Let be a number field, be a domain with fraction field of characteristic zero and be a representation such that is semisimple. If admits a finite monomorphism from a power series ring with coefficients in a -adic integer ring (resp. is an affinoid algebra over a -adic number field) and 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 is of density zero. If is a complete local Noetherian ring over with finite residue field of characteristic is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of form a Zariski-dense subset of , then we show that the set of ramified primes of 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. 2423–2429DOI 10.4171/DM/729