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The Q1 lattice equation, a member in the Adler-Bobenko-Suris list of 3Dconsistent lattices, is investigated. By using the multidimensionalconsistency, a novel Lax pair for Q1 equation is given, which can benonlinearised to produce integrable symplectic maps. Consequently, a Riemanntheta function expression for the discrete potential is derived with the helpof the Baker-Akhiezer functions. This expression leads to the algebro-geometricintegration of the Q1 lattice equation, based on the commutativity of discretephase flows generated from the iteration of integrable symplectic maps.

Algebro-geometric integration of the Q1 lattice equation via nonlinear integrable symplectic maps