The theory of tetragonal curves is applied to the study of discrete integrable systems. Based on the discrete Lenard equation, we derive a hierarchy of extended Volterra lattices associated with the discrete 4×4 matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the hierarchy of extended Volterra lattices, we introduce a tetragonal curve, a Baker-Akhiezer function, and meromorphic functions on it. We study algebro-geometric properties of the tetragonal curve and asymptotic behaviors of the Baker-Akhiezer function and meromorphic functions near the origin and two infinite points. The straightening out of various flows is precisely given by utilizing the Abel map and the meromorphic differential. We finally obtain Riemann theta function solutions of the entire hierarchy of extended Volterra lattices.
