  The X-ray and gamma-ray spectrum of rotation-powered millisecond pulsars is
investigated in a model for acceleration and pair cascades on open field lines
above the polar caps. Although these pulsars have low surface magnetic fields,
their short periods allow them to have large magnetospheric potential drops,
but the majority do not produce sufficient pairs to completely screen the
accelerating electric field. The accelerating particles maintain high Lorentz
factors and undergo cyclotron resonant absorption of radio emission, that
produces and maintains a large pitch angle, resulting in a strong synchrotron
component. The resulting spectra consist of several distinct components:
curvature radiation from primary electrons dominating from 1 - 100 GeV,
synchrotron radiation from primary and secondary electrons dominating up to
about 100 MeV, and much weaker inverse-Compton radiation from primary electrons
at 0.1 - 1 TeV. We find that the relative size of these components depends on
pulsar period, period derivative, and neutron star mass and radius with the
level of the synchrotron component also depending sensitively on the radio
emission properties. This model is successful in describing the observed X-ray
and gamma-ray spectrum of PSR J0218+4232 as synchrotron radiation, peaking
around 100 MeV and extending up to a turnover around several GeV. The predicted
curvature radiation components from a number of millisecond pulsars, as well as
the collective emission from the millisecond pulsars in globular clusters,
should be detectable with AGILE and GLAST. We also discuss a hidden population
of X-ray-quiet and radio-quiet millisecond pulsars which have evolved below the
pair death line, some of which may be detectable by telescopes sensitive above
1 GeV.
