  Modular inflation is the restriction to two fields of automorphic inflation,
a general group based framework for multifield scalar field theories with
curved target spaces, which can be parametrized by the comoving curvature
perturbation ${\cal R}$ and the isocurvature perturbation tensor $S^{IJ}$. This
paper describes the dynamics and observables of these perturbations and
considers in some detail the special case of modular inflation as an extensive
class of two-field inflation theories with a conformally flat target space. It
is shown that the nonmodular nature of derivatives of modular forms leads to
CMB observables in modular invariant inflation theories that are in general
constructed from almost holomorphic modular forms. The phenomenology of the
model of $j$-inflation is compared to the recent observational constraints from
the Planck satellite and the BICEP2/Keck Array data.
