  We consider two mod-p central series of the free group given by Stallings and
Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration
of the mapping class group we obtain two mod-p Johnson filtrations. Further, we
adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson
homomorphisms.
  We calculate the image of the first of these homomorphisms. We give
generators for the kernels of these homomorphisms as well. We restrict the
range of our mod-p Johnson homomorphisms using work of Morita. We finally prove
the announced result of Perron that a rational homology 3-sphere may be given
as a Heegaard splitting with gluing map coming from certain members of our
mod-p Johnson filtrations.
