Refinement of strong multiplicity one for automorphic representations of

We state a qualitative form of strong multiplicity one for . We derive refinements of strong multiplicity one for automorphic representations arising from Eisenstein series associated to a Borel subgroup on , and for the cuspidal representations on induced from idele class characters of cyclic extensions of prime degree. These results are in accordance with a conjecture of D. Ramakrishnan. We also show that Ramakrishnan's conjecture follows from a weak form of Ramanujan's conjecture. We state a conjecture concerning the structural aspects of refinements of strong multiplicity one for a pair of general automorphic representations.