Let E be a number field and X be a smooth geometrically connected variety
defined over a characteristic p finite field F_q. Given an n-dimensional pure
E-compatible system of semisimple \lambda-adic representations \rho_\lambda of
the fundamental group \pi_1(X) with connected algebraic monodromy groups
G_\lambda, we construct a common E-form G of all the groups G_\lambda. In the
absolutely irreducible case, we construct a common E-form i:G->GL_{n,E} of all
the tautological representations G_\lambda->GL_{n,E_\lambda} and a G-valued
adelic representation \rho_A^G of \pi_1(X) such that their composition is
isomorphic to the product representation of all \rho_\lambda. Moreover, if X is
a curve and the (absolute) outer automorphism group of G^der is trivial, then
the \lambda-components of \rho_A^G form an E-compatible system of
G-representations. Analogous rationality results in characteristic zero,
predicted by the Mumford-Tate conjecture, are obtained under some conditions
including ordinariness.Comment: 35 pages. Thm. 1.1(ii) is improved so that G sits in GL_{n,E