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 ${{\,\textrm{GL}\,}}_2({\mathbb {Q}}_p)$. 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.