In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type $(p,p,\cdots,p)$ between formal germs of $p$-adic curves and which generalises the formula proven in  in the case of Galois covers of degree $p$. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type $(p,p,\cdots,p)$ between $p$-adic schemes.
"Galois covers of type $(p,\cdots,p)$, vanishing cycles formula, and the existence of torsor structures." Osaka J. Math. 55 (2) 259 - 296, April 2018.