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.