  After delineating the physical regimes which vortex lattices encounter in
rotating Bose-Einstein condensates as the rotation rate, $\Omega$, increases,
we derive the normal modes of the vortex lattice in two dimensions at zero
temperature. Taking into account effects of the finite compressibility, we find
an inertial mode of frequency $\ge 2\Omega$, and a primarily transverse
Tkachenko mode, whose frequency goes from being linear in the wave vector in
the slowly rotating regime, where $\Omega$ is small compared with the lowest
compressional mode frequency, to quadratic in the wave vector in the opposite
limit. We calculate the correlation functions of vortex displacements and
phase, density and superfluid velocities, and find that the zero-point
excitations of the soft quadratic Tkachenko modes lead in a large system to a
loss of long range phase correlations, growing logarithmically with distance,
and hence lead to a fragmented state at zero temperature. The vortex positional
ordering is preserved at zero temperature, but the thermally excited Tkachenko
modes cause the relative positional fluctuations to grow logarithmically with
separation at finite temperature. The superfluid density, defined in terms of
the transverse velocity autocorrelation function, vanishes at all temperatures.
Finally we construct the long wavelength single particle Green's function in
the rotating system and calculate the condensate depletion as a function of
temperature.
