Rational fixed points for linear group actions

Corvaja, Pietro

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 4, pp. 561-597

We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$, an affine variety $V$ and a finite map $\pi :V\to G$, all defined over a finitely generated field $\kappa$ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi \left(V\right(\kappa \left)\right)$ contains a Zariski dense sub-semigroup $\Gamma \subset G\left(\kappa \right)$; namely, there must exist an unramified covering $p:\tilde{G}\to G$ and a map $\theta :\tilde{G}\to V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb{Q}$, $G={𝔾}_a$ is the additive group, we reobtain the original Hilbert Irreducibility Theorem. Our proof uses a new diophantine result, due to Ferretti and Zannier [9]. As a first application, we obtain (Theorem 1.1) a necessary condition for the existence of rational fixed points for all the elements of a Zariski-dense sub-semigroup of a linear group acting morphically on an algebraic variety. A second application concerns the characterisation of algebraic subgroups of ${GL}_N$ admitting a Zariski-dense sub-semigroup formed by matrices with at least one rational eigenvalue.

Classification : 11G35, 14G25

[1] J. Bernik, On groups and semigroups of matrices with spectra in a finitely generated field, Linear and Multilinear Algebra 53 (2005), 259-267. | Zbl | MR

[2] J. Bernik and J. Okniński, On semigroups of matrices with eigenvalue $1$ in small dimension, Linear Algebra Appl. 405 (2005), 67-73. | Zbl | MR

[3] E. Bombieri, D. Masser and U. Zannier, Intersecting a curve with algebraic subgroups of multiplicative groups, Int. Math. Res. Not. 20 (1999), 1119-1140. | Zbl | MR

[4] A. Borel, “Introduction aux groupes arithmétiques”, Hermann, Paris, 1969. | Zbl | MR

[5] A. Borel, “Linear Algebraic Groups”, 2nd Edition, GTM 126, Springer Verlag, 1997. | Zbl | MR

[6] C. W. Curtis and I. Reiner, “Representation Theory of Finite Groups and Associative Algebras”, John Wiley & Sons, 1962. | Zbl

[7] P. Dèbes, On the irreducibility of the polynomial $P(t^m,Y)$, J. Number Theory, 42 (1992), 141-157. | Zbl | MR

[8] R. Dvornicich and U. Zannier, Cyclotomic diophantine problems (Hilbert irreducibility and invariant sets for polynomial maps), Duke Math. J., to appear. | Zbl | MR

[9] A. Ferretti and U. Zannier, Equations in the Hadamard ring of rational functions, Ann. Scuola Norm. Sup. Cl. Sci. 6 (2007), 457-475. | Numdam | Zbl | MR | EuDML

[10] D. Hilbert, Ueber die Irreducibilität ganzer rationaler Functionen mit ganzzahligen Coefficienten, J. Reine Angew. Math. 110 (1892), 104-129. | JFM

[11] S. Lang, “Fundamentals of Diophantine Geometry”, Springer Verlag, 1984. | Zbl | MR

[12] M. Laurent, Equations diophantiennes exponentielles et suites récurrentes linéaires II, J. Number Theory 31 (1988), 24-53. | Zbl | MR

[13] D. W. Masser, Specializations of finitely generated subgroups of abelian varieties, Trans. Amer. Math. Soc. 311 (1989), 413-424. | Zbl | MR

[14] A. Van Der Poorten, Some facts that should be better known, especially about rational functions, In: “Number Theory and Applications (Banff, AB 1988)”, Kluwer Acad. Publ., Dordrecht (1989), 497-528. | Zbl | MR

[15] G. Prasad and A. Rapinchuk, Existence of irreducible $ℝ$-regular elements in Zariski-dense subgroups, Math. Res. Lett. 10 (2003), 21-32. | Zbl | MR

[16] G. Prasad and A. Rapinchuk, Zariski-dense subgroups and transcendental number theory, Math. Res. Lett. 12 (2005), 239-249. | Zbl | MR

[17] R. Rumely, Notes on van der Poorten proof of the Hadamard quotient theorem II, In: “Séminaire de Théorie des Nombres de Paris 1986-87”, Progress in Mathematics, Birkhäuser, 1988. | Zbl | MR

[18] J.-P. Serre, “Lectures on the Mordell-Weil Theorem”, 3rd Edition, Vieweg-Verlag, 1997. | Zbl | MR

[19] W. M. Schmidt, Linear Recurrence Sequences and Polynomial-Exponential Equations, In: “Diophantine Approximation”, F. Amoroso and U. Zannier (eds.), Proceedings of the C.I.M.E. Conference, Cetraro 2000, Springer LNM 1829, 2003. | Zbl | MR

[20] U. Zannier, A proof of Pisot’s $d$-th root conjecture, Ann. of Math. 151 (2000), 375-383. | Zbl | MR

[21] U. Zannier, “Some Applications of Diophantine Approximations to Diophantine Equations”, Forum Editrice, Udine, 2003. | Zbl