  Electron bilayers in a strong magnetic field exhibit insulating behavior for
a wide range of interlayer separation $d$ for total Landau level fillings
$\nu\leq 1/2$, which has been interpreted in terms of a pinned crystal. We
study theoretically the competition between many strongly correlated liquid and
crystal states and obtain the phase diagram as a function of quantum well width
and $d$ for several filling factors of interest. We predict that three crystal
structures can be realized: (a) At small $d$, the Triangular Ising
AntiFerromagnetic (TIAF) crystal is stabilized in which the particles overall
form a single-layer like triangular crystal while satisfying the condition that
no nearest-neighbor triangle has all three particles in the same layer. (b) At
intermediate $d$, a Correlated Square (CS) crystal is stabilized, in which
particles in each layer form a square lattice, with the particles in one layer
located directly across the centers of the squares of the other. (c) At large
$d$, we find a Bilayer Graphene (BG) crystal in which the A and B sites of the
graphene lattice lie in different layers. All crystals that we predict are
strongly correlated crystals of composite fermions; a theory incorporating only
electron Hartree-Fock crystals does not find any crystals besides the `trivial'
ones occurring at large interlayer separations for total filling factor
$\nu\leq1/3$ (when layers are uncorrelated and each layer is in the long
familiar single-layer crystal phase). The TIAF, CS and BG crystals come in
several varieties, with different flavors of composite fermions and different
interlayer correlations. The appearance of these exotic crystal phases adds to
the richness of the physics of electron bilayers in a strong magnetic field,
and also provides insight into experimentally observed bilayer insulator as
well as transitions within the insulating part of the phase diagram.
