  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 we 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 we
construct and study "genuinely PDE" differential modular forms. As further
applications we derive a Theorem of the Kernel and a Reciprocity Theorem for
arithmetic PDE Manin maps and also a finiteness Diophantine result for modular
parameterizations. We also prove structure results for the spaces of "PDE
differential modular forms defined on the ordinary locus." We also produce a
system of differential equations satisfied by our PDE modular forms based on
Serre and Euler operators.
