Abstract
Under a strong genericity condition, we prove the local analogue of the ghost conjecture of Bergdall and Pollack. As applications, we deduce in this case (a) a folklore conjecture of Breuil–Buzzard–Emerton on the crystalline slopes of Kisin’s crystabelline deformation spaces, (b) Gouvêa’s $\lfloor\frac{k-1}{p+1}\rfloor$-conjecture on slopes of modular forms, and (c) the finiteness of irreducible components of the eigencurves. In addition, applying combinatorial arguments by Bergdall and Pollack, and by Ren, we deduce as corollaries in the reducible and very generic case, (d) Gouvêa–Mazur conjecture, (e) a variant of Gouvêa’s conjecture on slope distributions, and (f) a refined version of Coleman–Mazur–Buzzard–Kilford spectral halo conjecture.
