Rational fixed points for linear group actions

Details Rational fixed points for linear group actions Rational fixed points for linear group actions

2006-10-23

arxiv.org/abs/math/0610661v2

Mathematics

Number Theory

35 pages, Plain Tex. A gap in the previous proof of Theorem 1.2 overcome, plus minor changes. Thanks to J. Bernik and the refe

Scientific paper

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 $\Gamma \subset G(k)$ has a rational fixed point in $X(k)$. We then deduce, under some mild technical assumptions, the existence of a rational map $G\to X$, 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.

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