Slopes of the U₇ operator acting on a space of overconvergent modular forms

Abstract

Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3) = ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ 7k−6 or χ 8−7k , depending on the embedding of Q(ζ) into C7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

The 3-adic eigencurve at the boundary of weight space

“…His work applies to arbitrary weight, but is restricted to conductor 25. Kilford and McMurdy [11] extend these results to p = 7 and conductor 49, where again the slopes occur in two arithmetic progressions.…”

Arithmetic properties of Fredholm series forp-adic modular forms

“…For higher slopes we get, using Proposition 3.11, that Theorem 3.10 then predicts that the slopes in S † z 2 χ (1) (scaled by 10) are given by the 55 arithmetic progressions with common difference 50 and starting terms: 0, 2, 3,4,5,5,6,7,8,10,10,11,12,13,14,14,15,16,17,19,19,20,21,22,23,23,24,25,27,28,28,29,30,31,32,32,33,35,36,37,37,38,39,40,41,41,43,44,45,46,46,47,48,49,50. Note that we've included in this list the contribution of the progression 10, 20, 30, .…”

“…The case p = 3 is due to Roe [19]. In the case where either p = 5 or p = 7 and N = 1, Kilford [13] and Kilford and McMurdy [14] verified part (c) for a single weight. More recently, Liu, Wan and Xiao have proven the analogous conjecture in the setting of overconvergent p-adic modular forms for a definite quaternion algebra [17, Theorems 1.3 and 1.5].…”

Slopes of Modular Forms and the Ghost Conjecture

“…In all the cases we know of, we independently verified that the ghost series determines the same list of slopes. The determination of the U p -slopes in these cases are due to, in order, Buzzard and Calegari [8], Buzzard and Kilford [10], Roe [22], Kilford [18] and Kilford and McMurdy [19]. (a) p = 2, N = 1, κ = 0, (b) p = 2, N = 1, v 2 (w κ ) < 3, (c) p = 3, N = 1, v 3 (w κ ) < 1, (d) p = 5, N = 1, κ of the form z k χ with χ conductor 25, and (e) p = 7, N = 1, κ ∈ W 0 ∪ W 2 of the form z k χ with χ conductor 49.…”