Rational fixed points for linear group actions

Corvaja, Pietro

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 561-597.

We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$, an affine variety $V$ and a finite map $\pi :V\to G$, all defined over a finitely generated field $\kappa$ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi \left(V\right(\kappa \left)\right)$ contains a Zariski dense sub-semigroup $\Gamma \subset G\left(\kappa \right)$; namely, there must exist an unramified covering $p:\tilde{G}\to G$ and a map $\theta :\tilde{G}\to V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb{Q}$, $G={𝔾}_a$ is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain (Theorem 1.1) a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of ${GL}_N$ admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.

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