  We investigate the effect of thermal fluctuations on the (mean-field)
Abrikosov phase. The lower critical dimension of the superconducting phase is
three, indicating the absence of the Abrikosov phase for dimensions d<3. Within
the d=3 vortex liquid, the phase correlation length along the magnetic field
direction grows exponentially rapidly as the temperature is lowered. For a
finite bulk system, there is a 3D-2D crossover effect when the phase
correlation length becomes comparable to the sample thickness. Such a crossover
effect takes place over a very narrow temperature interval and mimics the
``first order transition'' seen in experiments on clean YBCO and BSCCO
crystals. We calculate the jumps in the entropy, magnetization and specific
heat due to the crossover and find reasonably good agreement with experiments
on both YBCO and BSCCO.
