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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.

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
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