The slopes of local ghost series under direct sum.

The ghost conjecture is first provided by Bergdall and Pollack in [1,2] to study the U p -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 [10,11]. 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 (see Theorem 1.2). That answers a common question asked in both [2,10]. [ABSTRACT FROM AUTHOR]