The Density of Ramified Primes

Abstract

Let F be a number field, \mathcal{O} be a domain with fraction field \mathcal{K} of characteristic zero and \rho:\text{Gal}(\overline F/F)\to\text{GL}_n(\mathcal{O}) be a representation such that \rho\otimes\overline{\mathcal{K}} is semisimple. If \mathcal{O} admits a finite monomorphism from a power series ring with coefficients in a p -adic integer ring (resp. \mathcal{O} is an affinoid algebra over a p -adic number field) and \rho is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes of \rho is of density zero. If \mathcal{O} is a complete local Noetherian ring over \mathbb{Z}_p with finite residue field of characteristic p,\rho is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of \rho form a Zariski-dense subset of \text{Spec}\mathcal{O} , then we show that the set of ramified primes of \rho is of density zero. These results are analogues, in the context of big Galois representations, of a result of Khare and Rajan, and are proved relying on their result.