Torsion group schemes as iterative differential Galois groups

TL;DR

This paper explores how torsion group schemes of abelian schemes in positive characteristic can serve as iterative differential Galois groups, providing conditions and explicit examples for their realization.

Contribution

It establishes conditions under which torsion group schemes appear as iterative differential Galois groups and constructs explicit examples of such derivations.

Findings

Torsion group schemes can be realized as ID-Galois groups in positive characteristic.

Conditions are provided for when these torsion schemes occur as automorphism groups.

Explicit constructions of iterative derivations are demonstrated.

Abstract

We are considering iterative derivations on the function field L of abelian schemes in positive characteristic p>0, and give conditions when the torsion group schemes of this abelian scheme occur as ID-automorphism groups, i.e. are the ID-Galois groups of L over certain ID-subfields. For an explicit example, we even give a construction of (a family of) such iterative derivations.