[Paper Review] Rational fixed points for linear group actions

Pietro Corvaja|ArXiv.org|Oct 23, 2006

Advanced Differential Equations and Dynamical Systems4 citations

TL;DR

This paper establishes a geometric criterion for the existence of Zariski-dense sub-semigroups in linear algebraic groups over finitely generated fields of characteristic zero, showing that such sub-semigroups consist of elements with rational fixed points only if there exists a specific unramified covering and lifting map. The key contribution is a version of the Hilbert Irreducibility Theorem for linear group actions, generalizing classical results on rational eigenvalues and fixed points.

ABSTRACT

Let $k$ be a finitely generated field, let $X$ be an algebraic variety and $G$ a linear algebraic group, both defined over $k$. Suppose $G$ acts on $X$ and every element of a Zariski-dense semigroup $Γ\subset G(k)$ has a rational fixed point in $X(k)$. We then deduce, under some mild technical assumptions, the existence of a rational map $G o X$, defined over $k$, sending each element $g\in G$ to a fixed point for $g$. The proof makes use of a recent result of Ferretti and Zannier on diophantine equations involving linear recurrences. As a by-product of the proof, we obtain a version of the classical Hilbert Irreducibility Theorem valid for linear algebraic groups.

Motivation & Objective

• To generalize Hilbert's Irreducibility Theorem to the setting of linear algebraic group actions on algebraic varieties.
• To determine necessary and sufficient conditions under which a Zariski-dense sub-semigroup in a linear algebraic group over a finitely generated field consists of elements with rational fixed points.
• To characterize algebraic subgroups of GL_N admitting a Zariski-dense sub-semigroup of matrices with at least one rational eigenvalue.
• To establish a connection between the existence of rational fixed points and the existence of unramified coverings and lifting maps in the group structure.

Proposed method

• The proof relies on a new diophantine result by Ferretti and Zannier concerning rational points on varieties over finitely generated fields.
• The authors analyze the existence of rational fixed points via rational maps from the group G to the variety X, using the structure of algebraic group actions.
• They use the theory of Galois covers and thin sets to study the splitting fields of characteristic polynomials of group elements.
• The construction of a rational map w: G → X that is G-equivariant on its domain is central to proving the existence of fixed points for all elements in G(κ).
• The paper applies the Lie-Kolchin Theorem and properties of flag varieties to deduce solvability of certain algebraic groups under additional assumptions.
• It establishes equivalence between the existence of rational eigenvalues in a Zariski-dense semigroup and the existence of a character or rational map to projective space.

Experimental results

Research questions

• RQ1 Under what geometric conditions does a Zariski-dense sub-semigroup of a linear algebraic group over a finitely generated field consist of elements with rational fixed points on a given variety?
• RQ2 When does the existence of rational fixed points for all elements of a Zariski-dense semigroup imply the existence of a rational G-equivariant map from the group to the variety?
• RQ3 What is the precise algebraic-geometric condition that ensures all elements of a connected linear algebraic group over a finitely generated field have rational eigenvalues, given that this holds for a Zariski-dense sub-semigroup?
• RQ4 Can the existence of a Zariski-dense sub-semigroup of matrices with rational eigenvalues be characterized by the existence of a group homomorphism to a product of multiplicative groups?
• RQ5 Is the hypothesis that some group element has finitely many fixed points necessary for the existence of a rational fixed-point map?

Key findings

• A Zariski-dense sub-semigroup Γ ⊂ G(κ) consisting of elements with rational fixed points on a variety X exists if and only if there exists an unramified covering p: G̃ → G and a rational map θ: G̃ → V such that π∘θ = p, where π: V → G is a finite map.
• If X is projective and Γ ⊂ G(κ) is Zariski-dense with each γ ∈ Γ fixing a rational point in X, then every g ∈ G(κ) fixes a rational point in X(κ).
• For a connected algebraic subgroup G ⊂ GL_N defined over a finitely generated field κ, if a Zariski-dense sub-semigroup Γ ⊂ G(κ) consists of matrices with at least r rational eigenvalues (counting multiplicities), then every g ∈ G(κ) has at least r rational eigenvalues.
• There exists a character χ: G → G_m^r defined over κ such that the characteristic polynomial of g ∈ G is divisible by (T - χ_1(g))⋯(T - χ_r(g)) for all g ∈ G.
• The existence of a rational fixed-point map w: G → X implies that the Galois group of the splitting field of the characteristic polynomial of a generic group element cannot be isomorphic to the full Galois group unless the map π admits a section.
• In the case of the maximal flag variety, the existence of such a map implies that G is solvable, and hence admits a fixed complete flag under its action.