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 Γ 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 Zp. Abstract ghost seriesThe data needed for an abstract ghost series is two functions d, d new : Z −→ Z and an integer k 0 with 0 ≤ k 0 < δ. For notation, if n is an integer then set k n = k 0 + nδ and also define d p = 2d + d new . By the end of this section, we will make three assumptions (G), (LG), and (ND) on d and d new . (In Section 4 we introduce a fourth axiom (QL) which implies (G) and (LG).) We first assume: (G) lim n→∞ d(n) = ∞ and lim n→−∞ d(n) + d new (n) = −∞.

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

“…(1) The black dots represent data gathered from eigenforms that are twists of X 0 (11) modulo 11, including with multiplicity. (2) The larger blue dots represent all the data gathered in level Γ 0 (11). The plot is consistent with what we have already seen.…”

“…Figure 13 offers a second plot to visualize how ρ-filtered data sits within an unfiltered dataset. Namely, let ρ be the mod 11 representation associated with the elliptic curve X 0 (11). Then we consider all eigenforms f for which ρ f ≃ ρ ⊗ ω j for some j = 0, 1, .…”

“…Namely, we developed, between 2015 and 2019, a combinatorial model for slopes of a p that is called the ghost series. The ghost conjecture asserts that the ghost series predicts the slopes of weight k eigenforms, via its Newton polygons, whenever ρ is regular [10,11]. The main theorem of Liu-Truong-Xiao-Zhao is that the ghost conjecture holds under the hypotheses listed in Theorem 1.2.…”

“…Predicting slopes as the weight varies is known as the slope problem. It was the focus of work by Buzzard and Gouvêa in the early 2000's and the two authors' more recent "ghost conjecture" [20,42,10,11]. A major breakthrough on the slope problem was recently made by Liu, Truong, Xiao, and Zhao [51,52].…”

“…In fact, the authors themselves proved Gouvêa's distribution conjecture is a corollary of their ghost conjecture. See [11,Corollary 3.3]. Nonetheless, all but the smallest fraction of credit for Theorem 1.2 should be directed at Liu, Truong, Xiao, and Zhao, for their amazing work proving the ghost conjecture.…”