# The Density of Ramified Primes

### Jyoti Prakash Saha

Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India

## Abstract

Let $F$ be a number field, $O$ be a domain with fraction field $K$ of characteristic zero and $ρ:Gal(F/F)→GL_{n}(O)$ be a representation such that $ρ⊗K$ is semisimple. If $O$ admits a finite monomorphism from a power series ring with coefficients in a $p$-adic integer ring (resp. $O$ is an affinoid algebra over a $p$-adic number field) and $ρ$ 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 $ρ$ is of density zero. If $O$ is a complete local Noetherian ring over $Z_{p}$ with finite residue field of characteristic $p,ρ$ is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of $ρ$ form a Zariski-dense subset of $SpecO$, then we show that the set of ramified primes of $ρ$ 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.

## Cite this article

Jyoti Prakash Saha, The Density of Ramified Primes. Doc. Math. 24 (2019), pp. 2423–2429

DOI 10.4171/DM/729