We study the possibility of reentrant melting of a hexagonal columnar crystal of flexible charged polymers at high enough densities. The Lindemann criterion is employed in determining the melting point. Lattice fluctuations are calculated in the Debye model, and an analogy with the Abrikosov vortex lattice in superconductors is exploited in estimating both the elastic constants of the hexagonal lattice and the appropriate Lindemann constant. We also discuss the unusual functional integral describing the statistical mechanics of a single polymer in an Einstein cage model using the path-integral formulation. A crossover as a function of an external field along the column axis is also discussed.