Slopes of modular forms and reducible Galois representations, an oversight in the ghost conjecture

Abstract

The ghost conjecture, formulated by this article’s authors, predicts the list of p p -adic valuations of the non-zero a p a_p -eigenvalues (“slopes”) for overconvergent p p -adic modular eigenforms in terms of the Newton polygon of an easy-to-describe power series (the “ghost series”). The prediction is restricted to eigenforms whose Galois representation modulo p p is reducible on a decomposition group at p p . It has been discovered, however, that the conjecture is not formulated correctly. Here we explain the issue and propose a salvage.