  We study the Ginzburg-Landau equations on line bundles over non-compact
Riemann surfaces with constant negative curvature. We prove existence of
solutions with energy strictly less than that of the constant curvature
(magnetic field) one. These solutions are the non-commutative generalizations
of the Abrikosov vortex lattice of superconductivity. Conjecturally, they are
(local) minimizers of the Ginzburg-Landau energy. We obtain precise asymptotic
expansions of these solutions and their energies in terms of the curvature of
the underlying Riemann surface. Among other things, our result shows the
spontaneous breaking of the gauge-translational symmetry of the Ginzburg-Landau
equations.
