A local analogue of the ghost conjecture of Bergdall-Pollack

TL;DR

This paper formulates a local version of the ghost conjecture for GL_2(Q_p), analyzing its combinatorial properties and Newton polygon, with plans to prove it and explore arithmetic applications.

Contribution

It introduces a local analogue of the ghost conjecture based solely on representation theory, and studies its combinatorial and geometric properties.

Findings

Characterization of Newton polygon vertices

Proven integrality of slopes

Formulation of the local ghost conjecture

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.