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The local higher-derivative interactions that enter into the low-energyexpansion of the effective action of type IIB superstring theory with constantcomplex modulus generally violate the $U(1)$ R-symmetry of IIB supergravity by$q_U$ units. These interactions have coefficients that transform asnon-holomorphic modular forms under $SL(2, {\mathbb Z})$ transformations withholomorphic and anti-holomorphic weights $(w,-w)$, where $q_U=-2w$.  In this paper $SL(2, {\mathbb Z})$-covariance and supersymmetry are used todetermine first-order differential equations on moduli space that relate themodular form coefficients of classes of BPS-protected maximal $U(1)$-violatinginteractions that arise at low orders in the low-energy expansion. These arethe moduli-dependent coefficients of BPS interactions of the form $d^{2p}\mathcal{P}_n$ in linearised approximation, where $\mathcal{P}_n$ is theproduct 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 withsolutions that contain information concerning perturbative and non-perturbativecontributions to superstring amplitudes. For $p=3$ and $n\ge 6$ there are twoindependent modular forms, one of which has a vanishing tree-levelcontribution.  The analysis of super-amplitudes for $U(1)$-violating processes involvingarbitrary numbers of external fluctuations of the complex modulus leads to adiagrammatic derivation of the first-order differential relations and Laplaceequations satisfied by the coefficient modular forms. Combining this with a$SL(2, {\mathbb Z})$-covariant soft axio-dilaton limit that relates amplitudeswith different values of $n$ determines most of the modular invariantcoefficients, leaving a single undetermined constant.

Modular Forms and $SL(2, {\mathbb Z})$-covariance of type IIB superstring theory