Euler characteristic of crepant resolutions of specific modular quotient singularities

In this paper, we consider a generalization of the McKay correspondence in positive characteristic regarding the Euler characteristic of crepant resolutions of quotient singularities given by finite subgroups of the special linear group. As the main result, we prove that this generalization holds for groups with a specific semidirect product structure, using the wild McKay correspondence over finite fields as mass formulas. Furthermore, two additional examples with more complicated structures are also given. Based on our main result, we propose a conjectural form of the generalized McKay correspondence in the modular case.