  We calculate Coulomb breakup of the neutron rich nucleus $^{37}$Mg on a Pb
target at the beam energy of 244 MeV/nucleon within the framework of a finite
range distorted wave Born approximation theory that is extended to include the
effects of projectile deformation. In this theory, the breakup amplitude
involves the full wave function of the projectile ground state. Calculations
have been carried out for the total one-neutron removal cross section
$(\sigma_{-1n})$, the neutron-core relative energy spectrum, the parallel
momentum distribution of the core fragment, the valence neutron angular, and
energy-angular distributions. The calculated $\sigma_{-1n}$ has been compared
with the recently measured data to put constraints on the spin parity, and the
one-neutron separation energy ($S_n$) of the $^{37}$Mg ground state
($^{37}$Mg$_{gs}$). The dependence of $\sigma_{-1n}$ on the deformation of this
state has also been investigated. While a spin parity assignment of $7/2^-$ for
the $^{37}$Mg$_{gs}$ is ruled out by our study, neither of the $3/2^-$ and
$1/2^+$ assignments can be clearly excluded. Using the spectroscopic factor of
one for both the $3/2^-$ and $1/2^+$ configurations and ignoring the projectile
deformation effects, the $S_n$ values of $0.35 \pm 0.06$ MeV and $0.50 \pm
0.07$ MeV, respectively, are extracted for the two configurations. However, the
extracted $S_n$ is strongly dependent on the spectroscopic factor and the
deformation effects of the respective configuration. The narrow parallel
momentum distribution of the core fragment and the strong forward peaking of
the valence neutron angular distribution suggest a one-neutron halo
configuration in either of the $2p_{3/2}$ and $2s_{1/2}$ configurations of the
$^{37}$Mg ground state.
