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

Abstract

Abstract. We give a sufficient condition, namely "Buzzard irregularity", for there to exist a cuspidal eigenform which does not have integral p-adic slope.Résumé. Une remarque sur les pentes p-adiques non-entières des formes modulaires. On donne une condition suffisante, à savoir « irrégularité au sens de Buzzard », pour qu'il existe une forme parabolique propre de pente p-adique non-entière. Statement of resultLet p be a prime number. If k and M are integers then we write S k (Γ 0 (M )) for the space of weight k cusp forms of level Γ 0 (M ).

Slopes of Modular Forms and the Ghost Conjecture

“…We note though that the focus of our modification is not on the data. Rather, we observed systematic fractional slopes appearing in spaces of cuspforms with nebentype of conductor 59, and these observations dictated the placement of the additional zeros (compare with [4]). In particular, we are not simply artificially data fitting.…”

“…(The omitted regions are indicated with an open circles. 4 ) The picture over v 2 (w κ ) < 3 illustrates the result of Buzzard-Kilford [10]; over 3 < v < 4 you see pairs of parallel lines hinting at extra structure in the set of slopes, and so on. Our discussion also implies that similar pictures may be produced on discs v p (w κ − w k ) = v for a fixed integer k v / ∈ Z.…”

“…Below, we will produce fractional 2-adic slopes in certain spaces with character when N > 1. When p is not Γ 0 (N )-regular, see [4].…”