  We extend Faltings' finiteness criteria to determine the equivalence of two
$\ell$-adic, semisimple representations of the absolute Galois group of a
global field, to the context of potential equivalence. We observe that
potential equivalence of a representation $\rho$ is determined by an equality
of an $m$-power character $g\mapsto Tr(\rho(g^m))$ for some natural number $m$.
  We also discuss finiteness results for twist unramified representations.
