  In the framework of the weak coupling renormalization group technique we
examine the possible instabilities of the extended quasi-one-dimensional
electron gas model with both intrachain and interchain electron-electron
interactions, including the influence of umklapp scattering and the coupling of
spins to a magnetic field. In the limit of purely repulsive intrachain
interactions, we confirm the passage from singlet d-wave like superconductivity
to an inhomogeneous FFLO state under magnetic field. The passage is accompanied
by an anomalous increase of the upper critical field that scales with the
antinesting distance from the quantum critical point joining superconductivity
to antiferromagnetism in the phase diagram, as well as the strength of
interactions. Adding weak repulsive interchain interactions promotes the
passage from singlet to triplet $f$-wave superconductivity which is expanded
under field by the development of a triplet FFLO state with zero angular
momentum projection for the Cooper pairs. The connection between theory and
experiments on the anomalous upper critical field in the Bechgaard salts is
discussed.
