  Let X be a smooth curve over a finite field of characteristic p, let E be a
number field, and consider an E-compatible system of lisse sheaves on the curve
X. For each place lambda of E not lying over p, the lambda-component of the
system is a lisse E_lambda-sheaf on X, whose associated arithmetic monodromy
group is an algebraic group over the local field E_lambda. We use Serre's
theory of Frobenius tori and Lafforgue's proof of Deligne's conjecture to show
that when the E-compatible system is semisimple and pure of some integer
weight, the isomorphism type of the identity component of these monodromy
groups is ``independent of lambda''. More precisely: after replacing E by a
finite extension, there exists a connected split reductive algebraic group G_0
over the number field E such that for every place lambda of E not lying over p,
the identity component of the arithmetic monodromy group of the
lambda-component of the system is isomorphic to the group G_0 with coefficients
extended to the local field E_lambda.

Number Theory;Algebraic Geometry