On the reductions of certain two-dimensional crystabelline representations

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