A local analogue of the ghost conjecture of Bergdall-Pollack
Ruochuan Liu, Nha Xuan Truong, Liang Xiao, Bin Zhao
TL;DR
The paper develops a local, GL$_2(\mathbb{Q}_p)$-theoretic analogue of the ghost conjecture, defining a ghost series $G^{(\varepsilon)}_{\bar{\rho}}(w,t)$ that predicts the $U_p$-slopes of abstract $p$-adic and overconvergent forms via its Newton polygon. It builds a rigorous framework using Iwasawa algebras, abstract $p$-adic forms, and a detailed analysis of the $U_p$-action, obtaining explicit formulas for the multiplicities $m_n^{(\varepsilon)}(k)$ and showing key compatibilities with theta maps, Atkin--Lehner involutions, and $p$-stabilization, along with a robust ghost-duality mechanism. The authors prove several foundational properties and, crucially, establish a clear vertex structure for the Newton polygon in terms of near-Steinberg ranges, as well as integrality of the ghost-slopes at classical weights. These results provide a concrete, combinatorial model linking automorphic representation theory, Newton polygons, and $p$-adic families, with arithmetic consequences anticipated in a sequel (including connections to Kisin’s crystalline deformation spaces). The work sets the stage for proving the local ghost conjecture under mild hypotheses and for deriving applications to deformation theory and slope phenomena in the $p$-adic Langlands program.
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.