Saha, Jyoti Prakash

The Density of Ramified Primes

Doc. Math. 24, 2423-2429 (2019)
DOI: 10.25537/dm.2019v24.2423-2429
Communicated by Don Blasius


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.

Mathematics Subject Classification



Galois representations, ramification


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