$-adic valuations of the eigenvalues of the Hecke operators $T_p$. The study of slopes plays an important role in understanding the geometry of the eigencurves, introduced by Coleman and Mazur.

The study of the slope began in the 1990s when Gouvea and Mazur computed many numerical data and made several interesting conjectures. Later, Buzzard, Calegari, and other people made more precise conjectures by studying the space of overconvergent modular forms. Recently, Bergdall and Pollack introduced the ghost conjecture that unifies the previous conjectures in most cases. The ghost conjecture states that the slope can be predicted by an explicitly defined power series. We prove the ghost conjecture under a certain mild technical condition. In the pre-talk, I will explain an example in the quaternionic setting which was used as a testing ground for the proof.

This is joint work with Ruochuan Liu, Liang Xiao, and Bin Zhao.