On subgroups of semi-abelian varieties defined by difference equations

Details On subgroups of semi-abelian varieties defined by difference equations On subgroups of semi-abelian varieties defined by difference equations

2011-12-05

arxiv.org/abs/1112.0920v1

Mathematics

Logic

Scientific paper

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.

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