  The local higher-derivative interactions that enter into the low-energy
expansion of the effective action of type IIB superstring theory with constant
complex modulus generally violate the $U(1)$ R-symmetry of IIB supergravity by
$q_U$ units. These interactions have coefficients that transform as
non-holomorphic modular forms under $SL(2, {\mathbb Z})$ transformations with
holomorphic and anti-holomorphic weights $(w,-w)$, where $q_U=-2w$.
  In this paper $SL(2, {\mathbb Z})$-covariance and supersymmetry are used to
determine first-order differential equations on moduli space that relate the
modular form coefficients of classes of BPS-protected maximal $U(1)$-violating
interactions that arise at low orders in the low-energy expansion. These are
the moduli-dependent coefficients of BPS interactions of the form $d^{2p}
\mathcal{P}_n$ in linearised approximation, where $\mathcal{P}_n$ is the
product of $n$ fields that has dimension $=8$ with $q_U=8-2n$, and $p=0$, $2$
or $3$. These first-order equations imply that the coefficients satisfy $SL(2,
{\mathbb Z})$-covariant Laplace eigenvalue equations on moduli space with
solutions that contain information concerning perturbative and non-perturbative
contributions to superstring amplitudes. For $p=3$ and $n\ge 6$ there are two
independent modular forms, one of which has a vanishing tree-level
contribution.
  The analysis of super-amplitudes for $U(1)$-violating processes involving
arbitrary numbers of external fluctuations of the complex modulus leads to a
diagrammatic derivation of the first-order differential relations and Laplace
equations satisfied by the coefficient modular forms. Combining this with a
$SL(2, {\mathbb Z})$-covariant soft axio-dilaton limit that relates amplitudes
with different values of $n$ determines most of the modular invariant
coefficients, leaving a single undetermined constant.
