  Charge order in cuprate superconductors is a possible source of anomalous
electronic properties in the underdoped regime. Intra-unit cell charge ordering
tendencies point to electronic nematic order involving oxygen orbitals. In this
context we investigate charge instabilities in the Emery model and calculate
the charge susceptibility within diagrammatic perturbation theory. In this
approach, the onset of charge order is signalled by a divergence of the
susceptibility. Our calculations reveal three different kinds of order: a
commensurate ($q=0$) nematic order, and two incommensurate nematic phases with
modulation wavevectors that are either axial or oriented along the Brillouin
zone diagonal. We examine the nematic phase diagram as a function of the
filling, the interaction parameters, and the band structure. We also present
results for the excitation spectrum near the nematic instability, and show that
a soft nematic mode emerges from the particle-hole continuum at the transition.
The Fermi surface reconstructions that accompany the modulated nematic phases
are discussed with respect to their relevance for magneto-oscillation and
photoemission measurements. The modulated nematic phases that emerge from the
three-band Emery model are compared to those found previously in one-band
models.
