On subgroups of semi-abelian varieties defined by difference equations

TL;DR

The paper characterizes subgroups of semi-abelian varieties defined by difference equations, showing they are essentially derived from isogenies and establishing stability properties in model theory, especially in positive characteristic.

Contribution

It extends known results from characteristic zero to positive characteristic, providing new structural and model-theoretic insights for semi-abelian varieties and their definable subgroups.

Findings

Finite equivariant maps arise from isogenies.

Definable subgroups are stable and stably embedded.

Results hold specifically in positive characteristic.

Abstract

Consider the algebraic dynamics on a torus T=G_m^n given by a matrix M in GL_n(Z). Assume that the characteristic polynomial of M is prime to all polynomials X^m-1. We show that any finite equivariant map from another algebraic dynamics onto (T,M) arises from a finite isogeny T \to T. A similar and more general statement is shown for Abelian and semi-abelian varieties. In model-theoretic terms, our result says: Working in an existentially closed difference field, we consider a definable subgroup B of a semi-abelian variety A; assume B does not have a subgroup isogenous to A'(F) for some twisted fixed field F, and some semi-Abelian variety A'. Then B with the induced structure is stable and stably embedded. This implies in particular that for any n>0, any definable subset of B^n is a Boolean combination of cosets of definable subgroups of B^n. This result was already known in characteristic 0 where indeed it holds for all commutative algebraic groups ([CH]). In positive characteristic, the restriction to semi-abelian varieties is necessary.