A Difference Version of Nori's Theorem

TL;DR

This paper extends Nori's Theorem to the setting of Frobenius difference equations, showing that any semisimple, simply-connected algebraic group over F_q can be realized as a difference Galois group over certain function fields, using bounds on Galois group schemes.

Contribution

It develops bounds on Galois group schemes of Frobenius difference equations and proves a difference analogue of Nori's Theorem for realizing algebraic groups as Galois groups.

Findings

Every semisimple, simply-connected algebraic group over F_q can be realized as a difference Galois group.

Bounds on Galois group schemes are established for Frobenius difference equations.

The result generalizes Nori's Theorem to a difference algebra setting.

Abstract

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(F_q) occurs as (finite) Galois group over F_q(s).