On the reductions of certain two-dimensional crystabelline representations

Abstract

Crystabelline representations are representations of the absolute Galois group \(\smash {G_{\mathbb {Q}_p}}\) over \(\smash {\smash {\overline{\mathbb {Q}}_p}}\) that become crystalline on \(\smash {G_{F}}\) for some abelian extension \(\smash {F/\mathbb {Q}_p}\). Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by \(\smash {p^2}\) is crystabelline. In this article, we study the connection between the slopes of two-dimensional crystabelline representations and the reducibility of their modulo \(\smash {p}\) reductions. This question is inspired by a theorem by Buzzard and Kilford which implies that the slopes on the boundary of the \(\smash {2}\)-adic eigencurve of tame level \(\smash {1}\) are integers (and in arithmetic progression); an analogous theorem by Roe which says that the same is true for the \(\smash {3}\)-adic eigencurve; Coleman’s halo conjecture and the ghost conjecture which give predictions about the slopes on the \(\smash {p}\)-adic eigencurve of general tame level; and Hodge theoretic conjectures by Breuil, Buzzard, Emerton, and Gee which indicate that there is a connection between all of these and the slopes of two-dimensional crystabelline representations whose reductions modulo \(\smash {p}\) are reducible. We prove that the reductions of certain two-dimensional crystabelline representations with slopes in \(\smash {(0,\frac{p-1}{2})\backslash \mathbb {Z}}\) are usually irreducible, with the exception of a small region where the slopes are half-integers and reducible representations do occur.