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
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Abstract

Let FF be a number field, O\mathcal{O} be a domain with fraction field K\mathcal{K} of characteristic zero and ρ:Gal(F/F)GLn(O)\rho:\text{Gal}(\overline F/F)\to\text{GL}_n(\mathcal{O}) be a representation such that ρK\rho\otimes\overline{\mathcal{K}} is semisimple. If O\mathcal{O} admits a finite monomorphism from a power series ring with coefficients in a pp-adic integer ring (resp. O\mathcal{O} is an affinoid algebra over a pp-adic number field) and ρ\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 ρ\rho is of density zero. If O\mathcal{O} is a complete local Noetherian ring over Zp\mathbb{Z}_p with finite residue field of characteristic p,ρp,\rho is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of ρ\rho form a Zariski-dense subset of SpecO\text{Spec}\mathcal{O}, then we show that the set of ramified primes of ρ\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.

Cite this article

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

DOI 10.4171/DM/729