  We study the diagonalization problem of certain discrete quantum integrable
models by the method of Baxter's T-Q relation from the algebraic geometry
aspect. Among those the Hofstadter type model (with the rational magnetic
flux), discrete quantum pendulum and discrete sine-Gordon model are our main
concern in this report. By the quantum inverse scattering method, the Baxter's
T-Q relation is formulated on the associated spectral curve, a high genus
Riemann surface in general, arisen from the study of the spectrum problem of
the system. In the case of degenerated spectral curve where the spectral
variables lie on rational curves, we obtain the complete and explicit solution
of the T-Q polynomial equation associated to the model, and the intimate
relation between the Baxter's T-Q relation and algebraic Bethe Ansatz is
clearly revealed. The algebraic geometry of a general spectral curve attached
to the model and certain qualitative properties of solutions of the Baxter's
T-Q relation are discussed incorporating the physical consideration.
