Rational fixed points for linear group actions.

*(English)*Zbl 1207.11067The 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.

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.

Reviewer: Martin Bright (Warwick)