For a general reductive group G, the multiplicity one theorem fails, but there is hope for the following ``global rigidity conjecture'': given an automorphic \(\pi =\otimes \pi_{\nu}\) and a finite set of places V, there are only finitely many automorphic \(\pi '=\otimes \pi '_{\nu}\) with \(\pi_{\nu}\) equivalent to \(\pi '_{\nu}\) for \(\nu\) outside V. One approach to proving this conjecture in certain cases would be to use the trace formula to compare G with another group G' for which the rigidity is known. This paper sets up the machinery needed for such a proof, and applies it in the particular case of the three-dimensional unitary group. For higher dimensions the argument would rely on the matching of stable orbital integrals.