A uniform bound for inertially equivalent, pure $\mathrm{\ell }$-adic representations: an extension of Faltings' theorem

We introduce a notion of inertial equivalence for integral $\mathrm{\ell }$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $\mathrm{\ell }$-adic representations of the absolute Galois group of a global field lifting a fixed absolutely irreducible residual representation and with given inertial type outside a fixed finite set of places is uniformly bounded independent of the inertial type.