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We establish an infinite family of solutions in terms of elliptic functionsof the lattice Boussinesq systems by setting up a direct linearisation scheme,which provides the solution structure for those equations in the elliptic case.The latter, which contains as main structural element a Cauchy kernel on thetorus, is obtained from a dimensional reduction of the elliptic directlinearisation scheme of the lattice Kadomtsev-Petviashvili equation, whichrequires the introduction of a novel technical concept, namely the "ellipticcube root of unity". Thus, in order to implement the reduction we define, moregenerally, the notion of {\em elliptic $N^{\rm th}$ root of unity}, and discusssome of its properties in connection with a special class of elliptic additionformulae. As a particular concrete application we present the class of elliptic$N$-soliton solutions of the lattice Boussinesq systems.

Elliptic solutions of Boussinesq type lattice equations and the elliptic Nth root of unity