The slope-invariant of local ghost series under direct sum

TL;DR

This paper establishes a necessary and sufficient condition for local ghost series to maintain their Newton polygon under direct sum operations, advancing understanding of the ghost conjecture in modular forms.

Contribution

It provides a precise criterion for the slope-invariance of local ghost series under direct sum, addressing a key question in the theory.

Findings

Characterization of slope-invariance for local ghost series

Resolution of a question posed in prior works

Enhanced understanding of Newton polygons in modular forms

Abstract

The ghost conjecture is first provided by Bergdall and Pollack in [BP-1,BP-2] to study the Up-slopes of spaces of modular forms, which, so far, has already brought plenty of important results. The local version of this conjecture under genericity condition has been solved by Liu-Truong-Xiao-Zhao in [LTXZ-1, LTXZ-2]. In the current paper, we prove a necessary and sufficient condition for a sequence of local ghost series to satisfy that their product has the same Newton polygon to the ghost series build from the direct sum of their associated modules. That answers a common question asked in both [BP2,LTXZ-1].