I will report on an on-going joint project with Ruochuan Liu and Bin Zhao, studying the p-adic slopes of modular forms. Recently, Bergdall and Pollack, based on computer calculation, raised a very interesting conjecture on the slopes of overconvergent modular forms, which asserts that the Newton polygons of the characteristic power series of Up are the same as the Newton polygons of another explicit characteristic power series, which they call ghost series. This conjecture would imply many well-known conjectures regarding slopes of modular forms, like Gouvea’s conjecture, Gouvea-Mazur conjecture, and etc. I will discuss a reformulation using completed homology, and give some theoretical evidence to this conjecture, and possibly some other applications of the conjecture.

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