The Local Ghost Theorem in the Très Ramifié Case

Source abstract

Let $p\geq 11$ and let \[ \barρ\cong \left(\begin{smallmatrix}ω&*\0&1\end{smallmatrix}\right) \] be a tr`es ramifi'ee nonsplit representation. For a primitive projective-augmented module of the corresponding Steinberg type, we determine the associated Iwahori and unramified dimensions and construct its ghost series. We prove that, at every point of the relevant component of $p$-adic weight space, the Newton polygon of the characteristic power series of the $U_p$-operator agrees with that of the ghost series. This establishes the local ghost conjecture in the tr`es ramifi'ee case. The main new feature is that the Iwahori power basis is indexed by a single arithmetic progression, so the Iwahori dimension may be odd. We establish the corresponding forms of ghost duality, the characterization of vertices by near-Steinberg ranges, and the finite-minor estimates required in the proof, treating both integral and half-integral centres and the fixed central index of the Atkin--Lehner involution. The global applications in the tr`es ramifi'ee case and the analogous local and global results in the peu ramifi'ee case are the subject of ongoing work.