  We study s-wave superconductivity in the attractive Hubbard model in an
applied magnetic field and assume the extreme Pauli limit where the orbital
critical field is much greater than the Zeeman critical field. We work at a
coupling corresponding roughly to the peak $T_c$ in the BCS to BEC crossover
window and retain the crucial amplitude and phase fluctuations. At low field,
as expected, the superconductor undergoes a second order thermal transition to
the normal state, and is only weakly magnetized near $T_c$. At intermediate
fields the thermal transition is still second order, but the magnetization is
significantly larger, characteristic of a `breached pair' state. At strong
field, the thermal transition is first order and our Monte Carlo reveals the
presence of a metastable Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. At even
higher fields we observe the true FFLO ground state. We present the full
`field-temperature' phase diagram of this strong coupling problem, revealing
$T_c$ scales an order of magnitude below the mean field estimate, compute the
superconducting and magnetic order parameters, and map out the directly
measurable magnetic structure factor. We compare these trends to results on the
Pauli limited heavy fermion CeCoIn$_5$, and the cold atomic Fermi gas at
unitarity. This paper focuses primarily on the homogeneous superconducting
state, another deals with thermal effects in the FFLO regime.
