Frobenius base change of torsors

We study the Frobenius base change of a torsor under a smooth algebraic group over a field of positive characteristic by relating it to the pushforward of the torsor under the Frobenius homomorphism. As an application, we determine the change of the multiplicity of a closed fiber of an elliptic surface by purely inseparable base changes with respect to the base curve in the case where the generic fiber is supersingular.