On Convergence of Stringy Motives of Wild $p^n$-Cyclic Quotient Singularities

The wild McKay correspondence, a variant of the McKay correspondence in positive characteristics, shows that stringy motives of quotient varieties equal some motivic integrals on the moduli space of of the Galois covers of a formal disk. In this paper, we determine when the integrals converge for the the case of cyclic groups of prime power order. As an application, we give a criterion for the quotient variety being canonical or log canonical.