Differential existential closedness for the -function

We prove the Existential Closedness conjecture for the differential equation of the j j -function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the j j -function have solutions. Its consequences include a complete axiomatisation of j j -reducts of differentially closed fields, a dichotomy result for strongly minimal sets in those reducts, and a functional analogue of the Modular Zilber-Pink with Derivatives conjecture.