The Density of Ramified Primes

Jyoti Prakash Saha

Documenta mathematica, Tome 24 (2019), pp. 2423-2429

Résumé

Let F be a number field, O be a domain with fraction field K of characteristic zero and ρ:Gal(F/F)→GLn​(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 Zp​ 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.

Détail

DOI : 10.4171/dm/729

Classification : 11F80
Mots-clés : Galois representations, ramification