Slopes of modular forms and the ghost conjecture, II

TL;DR

This paper generalizes the ghost series conjecture for slopes of overconvergent modular forms, introducing abstract ghost series applicable to various subspaces and providing numerical evidence and distributional results.

Contribution

It constructs abstract ghost series for different subspaces of overconvergent modular forms, extending the conjecture to cases with fixed residual representations and proving related distributional properties.

Findings

Abstract ghost series encode slopes for various subspaces.

Numerical evidence supports the generalized conjecture.

Slopes satisfy distributional properties at classical weights.

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 Gouv\^ea) while the slopes form unions of arithmetic progressions at all weights not in $\mathbf{Z}_p$.