  We report progress in constructing Boltzmann weights for integrable
3-dimensional lattice spin models. We show that a large class of vertex
solutions to the modified tetrahedron equation can be conveniently
parameterized in terms of N-th roots of theta-functions on the Jacobian of a
compact algebraic curve. Fay's identity guarantees the Fermat relations and the
classical equations of motion for the parameters determining the Boltzmann
weights. Our parameterization allows to write a simple formula for fused
Boltzmann weights R which describe the partition function of an arbitrary open
box and which also obey the modified tetrahedron equation. Imposing periodic
boundary conditions we observe that the R satisfy the normal tetrahedron
equation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov model
and the Chessboard model as special cases.
