Lifting problem for minimally wild covers of Berkovich curves

This work continues the study of residually wild morphisms $f\colon Y\to X$ of Berkovich curves initiated by Cohen, Temkin and Trushin in [CTT16]. The different function $\delta_f$ introduced in [CTT16] is the primary discrete invariant of such covers. When $f$ is not residually tame, it provides a non-trivial enhancement of the classical invariant of $f$ consisting of morphisms of reductions $\widetilde{f}\colon \widetilde{Y}\to\widetilde{X}$ and metric skeletons $\Gamma_f\colon \Gamma_Y\to\Gamma_X$. In this paper we interpret $\delta_f$ as the norm of the canonical trace section $\tau_f$ of the dualizing sheaf $\omega_f$, and introduce a finer reduction invariant $\widetilde{\tau}_f$, which is (loosely speaking) a section of $\omega_{\widetilde{f}}^{\rm log}$. Our main result generalizes a lifting theorem of Amini-Baker-Brugall\'e-Rabinoff from the case of residually tame morphism to the case of minimally residually wild morphisms. For such morphisms we describe all restrictions the datum $(\widetilde{f},\Gamma_f,\delta|_{\Gamma_Y},\widetilde{\tau}_f)$ satisfies, and prove that, conversely, any quadruple satisfying these restrictions can be lifted to a morphism of Berkovich curves.