Robert Pollack will speak on

Slopes of modular forms and the ghost conjecture

Abstract: The Fourier coefficients of cuspidal modular forms are subtle invariants which contain a wealth of arithmetic information. Even bounding the size of these coefficients involve very deep mathematics -- the best bounds follow from Deligne's proof of the Weil conjectures. In this talk, rather than looking at complex absolute values, we will instead focus on the p-adic size of p-th Fourier coefficient of an eigenform. We give a conjectural description of the variation of these sizes over all weights (classical and p-adic). This conjecture (the ghost conjecture) then has implications regarding the shape and structure of the eigencurve. This is a joint project with John Bergdall.