Quasi-periodic solutions to three-component Volterra lattice equations

We investigate a discrete integrable hierarchy of three-component Volterra-type lattice equations arising from a 4×4 matrix spectral problem. The hierarchy is generated by employing the zero-curvature condition together with Lenard recursion relations. To explore its algebro-geometric aspects, we analyze the characteristic polynomial of the associated Lax matrix and construct a tetragonal spectral curve. The geometry of the corresponding four-sheeted compact Riemann surface, including its genus, holomorphic differentials, and Riemann theta functions, is then examined. Within the framework of algebro-geometric integration, we further study the analytic structure of the Baker-Akhiezer function and related meromorphic functions. As a result, explicit quasi-periodic solutions of the entire hierarchy are derived in terms of Riemann theta functions. This work provides a systematic algebro-geometric characterization of the hierarchy and enriches the theory of integrable lattice equations.