A formalism of arithmetic partial differential equations (PDEs) is being developed in which one considers several arithmetic differentiations at one fixed prime. In this theory, solutions can be defined in algebraically closed \(p\)-adic fields. As an application, the authors show that for at least two arithmetic directions every elliptic curve possesses a non-zero arithmetic PDE Manin map of order 1; such maps do not exist in the arithmetic ODE case. Similarly, the authors construct and study “genuinely PDE” differential modular forms.

As further applications, the authors derive a theorem of the kernel and a reciprocity theorem for arithmetic PDE Manin maps and also a finiteness Diophantine result for modular parameterizations. The authors also prove structure results for the spaces of “PDE differential modular forms defined on the ordinary locus”. They also produce a system of differential equations satisfied by their PDE modular forms based on Serre and Euler operators.