Slopes of modular forms and the ghost conjecture

In this talk, we present a conjecture on slopes of p-adic modular. We write down a relatively
simple and explicit power series over weight space and conjecture, in the Buzzard-regular case,
that this power series knows the slopes of the U_p operator acting on all spaces of overconvergent modular forms.  Precisely, we conjecture that the Newton polygon of our series evaluated at a
weight k (classical or not) matches the Newton polygon of the characteristic power series of
U_p acting on weight k overconvergent modular forms.  We call this power series the "ghost
series" as its spectral curve hovers around the true spectral curve.

In this talk, we will explain this ghost conjecture and its connections to other conjectures on
slopes and discuss implications for the shape and structure of the eigencurve.  This is a joint
project with John Bergdall.