  In this paper we study the relation between the oriented rational homology
3-spheres and the mod-d Torelli group (d integer). We formulate a criterion to
decide whenether a rational homology 3-sphere can be constructed from a
Heegaard splitting with gluing map an element of the mod-d Torelli group. Then
we give a tool to construct invariants of rational homology 3-spheres from
families of trivial 2-cocycles on the mod-d Torelli groups. Later we apply this
tool to find invariants of rational homology 3-spheres whose associated
2-cocycle come from the abelianization of the mod-p Torelli group (p prime)
obtaining new invariants. Finally, we disprove the conjectured extension of the
Casson invariant modulo p to the rational homology 3-spheres by B. Perron.
