Independence of ℓ in Lafforgue's theorem

AbstractLet X be a smooth curve over a finite field of characteristic p, let ℓ≠p be a prime number, and let L be an irreducible lisse Q̄ℓ-sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number ℓ′≠p, there exists an irreducible lisse Q̄ℓ′-sheaf L′ on X which is compatible with L, in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for L and L′ are equal. We prove an “independence of ℓ” assertion on the fields of definition of these irreducible ℓ′-adic sheaves L′: namely, that there exists a number field F such that for any prime number ℓ′≠p, the Q̄ℓ′-sheaf L′ above is defined over the completion of F at one of its ℓ′-adic places