“…For instance, if two integral irreducible Q ac ℓ -representations of GL(n, F ) become isomorphic modulo ℓ, then Z acts on these representations by scalars congruent modulo ℓ. 59 It is a slight modification of the Breuil and Schneider correspondence transferred to ℓ-adic representations; the socle of V is the maximal semi-simple subrepresentation of V 60 ρ identifies with a representation Gal(F ac /F ) → GL(n, E) 61 For an example of a local p-adic Langlands correspondence in families for GL (2,Qp), see Ildar Gaisin and Joaquin Rodrigues Jacinto [74] For any R and G quasi-split, Dat, Helm, Kurinczuk and Moss [51] studied the scheme of Langlands parameters of G with coefficient the smallest possible ring R = Z[1/p] for a local Langlands correspondence in families. In particular, this allows to study chain of congruences of Langlands parameters modulo several different primes.…”