The local higher-derivative interactions that enter into the low-energy expansion of type IIB superstring theory with constant complex modulus generally violate the U(1) R-symmetry of IIB supergravity. The coefficients of these interactions are non-holomorphic modular forms under the SL(2, Z) transformation. The special ones are the BPS-protected interactions. They are highly constrained by the maximal supersymmetry and SL(2, Z) symmetry. These constraints can be most efficiently implemented using on-shell amplitudes. The analysis leads to differential equations, which determine uniquely all these modular forms of the BPS-protected interactions. The talk will be based on https://arxiv.org/abs/1904.13394.