On subgroups of semi-abelian varieties defined by difference equations

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.