Purely Arithmetic PDEs Over a p-Adic Field: δ-Characters and δ-Modular Forms

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.