The slopes of local ghost series under direct sum (Q6608726)

The ghost conjecture, provided by \textit{J. Bergdall} and \textit{R. Pollack} [Int. Math. Res. Not. 2019, No. 4, 1125--1144 (2019; Zbl 1451.11035); Trans. Am. Math. Soc. 372, No. 1, 357--388 (2019; Zbl 1451.11034)], unifies several important conjectures regarding slopes. In the paper under the review, the author studies related ghost series. Let \(\widetilde{H_1}\) and \(\widetilde{H_2}\) be \(\mathcal{O} [[ K_p ]]\)-projective augmented modules of the so-called related type. We denote by \(G_1(w, -)\), \(G_2(w, -)\), and \(G_3(w, -)\) the ghost series corresponding to \(\widetilde{H_1}\), \(\widetilde{H_2}\) and \(\widetilde{H_1} \oplus \widetilde{H_2}\), respectively. Let \(\mathfrak{m}_{\mathbb{C}_p}\) denote the maximal ideal in \(\mathcal{O}_{\mathbb{C}_p}\).\N\NIn the paper under the review, the author obtains a necessary and sufficient condition under which the Newton polygons of \((G_1 \cdot G_2)(w_{\star}, -)\) and of \(G_3(w_{\star}, -)\) are the same for every \(w_{\star} \in \mathfrak{m}_{\mathbb{C}_p}\), for finitely many \(\mathcal{O} [[ K_p ]]\)-projective augmented modules of \(\varepsilon\)-related and generic types.