  The main result of the paper is a reciprocity law which proves that
compatible systems of semisimple, abelian mod $p$ representations (of arbitrary
dimension) of absolute Galois groups of number fields, arise from Hecke
characters. In the last section of the paper analogs for Galois groups of
function fields of these results are explored, and a question is raised whose
answer will require developments in transcendence theory in characteristic $p$.
