An explicit computation of the Hecke operator and the ghost conjecture
Nha Xuan Truong
TL;DR
This work explicitly analyzes the $U_5$-operator on overconvergent automorphic forms for a definite quaternion algebra, proving that the upper-left $n\times n$ minors of the associated matrix have nonzero corank and exhibit a ghost-like pattern. It provides explicit matrix formulas for $U_5$, decomposes the spaces into four weight-parameter blocks, and uses SAGE computations to observe unimodal corank patterns mirroring ghost zeros. The authors formulate a quaternionic variant of the ghost conjecture, showing divisibility properties of minors by ghost-series coefficients and establishing a slope bound analogous to Gou\'eva-type predictions, improving understanding of $p$-adic slopes in this setting. These results pave the way for proving the ghost conjecture in this context via Jacquet-Langlands transfer and offer a concrete computational framework for comparing classical and overconvergent automorphic forms.
Abstract
In this paper, we investigate the Hecke operator at p = 5 and show that the upper minors of the matrix have non zero corank and, interestingly, follow the same ghost pattern in the Ghost conjecture of Bergdall and Pollack. Due to this facts, we conjecture that the slope of Hecke action in this case can be computed using an appropriate variant of ghost series. Assume this result, we achieve an upper bound for the slopes that is similar to the Gouvea's (k-1)/(p+1) conjecture.