  We study the ground state structure of electronic-like bilayers, where
different phases compete upon changing the inter-layer separation or particle
density. New series representations with exceptional convergence properties are
derived for the exact Coulombic energies under scrutiny. The complete phase
transition scenario --including critical phenomena-- can subsequently be worked
out in detail, thereby unifying a rather scattered or contradictory body of
literature, hitherto plagued by the inaccuracies inherent to long range
interaction potentials.
