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We report progress in constructing Boltzmann weights for integrable3-dimensional lattice spin models. We show that a large class of vertexsolutions to the modified tetrahedron equation can be convenientlyparameterized in terms of N-th roots of theta-functions on the Jacobian of acompact algebraic curve. Fay's identity guarantees the Fermat relations and theclassical equations of motion for the parameters determining the Boltzmannweights. Our parameterization allows to write a simple formula for fusedBoltzmann weights R which describe the partition function of an arbitrary openbox and which also obey the modified tetrahedron equation. Imposing periodicboundary conditions we observe that the R satisfy the normal tetrahedronequation. The scheme described contains the Zamolodchikov-Baxter-Bazhanov modeland the Chessboard model as special cases.Comment: 24 pages, 5 figures, typos correcte

Theta-function parametrization and fusion for 3D integrable Boltzmann weights