Rigidity and Frobenius Structure

TL;DR

This paper demonstrates that certain rigid irreducible differential equations on the projective line admit Frobenius structures after a finite extension, linking rigidity conditions to p-adic properties.

Contribution

It establishes a connection between Katz's rigidity and the existence of Frobenius structures for differential equations over the projective line.

Findings

Rigid irreducible differential equations have Frobenius structures for some prime p.

Frobenius structures exist under specific necessary conditions related to p.

The results extend understanding of p-adic properties of differential equations.

Abstract

We show that an irreducible ordinary differential equation on the projective line has a Frobenius structure for a power of some prime p if it is rigid in the sense of Katz and satisfies some other reasonable (and necessary) conditions relative to the prime p.