Slopes of 2-adic overconvergent modular forms with small level

Abstract

Abstract. Let τ be the primitive Dirichlet character of conductor 4, let χ be the primitive even Dirichlet character of conductor 8 and let k be an integer. We show that the U 2 operator acting on cuspidal overconvergent modular forms of weight 2k − 1 and character τ has slopes in the arithmetic progression {2, 4, . . . , 2n, . . . }, and the U 2 operator acting on cuspidal overconvergent modular forms of weight k and character χ · τ k has slopes in the arithmetic progression {1, 2, . . . , n, . . . }.We also show that the characteristic polynomials of the Hecke operators U 2 and T p acting on the space of classical cusp forms of weight k and character either τ or χ · τ k split completely over Q 2 .

On the slopes of the {U_5} operator acting on overconvergent modular forms

“…We see that the slopes at weight 0 do not fall into the arithmetic progressions found in [11] and [4]; we note that it appears that the behaviour of the slopes at the boundary of weight-space, which we consider in this paper, is rather different to the behaviour of the slopes at the centre of weight-space considered in [3].…”

“…In Buzzard-Calegari [3], the following theorem is proved, using similar techniques to those in [16], [11] and [4]: Theorem 6. The slopes of the U 2 operator acting on 2-adic overconvergent modular forms of weight 0 are…”

“…It also uses methods developed by Smithline in his thesis [16] and applied by him in [17], which were then also used in the author's paper [11] and in the paper of the author with Buzzard [4].…”

“…This section follows Section 3 of [11] in its layout and direction; more details on the specific steps can be found there.…”

“…Using arguments exactly similar to those in [11], we define the affinoid subdomain Z 0 (5 m ) of X 0 (5 m ) to be the connected component containing the cusp ∞ of the set of points t = (C, P ) in X 0 (5 m ) which have v 5 (E 4 (t)) = 0; we note that v 5 (E 4 (t)) = 0 means either that the point t corresponds to an ordinary elliptic curve or that t is a cusp.…”