  By introducing Lenard recursion equations, we derive a general coupled
nonlinear Sch$\mathrm{\ddot{o}}$dinger (CNLS) hierarchy associated with
well-known Manakov system and Sasa-Satsuma system. Based on the characteristic
polynomial of Lax matrix for CNLS hierarchy, we obtain a third order algebraic
curve $\mathcal{K}_{m-2}$ of arithmetic genus $m-2$, from which we establish
the associated Baker-Ahhiezer functions, meromorphic function and Dubrovin-type
equations for analogs of Dirichlet and Neumann divisors. Using these results
and the theory of algebraic curve, we obtain the explicit theta function
representations of the Baker-Ahhiezer functions, the meromorphic function, and
in particular, of the algebro-geometric solutions for the entire CNLS
hierarchy.
