The trigonometric quantized Knizhnik Zamalodchikov ( qKZ ) equation associated with the quantum group Uq (Sl₂) is a system of linear difference equations with values in a tensor product of Uq (Sl₂) Verma modules. We solve the equation in terms of multidimensional q -hypergeometric functions and definie a natural isomorphism of the space of solutions and the tensor product of the corresponding evaluation Verma modules over the elliptic quantum group E _p,lambdà(Sl₂) where parameters p and lambdà are related to the parameter q of the quantum group Uq (Sl₂) and the step p of the qKZ equation via p = e^2mp and q = e^-2mlambdà .

We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the dynamical elliptic R -matrices. This description of the transition functions gives a connection between representation theories of the quantum loop algebra Uq (Sl₂) and the elliptic quantum group E_{micron, lambdà} (Sl₂) and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation.

In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation.