INDEPENDENCE OF l IN LAFFORGUE ’ S THEOREM

Let X be a smooth curve over a finite field of characteristic p , let l 6= p be a prime number, and let L be an irreducible lisse Ql -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number l 6= p , there exists an irreducible lisse Ql′ -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 6= p , the Ql′ -sheaf L above is defined over the completion of F at one of its l -adic places.