INDEPENDENCE OF l OF MONODROMY GROUPS

Let X be a smooth curve over a finite field of characteristic p, let E be a number field, and let L = {Lλ} be an E-compatible system of lisse sheaves on the curve X . For each place λ of E not lying over p, the λ-component of the system L is a lisse Eλ-sheaf Lλ on X , whose associated arithmetic monodromy group is an algebraic group over the local field Eλ. We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E-compatible system L is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ”. More precisely, after replacing E by a finite extension, there exists a connected split reductive algebraic group G0 over the number field E such that for every place λ of E not lying over p, the identity component of the arithmetic monodromy group of Lλ is isomorphic to the group G0 with coefficients extended to the local field Eλ.