# zbMATH — the first resource for mathematics

Rational fixed points for linear group actions. (English) Zbl 1207.11067
The main result of this article is as follows. Let $$\kappa$$ be a field of characteristic zero, finitely generated over the field of rational numbers. Let $$G$$ be a connected algebraic group over $$\kappa$$, and $$X$$ a variety over $$\kappa$$ on which $$G$$ acts $$\kappa$$-morphically.
Theorem 1.1: Let $$\Gamma \subset G(\kappa)$$ be a Zariski-dense sub-semigroup. If the following two conditions are satisfied:
(a) for every element $$\gamma \in \Gamma$$ there exists a rational point $$x_\gamma \in X(\kappa)$$ fixed by $$\gamma$$; and
(b) there exists at least one element $$g \in G$$ with only finitely many fixed points;
then there exists a rational map $$w: G \to X$$, defined over $$\kappa$$, such that for each $$g$$ in the domain of $$w$$, $$g(w(g)) = w(g)$$. If moreover $$X$$ is projective, then each element $$g \in G(\kappa)$$ has a rational fixed point in $$X(\kappa)$$.
As a step in the proof of Theorem 1.1, the author obtains the following generalisation of Hilbert’s Irreducibility Theorem:
Theorem 1.6: Let $$\kappa$$ and $$G$$ be as before. Let $$V$$ be a smooth affine algebraic variety of the same dimension as $$G$$, and $$\pi: V \to G$$ a finite map, both defined over $$\kappa$$. Let $$\Gamma \subset G(\kappa)$$ be a Zariski-dense semigroup. If $$\Gamma$$ is contained in the set $$\pi(V(\kappa))$$, then there exists an irreducible component $$V'$$ of $$V$$ such that the restriction $$\pi_{|V'}: V' \to G$$ is an unramified cover. In particular, $$V'$$ has the structure of an algebraic group over $$\kappa$$.
Taking $$\kappa$$ to be the field of rational numbers, $$G={\mathbb G}_a$$ and $$\Gamma = {\mathbb N}$$, one recovers Hilbert’s original theorem.
There are also a number of related theorems and corollaries described in the extensive and useful introduction, together with several examples.
Section 2 contains some technical results on specialisations of finitely generated rings. Section 3 contains the main ingredients of the proof of Theorem 1.1, which concern exponential polynomials. The most important result here is Theorem 3.6, on exponential polynomials in several variables; the proof uses a result of A. Ferretti and U. Zannier [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 6, No. 3, 457–475 (2007; Zbl 1150.11008)]. Section 4 collects some auxiliary geometrical results, and Section 5 gathers these ingredients together to prove the main theorems.

##### MSC:
 11G35 Varieties over global fields 14G25 Global ground fields in algebraic geometry 11E57 Classical groups 20G30 Linear algebraic groups over global fields and their integers
##### Keywords:
Algebraic groups; Hilbert irreducibility
Full Text: