Slopes of modular forms and the ghost conjecture, II

Abstract:

In a previous article we constructed an entire power series over $p$-adic weight space (the ghost series) and conjectured, in the $\Gamma _0(N)$-regular case, that this series encodes the slopes of overconvergent modular forms of any $p$-adic weight. In this paper, we construct abstract ghost series which can be associated to various natural subspaces of overconvergent modular forms. This abstraction allows us to generalize our conjecture to, for example, the case of slopes of overconvergent modular forms with a fixed residual representation that is locally reducible at $p$. Ample numerical evidence is given for this new conjecture. Further, we prove that the slopes computed by any abstract ghost series satisfy a distributional result at classical weights (consistent with conjectures of Gouvêa) while the slopes form unions of arithmetic progressions at all weights not in $\mathbf {Z}_p$.