  We study the relationship between potential equivalence and character theory;
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 $m$. Using this, we extend Faltings' finiteness criteria to
determine the equivalence of two $\ell$-adic, semisimple representations of the
absolute Galois group of a number field, to the context of potential
equivalence.
  We also discuss finiteness results for twist unramified representations.
