Rational fixed points for linear group actions
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 4, p. 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 $\pi \left(V\left(\kappa \right)\right)$ contains a Zariski dense sub-semigroup $\Gamma \subset G\left(\kappa \right)$; namely, there must exist an unramified covering $p:\stackrel{˜}{G}\to G$ and a map $\theta :\stackrel{˜}{G}\to V$ such that $\pi \circ \theta =p$. In the case $\kappa =ℚ$, $G={𝔾}_{a}$ 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.

Classification:  11G35,  14G25
@article{ASNSP_2007_5_6_4_561_0,
author = {Corvaja, Pietro},
title = {Rational fixed points for linear group actions},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 6},
number = {4},
year = {2007},
pages = {561-597},
zbl = {1207.11067},
mrnumber = {2394411},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2007_5_6_4_561_0}
}

Corvaja, Pietro. Rational fixed points for linear group actions. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 6 (2007) no. 4, pp. 561-597. http://www.numdam.org/item/ASNSP_2007_5_6_4_561_0/

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