A local analogue of the ghost conjecture of Bergdall-Pollack

Abstract

We formulate a local analogue of the ghost conjecture of Bergdall and Pollack, which essentially relies purely on the representation theory of GL_2(Q_p). We further study the combinatorial properties of the ghost series as well as its Newton polygon, in particular, giving a characterization of the vertices of the Newton polygon and proving an integrality result of the slopes. In a forthcoming sequel, we will prove this local ghost conjecture under some mild hypothesis and give arithmetic applications.

We change several notations from the first version. We change 'weak arithmetic module' to 'K_p-projective augmented module' in Definition 2.3 and change 'arithmetic p-adic forms' to 'abstract p-adic forms' in section 2.4. We also add an example (5.13) to explain the meaning of the function Δdefined in 5.1