The authors study the algebro-geometric solutions of the Q1 quadrilateral lattice equation, \[\beta_1^2 (\tilde{\bar{u}}-\tilde{u})(\bar{u} - u) - \beta_2^2(\bar{\tilde{u}}- \bar{u})(\tilde{u} - u) + \delta^2 \beta_1^2\beta_2^2(\beta_1^2 - \beta_2^2) = 0. \] Here the functions \(u = u_{m,n}\) depend on two discrete independent variables \(m, n \in\mathbb{Z}\) forming a regular lattice with coordinates \((m,n)\in\mathbb{Z}^2\). Elementary lattice shifts are denoted by \(\tilde{u}= u_{m + 1,n}\), \(\bar{u}= u_{m,n+1}\). The above equation is a member in the well-known Adler-Bobenko-Suris (ABS) list of 3D consistent lattices, where \(\beta_1\), \(\beta_2\) are (lattice) parameters associated with the two lattice directions, while \(\delta \) is a fixed parameter.
A novel Lax pair for the Q1 equation is given. By the method of multidimensional consistency a basic discrete spectral problem is derived: \(\tilde{\chi}={\mathcal{D}}^{(\beta )}(\lambda ,b)\chi\), with \[ {\mathcal{D}}^{(\beta )}(\lambda ,b) = 1/B \begin{pmatrix} \lambda b & \lambda^2\delta^2\beta+\beta B^2\\ \beta & \lambda b \end{pmatrix}, \] where \(B=(b^2-\delta^2\beta^2)^{1/2}\) and \(\beta b = \tilde{u}-u\). This can be nonlinearized to produce integrable symplectic maps. Having this in mind, a Riemann theta function expression for the discrete potential is derived with the aid of the Baker-Akhiezer functions. This expression leads to the algebro-geometric integration of the Q1 lattice equation, based on the commutativity of discrete phase flows generated from the iteration of integrable symplectic maps. The integrability of the symplectic maps is studied with the aid of the \(r\)-matrix formulation.