Difference algebraic subgroups of commutative algebraic groups over finite fields

We study the question of which torsion subgroups of commutative algebraic groups over finite fields are contained in modular difference algebraic groups for some choice of a field automorphism. We show that if G is a simple commutative algebraic group over a finite field of characteristic p, ` is a prime different from p, and for some difference closed field (K, σ) the `-primary torsion of G(K) is contained in a modular group definable in (K, σ), then it is contained in a group of the form {x ∈ G(K) : σ(x) = [a](x)} with a ∈ N \ pN. We show that no such modular group can be found for many G of interest. In the cases that such equations may be found, we recover an effective version of a theorem of Boxall.