  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.
