  We prove small data energy estimates of all orders of differentiability
between past null infinity and future null infinity of de Sitter space for the
conformally invariant Maxwell-scalar field system. This allows us to construct
bounded and invertible, but nonlinear, scattering operators taking past
asymptotic data to future asymptotic data. We also deduce exponential decay
rates for solutions with data having at least two derivatives, and for more
regular solutions discover an asymptotic decoupling of the scalar field from
the charge. The construction involves a carefully chosen complete gauge fixing
condition which allows us to control all components of the Maxwell potential,
and a nonlinear Gr\"onwall inequality for higher order estimates.
