  We show that the construction of Gabor frames in $L^{2}(\mathbb{R})$ with
generators in $\mathbf{S}_{0}(\mathbb{R})$ and with respect to time-frequency
shifts from a rectangular lattice $\alpha\mathbb{Z}\times\beta\mathbb{Z}$ is
equivalent to the construction of certain Gabor frames for $L^{2}$ over the
adeles over the rationals and the group $\mathbb{R}\times\mathbb{Q}_{p}$.
Furthermore, we detail the connection between the construction of Gabor frames
on the adeles and on $\mathbb{R}\times\mathbb{Q}_{p}$ with the construction of
certain Heisenberg modules.
