Differential overconvergence

TL;DR

The paper demonstrates that certain fundamental differential functions in arithmetic differential equations originate from ramified contexts, revealing an overconvergence property and enabling the construction of new unramified functions from ramified ones.

Contribution

It establishes a link between ramified and unramified differential functions, showing how overconvergence properties can be used to generate new functions in arithmetic differential equations.

Findings

Basic differential functions can be derived from ramified situations.

Overconvergence property connects ramified and unramified functions.

New unramified differential functions can be constructed from ramified ones.

Abstract

We prove that some of the basic differential functions appearing in the (unramified) theory of arithmetic differential equations, especially some of the basic differential modular forms in that theory, arise from a "ramified situation". This property can be viewed as a special kind of overconvergence property. One can also go in the opposite direction by using differential functions that arise in a ramified situation to construct "new" (unramified) differential functions.