Slopes of Modular Forms and the Ghost Conjecture

Abstract

We formulate a conjecture on slopes of overconvergent padic cusp forms of any p-adic weight in the Γ0(N )-regular case. This conjecture unifies a conjecture of Buzzard on classical slopes and more recent conjectures on slopes "at the boundary of weight space".

A remark on non-integral p-adic slopes for modular forms

“…He conjectures that if p is Γ 0 (N )-regular then this list is exactly the list of slopes of T p acting on S k (Γ 0 (N )). The authors of the present work also have made a separate conjecture ( [3]) which predicts the U p -slopes of all p-adic modular forms of tame level Γ 0 (N ) still assuming that p is Γ 0 (N )-regular. The two conjectures are consistent with each other experimentally, but have not yet been shown to be consistent in general.…”

“…Buzzard's conjecture clearly implies that every slope is an integer. (This implication is not at all clear from the conjectures in [3].) It is worth asking if the integrality of slopes is characteristic of Γ 0 (N )-regularity.…”

Slopes of modular forms and the ghost conjecture, II

“…The proofs of the results described in the previous paragraph provides proofs of the unjustified statements found in [3]. We refer the reader to our previous article for a more robust discussion about slopes of modular forms, historical notes, and what is currently known about the ghost conjecture.…”

“…In previous work ( [3]) we formulated a conjecture, called the ghost conjecture, on the slopes of overconvergent p-adic cuspforms. The form of the conjecture is the following.…”

“…1 'Gouvêa distribution' when the ghost series is as in [3] or is attached to a fixed mod p Galois representation ρ. We further explore the combinatorics of what we called 'global halos' in [3] (generalizing Coleman's conjectural 'spectral halos'). As intimated in previous work, the presence of halos is a structural feature of the ghost series.…”