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 $\rho :\mathrm{Gal}(\overline{F}/F)\to {\mathrm{GL}}_n(\mathcal{O})$ 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 $\mathrm{Spec}\mathcal{O}$, 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.