Irreducible components of crystabelline deformation rings: the wildly ramified case

Abstract

Let $p\geq11$ be a prime, and let $\barρ:\Gal(\bar\QQ_p/\QQ_p)\rightarrow\GL_2(\bar\FF_p)$ be a reducible and very generic representation. In this paper, we use ghost series \cite{Lghost2} to study the irreducible components of Kisin's potentially crystalline deformation rings of $\barρ$. For wildly ramified principal series inertial types and arbitrary regular Hodge--Tate weights, we distinguish the components by the crystabelline slopes on their generic fibers and determine the total number of irreducible components. Combining this with an explicit formula for the total Hilbert--Samuel multiplicity of the special fiber obtained from the Breuil--Mézard conjecture, we compute the Hilbert--Samuel multiplicity of the special fiber of each component. Every such multiplicity is one when $\barρ$ is non-split, and is either one or two when $\barρ$ is split, according to whether the generic fiber of the corresponding component parametrizes reducible or irreducible Galois representations.