Purely arithmetic PDE's over a p-adic field I: delta-characters and delta-modular forms

TL;DR

This paper develops a formalism for arithmetic PDEs over p-adic fields, introducing delta-characters and delta-modular forms, and demonstrates their applications in elliptic curves, modular forms, and Diophantine problems.

Contribution

It introduces a novel formalism for arithmetic PDEs over p-adic fields, including delta-characters and delta-modular forms, with new results on elliptic curves and modular forms.

Findings

Existence of non-zero arithmetic PDE Manin maps for elliptic curves in multiple directions

Construction of genuine PDE differential modular forms

Finiteness results for modular parameterizations

Abstract

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.