  We construct families of TQFT's over the finite field Z/pZ starting from an
integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to
describe the constant order contributions of the cyclotomic integer expansions
of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is
intimately related to the p-modular representation theory of the symmetric
groups S_n. We construct resolutions of simple p-modular TQFT's and S_n-modules
over Z/pZ using powers of Lefschetz operators. These yield expansions of the
irreducible p-modular S_n-characters in terms of the ordianry ones as well as
expressions for the Alexander Polynomial of a 3-manifold evaluated at a p-th
root of unity in terms of traces in the irreducible modular TQFT's over
covering cobordisms. Together with identifications with the constant orders of
quantum TQFT's this results, e.g., in non-trivial criteria for a group to be a
3-manifold group.
